Introduction
The Razak–Jacelon algebra W \mathcal{W} W C ∗ C^* C ∗ [21 ] . Tensoring by W \mathcal{W} W K K K
The invariant and the main theorem
For a C ∗ C^* C ∗ C C C T ( C ) T(C) T ( C ) τ : C + → [ 0 , ∞ ] \tau:C_+\to[0,\infty] τ : C + → [ 0 , ∞ ]
τ ( 0 ) = 0 , τ ( a + b ) = τ ( a ) + τ ( b ) , τ ( r a ) = r τ ( a ) ( r > 0 ) , \tau(0)=0,\qquad\tau(a+b)=\tau(a)+\tau(b),\qquad\tau(ra)=r\tau(a)\quad(r>0), τ ( 0 ) = 0 , τ ( a + b ) = τ ( a ) + τ ( b ) , τ ( r a ) = r τ ( a ) ( r > 0 ) , together with τ ( x ∗ x ) = τ ( x x ∗ ) \tau(x^*x)=\tau(xx^*) τ ( x ∗ x ) = τ ( x x ∗ ) C + C_+ C + I I I
τ I ( a ) = { 0 , a ∈ I + , ∞ , a ∈ C + ∖ I + . \tau_I(a)=\begin{cases}0,&a\in I_+,\\ \infty,&a\in C_+\setminus I_+.\end{cases} τ I ( a ) = { 0 , ∞ , a ∈ I + , a ∈ C + ∖ I + . The canonical topology is specified by
τ i ⟶ τ ⟺ lim sup i τ i ( ( a − ε ) + ) ≤ τ ( a ) ≤ lim inf i τ i ( a ) ( a ∈ C + , ε > 0 ) . (1) \tau_i\longrightarrow\tau\quad\Longleftrightarrow\quad\limsup_i\tau_i((a-\varepsilon)_+)\leq\tau(a)\leq\liminf_i\tau_i(a)\quad(a\in C_+,\ \varepsilon>0). \tag*{(1)} τ i ⟶ τ ⟺ i lim sup τ i (( a − ε ) + ) ≤ τ ( a ) ≤ i lim inf τ i ( a ) ( a ∈ C + , ε > 0 ) . (1) An isomorphism of these topological cones is a homeomorphism preserving addition, the zero weight, and multiplication by every positive real scalar.
Theorem 1.1. Let A A A B B B C ∗ C^* C ∗ T ( A ) T(A) T ( A ) T ( B ) T(B) T ( B )
A ⊗ W ⊗ K ≅ B ⊗ W ⊗ K , A\otimes\mathcal{W}\otimes\mathcal{K}\cong B\otimes\mathcal{W}\otimes\mathcal{K}, A ⊗ W ⊗ K ≅ B ⊗ W ⊗ K , where K = K ( ℓ 2 ( N ) ) \mathcal{K}=\mathcal{K}(\ell^2(\mathbb{N})) K = K ( ℓ 2 ( N ))
The conclusion is an actual C ∗ C^* C ∗ F : T ( B ⊗ W ⊗ K ) → T ( A ⊗ W ⊗ K ) F:T(B\otimes\mathcal{W}\otimes\mathcal{K})\to T(A\otimes\mathcal{W}\otimes\mathcal{K}) F : T ( B ⊗ W ⊗ K ) → T ( A ⊗ W ⊗ K ) Φ \Phi Φ τ ∘ Φ = F ( τ ) \tau\circ\Phi=F(\tau) τ ∘ Φ = F ( τ ) τ \tau τ
The presence of (1.1) makes the invariant sensitive to this ideal structure. For example,
I ⊆ J ⟺ τ I + τ J = τ I . I\subseteq J \quad\Longleftrightarrow\quad\tau_I+\tau_J=\tau_I. I ⊆ J ⟺ τ I + τ J = τ I . The finite and zero ideals of a general weight are also encoded by the two limiting operations obtained by letting its positive scalar multiple tend to zero and to infinity. These features will let us localize each approximation on the ideal where its controlling trace is finite.
History and significance
Razak’s classification of simple inductive limits of stably projectionless building blocks made the cone of traces, together with its scale, an effective classification invariant [27 ] . Jacelon’s construction singled out the simple monotracial algebra W \mathcal{W} W [21 ] . The existence, uniqueness, and approximate-intertwining organization of those classification arguments also provides the broad method used here, with new local estimates needed for arbitrary ideals.
The trace-cone question is due to Leonel Robert. It was recorded in Santiago’s 2012 conference abstract [30 ] and appears as Problem LXVIII in the survey of Schafhauser, Tikuisis, and White [32 ] . Their formulation retains every extended trace, including the zero/infinity ideal weights. Their May 2026 discussion describes the known simple and traceless cases and the difficulty already presented by one proper nonzero ideal. Theorem 1.1 gives a positive answer to that problem for the full cone and the unrestricted class of separable nuclear algebras in its statement.
Elliott, Robert, and Santiago developed the compact topological cone of extended lower-semicontinuous traces and its relation to functionals on the Cuntz semigroup [14 ] . Robert subsequently identified tensoring with W \mathcal{W} W [28 ] . Thus the cone in (1) determines the Cuntz semigroup of the stabilized algebra, including its ideals and positive real scalar multiplication. The remaining task is to realize this invariant by algebra maps and prove their uniqueness.
In the tracial simple case, the work of Elliott and Niu [13 ] and the published classification of Elliott, Gong, Lin, and Niu [12 ] , Theorem 7.5 establish classification by scaled traces under the stated K K KK K K Z \mathcal{Z} Z [3 ] , Theorem A. Nawata gave another proof that A ⊗ W ≅ W A\otimes\mathcal{W}\cong\mathcal{W} A ⊗ W ≅ W A A A [25 ] , Corollary 6.2(i). These simple-algebra hypotheses are not imposed on A A A B B B A ⊗ W A\otimes\mathcal{W} A ⊗ W [26 ] , Theorem 1.3 together with W ⊗ Z ≅ W \mathcal{W}\otimes\mathcal{Z}\cong\mathcal{W} W ⊗ Z ≅ W [21 ] , Corollary 6.3. The classification argument below does not use this nuclear-dimension bound.
The traceless case has a different structure: when every extended trace takes only zero and infinity, the trace cone reduces to ideal information. Rørdam’s absorption criterion [29 ] , Theorem 5.2 and Kirchberg’s ideal-lattice classification, with a complete proof in [15 ] , Theorem 6.13, explain the known traceless branch. The present argument must also retain finite trace data and distinguish the regions where a weight is finite from those where it is infinite. That distinction is necessary when finite and infinite subquotients occur in the same algebra.
Restricted nonsimple precedents include Lin and Ng’s classification of certain essential extensions by W \mathcal{W} W K K K [24 ] . The general cone-only question allows arbitrary ideals.
The main analytic tools have separate roles. Ciuperca, Giordano, Ng, and Niu give unital von Neumann algebra uniqueness from support-rank data [7 ] ; we combine this with trace caps and rational averaging to obtain local rank estimates. The stable-uniqueness methods of Dadarlat and Eilers [10 ] and the ideal-related absorption and multiplier results of Gabe [16 ] , Sections 11–13 remove the resulting errors in norm. Those intermediate results apply here without pure infiniteness of the auxiliary targets. In the existence argument, Connes’s hyperfiniteness theorem [8 ] , nuclear completely positive approximation [6 ] , and measured-field methods [34 ] provide the classical inputs. The fixed controllers, lower ideal-support estimates, and fine-label correction that connect these tools are proved below.
The strategy of lifting tracial data and then correcting a map through an extension has a methodological precedent in Schafhauser’s proof of the Tikuisis–White–Winter theorem [31 ] . His setting uses a faithful amenable trace and the universal coefficient theorem (UCT); the present local error ideals and support estimates address a different invariant, with no UCT assumption. The use of quotient cone approximations and carriers in prescribed ideals also has a precedent in Bosa–Gabe–Sims–White [2 ] . Here the small UHF fractions must satisfy both upper trace bounds and lower comparisons into fixed positive cuts. The suspension-and-corner passage follows the method in [15 ] and [16 ] ; its trace normalization is proved below for all extended weights.
The construction
Fix
P = A ⊗ W ⊗ K , Q = B ⊗ W ⊗ K . P = A \otimes\mathcal{W} \otimes\mathcal{K}, \qquad Q = B \otimes\mathcal{W} \otimes\mathcal{K}. P = A ⊗ W ⊗ K , Q = B ⊗ W ⊗ K . For a C ∗ C^* C ∗ C C C Cu ( C ) \operatorname{Cu}(C) Cu ( C ) [ a ] [a] [ a ] C ⊗ K C \otimes\mathcal{K} C ⊗ K a ≾ b a \precsim b a ≾ b v n ∗ b v n → a v_n^* b v_n \to a v n ∗ b v n → a ( v n ) (v_n) ( v n ) a a a d τ ( a ) = lim k τ ( a 1 / k ) d_\tau(a) = \lim_{k} \tau(a^{1/k}) d τ ( a ) = lim k τ ( a 1/ k ) [ a ] [a] [ a ]
The given cone isomorphism yields compatible isomorphisms
F : T ( Q ) ⟶ T ( P ) , G : Cu ( P ) ⟶ Cu ( Q ) , d τ ( G [ a ] ) = d F τ ( a ) . F \colon T(Q) \longrightarrow T(P), \qquad G \colon\operatorname{Cu}(P) \longrightarrow\operatorname{Cu}(Q), \qquad d_\tau(G[a]) = d_{F\tau}(a). F : T ( Q ) ⟶ T ( P ) , G : Cu ( P ) ⟶ Cu ( Q ) , d τ ( G [ a ]) = d F τ ( a ) . We identify the primitive ideal spaces through this correspondence and denote the common space by X X X U ⊆ X U \subseteq X U ⊆ X P ( U ) P(U) P ( U ) Q ( U ) Q(U) Q ( U ) Q Q Q
D = Q ∞ = ℓ ∞ ( Q ) / c 0 ( Q ) . D = Q_\infty= \ell^\infty(Q)/c_0(Q). D = Q ∞ = ℓ ∞ ( Q ) / c 0 ( Q ) . We regard Q Q Q D D D ω \omega ω N \mathbb{N} N τ n ∈ T ( Q ) \tau_n \in T(Q) τ n ∈ T ( Q )
ρ ( [ ( x n ) ] ) = sup ε > 0 lim n → ω τ n ( ( x n − ε ) + ) , x n ≥ 0. \rho([(x_n)]) = \sup_{\varepsilon> 0} \lim_{n \to\omega} \tau_n((x_n - \varepsilon)_+), \qquad x_n \ge0. ρ ([( x n )]) = ε > 0 sup n → ω lim τ n (( x n − ε ) + ) , x n ≥ 0. Section 3 proves that this defines an extended trace. We use Y Y Y R \mathbb{R} R m m m ν ∈ T ( P ) \nu\in T(P) ν ∈ T ( P ) Fin ( ν ) \operatorname{Fin}(\nu) Fin ( ν ) ν \nu ν m ⊗ ν m \otimes\nu m ⊗ ν C 0 ( Y ) ⊗ Fin ( ν ) C_0(Y) \otimes\operatorname{Fin}(\nu) C 0 ( Y ) ⊗ Fin ( ν ) ( ∫ f d m ) ν ( a ) (\int f\,dm)\nu(a) ( ∫ f d m ) ν ( a ) 0 ⋅ ∞ = 0 0 \cdot\infty= 0 0 ⋅ ∞ = 0 model is a homomorphism p : C 0 ( Y ) ⊗ P → D p \colon C_0(Y) \otimes P \to D p : C 0 ( Y ) ⊗ P → D
ρ ∘ p = m ⊗ F ( ρ ∣ Q ) \rho\circ p = m \otimes F(\rho|_Q) ρ ∘ p = m ⊗ F ( ρ ∣ Q ) for every such regularized limit ρ \rho ρ C 0 ( Y ) ⊗ P C_0(Y) \otimes P C 0 ( Y ) ⊗ P Y × X Y \times X Y × X X X X
The main difficulty is to preserve both the finite and the infinite parts of (1.3) . Fix a regularized limit ρ \rho ρ σ = ρ ∣ Q \sigma= \rho|_Q σ = ρ ∣ Q Q ( V ) = Fin ( σ ) Q(V) = \operatorname{Fin}(\sigma) Q ( V ) = Fin ( σ ) C 0 ( Y ) ⊗ P ( V ) C_0(Y) \otimes P(V) C 0 ( Y ) ⊗ P ( V ) ρ \rho ρ C 0 ( Y ) ⊗ P ( V ) C_0(Y) \otimes P(V) C 0 ( Y ) ⊗ P ( V ) (1.3) are infinite on every positive element outside it. This two-part recovery is the reason for the following constructions.
Local error ideals and stabilizers. For each open U ⊆ X U \subseteq X U ⊆ X J ( U ) ⊆ D J(U) \subseteq D J ( U ) ⊆ D c c c c ≾ ( b − δ ) + c \precsim(b - \delta)_+ c ≾ ( b − δ ) + b ∈ Q ( U ) + b \in Q(U)_+ b ∈ Q ( U ) + δ > 0 \delta> 0 δ > 0 Q ( U ) Q(U) Q ( U )
Tracial approximation with lower ideal support. Convex separation and von Neumann algebra approximation construct completely positive cone maps with the required first and second trace moments. The construction simultaneously imposes norm tests whose positive averages force lower rank comparisons. The comparison constant is independent of the averaging multiplicities. This lower signal determines the part of the cone on which a weight must remain infinite. Factor representations and their measurable assembly are used only through cut tests controlled by fixed finite-rank constants.
Correction with all ideal labels retained. A small stabilizer places the multiplicative errors in a sigma-unital hereditary target. Module Stinespring dilation and absorption then correct the cone map. The support estimate is imposed on the entire projected ideal of each input. It therefore survives scaling of the cone variable and makes every dilation weakly equivariant for the finer primitive-space labels. Compact corrections can be compressed to finitely many terms of the original diagonal repeat, with an explicit tail estimate. A second small homomorphism then dominates the correction itself. Adding it back retains the lower ideal comparison, so the finite and infinite parts combine to give the full model equation in Section 7.
Uniqueness and passage to actual maps. For two models with the same data, tracial alignment first makes their difference locally small. A stabilizer drawn from a spare model then allows ideal-related stable uniqueness to remove that difference in norm. The spare is supplied by splitting each model into two UHF halves and replacing the halves successively. This argument, in Section 5, assumes that the two models are given; it does not depend on the later existence theorem. Once corrected cone models exist, an exhaustion of the line gives models with Lebesgue data. Translation covariance and a full crossed-product projection give a model for P P P P → Q P \to Q P → Q Q → P Q \to P Q → P Figure 1 displays these dependencies. In particular, the uniqueness theorem is conditional on a pair of models; it is available when the existence construction later supplies them.
Figure 1. Principal model-construction dependencies. The existence and uniqueness arguments meet in the passage from sequence models to actual maps. Shared auxiliary inputs are not drawn separately: the trace-cap and barycenter lemmas in Section 5 are used in Section 6; the absorption and hereditary-support lemmas of Section 5 are used in Section 7. Norm uniqueness is a statement about models when they are given; its proof does not assume the cone-existence theorem. The dashed arrow records its further use in actual-map extraction and the final two-sided intertwining.
Organization and conventions
Theorem 2.1 and Proposition 2.2 collect the structural input. The sequence comparisons and local diagonalization are proved in Proposition 3.6 and Lemma 3.10. The stabilizer construction is Theorem 4.1, and norm uniqueness is Theorem 5.1. The two existence stages are Theorems 6.1 and 7.6; the final passage to an isomorphism is Theorem 8.5.
All algebras and Hilbert spaces are complex. A completely positive contractive map is abbreviated c.p.c. Traces on finite matrix amplifications use the unnormalized matrix trace. The universal UHF algebra is denoted by Q \mathcal{Q} Q Q Q Q V ∈ U V \in U V ∈ U X X X
The invariant, its ideals, and matrix conventions
The trace cone supplies three inputs to the construction: pointwise comparison of Cuntz classes in Q Q Q P P P Q Q Q Q Q Q
Ranks and realification
For a ∈ ( C ⊗ K ) + a \in(C \otimes\mathcal{K})_+ a ∈ ( C ⊗ K ) + τ ∈ T ( C ) \tau\in T(C) τ ∈ T ( C )
d τ ( a ) = lim k → ∞ τ ( a 1 / k ) ( 0 ≤ a ≤ 1 ) . (2) d_{\tau}(a) = \lim_{k \to\infty} \tau\left(a^{1/k}\right) \qquad(0 \leq a \leq1). \tag*{(2)} d τ ( a ) = k → ∞ lim τ ( a 1/ k ) ( 0 ≤ a ≤ 1 ) . (2) Rescaling an element to be contractive does not change its rank. The resulting function depends only on its Cuntz class. We write d τ ( x ) d_\tau(x) d τ ( x ) x ∈ Cu ( C ) x \in\operatorname{Cu}(C) x ∈ Cu ( C )
d τ ( a ) = sup ε > 0 d τ ( ( a − ε ) + ) , τ ( a ) = ∫ 0 ∥ a ∥ d τ ( ( a − t ) + ) d t . (3) d_\tau(a)=\sup_{\varepsilon>0}d_\tau((a-\varepsilon)_+),\qquad\tau(a)=\int_0^{\lVert a\rVert}d_\tau((a-t)_+)\,dt. \tag*{(3)} d τ ( a ) = ε > 0 sup d τ (( a − ε ) + ) , τ ( a ) = ∫ 0 ∥ a ∥ d τ (( a − t ) + ) d t . (3) These identities use extended nonnegative arithmetic and apply to all the weights in (1) . We use the following form of the trace and realification theorems. The assertion about the functional cone is [14 ] , Proposition 4.2, Remark 4.3, and Theorem 4.4; compactness is [14 ] , Theorem 3.7. The realification assertions are [28 ] , Proposition 3.1.1 and Theorems 3.2.1 and 5.1.2.
Theorem 2.1 (Trace realification). For every C ∗ C^* C ∗ C C C (1) is compact Hausdorff, and it is second countable when C C C C C C τ ↦ d τ \tau\mapsto d_\tau τ ↦ d τ Cu ( C ) \operatorname{Cu}(C) Cu ( C )
For every C ∗ C^* C ∗ C C C
Cu ( C ⊗ W ) ≅ Cu ( C ) R . \operatorname{Cu}(C\otimes\mathcal{W})\cong\operatorname{Cu}(C)_{\mathbb{R}}. Cu ( C ⊗ W ) ≅ Cu ( C ) R . Realification is determined functorially by the topological functional cone. Its realization as functions on that cone has pointwise order, pointwise addition, and multiplication by positive real scalars. It has the same functional cone as Cu ( C ) \operatorname{Cu}(C) Cu ( C )
Consequently the given trace-cone isomorphism induces isomorphisms
F : T ( Q ) ⟶ T ( P ) , G : Cu ( P ) ⟶ Cu ( Q ) , d τ ( G x ) = d F τ ( x ) . (4) F:T(Q)\longrightarrow T(P),\qquad G:\operatorname{Cu}(P)\longrightarrow\operatorname{Cu}(Q),\qquad d_\tau(Gx)=d_{F\tau}(x). \tag*{(4)} F : T ( Q ) ⟶ T ( P ) , G : Cu ( P ) ⟶ Cu ( Q ) , d τ ( G x ) = d F τ ( x ) . (4) In particular, for x , y ∈ Cu ( Q ) x,y\in\operatorname{Cu}(Q) x , y ∈ Cu ( Q )
x ≤ y ⟺ d τ ( x ) ≤ d τ ( y ) for every τ ∈ T ( Q ) . (5) x\leq y\quad\Longleftrightarrow\quad d_\tau(x)\leq d_\tau(y)\quad\text{for every }\tau\in T(Q). \tag*{(5)} x ≤ y ⟺ d τ ( x ) ≤ d τ ( y ) for every τ ∈ T ( Q ) . (5) Multiplication by every positive real scalar is an order automorphism preserving increasing suprema and compact containment.
Here the function realization is Robert’s cone L ( F ( Cu ( C ) ) ) L(F(\operatorname{Cu}(C))) L ( F ( Cu ( C ))) F ( Cu ( C ) ) F(\operatorname{Cu}(C)) F ( Cu ( C )) T T T L ( T ) L(T) L ( T ) f = sup n f n f=\sup_n f_n f = sup n f n f n : T → [ 0 , ∞ ] f_n:T\to[0,\infty] f n : T → [ 0 , ∞ ] 0 < ε n < 1 0<\varepsilon_n<1 0 < ε n < 1
f n ≤ ( 1 − ε n ) f n + 1 , f n is continuous at every τ with f n + 1 ( τ ) < ∞ . f_n\leq(1-\varepsilon_n)f_{n+1},\qquad f_n\text{ is continuous at every }\tau\text{ with }f_{n+1}(\tau)<\infty. f n ≤ ( 1 − ε n ) f n + 1 , f n is continuous at every τ with f n + 1 ( τ ) < ∞. Continuity here is on all of T T T [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] L ( T ( C ) ) L(T(C)) L ( T ( C )) G G G (4) . We fix the normalized trace of W \mathcal{W} W K \mathcal{K} K [18 ] , Theorem 5.11 and the Blanchard–Kirchberg reduction recalled in [14 ] , Remark 4.3. We use the extended nonunital interface stated there. Thus the functional cone used here is exactly the invariant of the problem.
We also record the tensor-factor properties
W ⊗ Q ≅ W , W ⊗ W ≅ W , KK ( W , W ) = 0. (6) \mathcal{W} \otimes\mathbb{Q} \cong\mathcal{W}, \qquad\mathcal{W} \otimes\mathcal{W} \cong\mathcal{W}, \qquad\operatorname{KK}(\mathcal{W},\mathcal{W})=0. \tag*{(6)} W ⊗ Q ≅ W , W ⊗ W ≅ W , KK ( W , W ) = 0. (6) The first property follows from [21 ] , Proposition 6.2; the remaining properties are recorded in [25 ] , Section 2.1 and Corollary 6.2. They are properties of the tensor factor W \mathcal{W} W A A A B B B P P P Q Q Q
For completeness, the tensor-square and approximate-inner-flip properties underlying realification have accessible proofs in these same sources. Choose β : W ⊗ W → W \beta:\mathcal{W}\otimes\mathcal{W}\to\mathcal{W} β : W ⊗ W → W [25 ] , Corollary 6.2(i). Conjugating the flip by β \beta β W \mathcal{W} W [21 ] , Corollary 4.6. Pulling the implementing unitaries back through the unitized β \beta β W ⊗ W \mathcal{W}\otimes\mathcal{W} W ⊗ W [21 ] , Remark 5.2. Thus the tensor-square isomorphism and the approximate inner flip needed for realification follow from properties of the factor itself.
Finite ideals, zero ideals, and open supports
For τ ∈ T ( C ) \tau\in T(C) τ ∈ T ( C ) Fin ( τ ) \operatorname{Fin}(\tau) Fin ( τ ) τ \tau τ Ker ( τ ) \operatorname{Ker}(\tau) Ker ( τ ) τ \tau τ Fin ( τ ) \operatorname{Fin}(\tau) Fin ( τ ) τ ( a ) = ∞ \tau(a)=\infty τ ( a ) = ∞ a ∈ C + ∖ Fin ( τ ) a\in C_{+}\setminus\operatorname{Fin}(\tau) a ∈ C + ∖ Fin ( τ )
a ∈ Fin ( τ ) ⟺ d τ ( ( a − ε ) + ) < ∞ for every ε > 0. (7) a \in\operatorname{Fin}(\tau) \Longleftrightarrow d_{\tau}((a-\varepsilon)_{+})<\infty\quad\text{for every } \varepsilon>0. \tag*{(7)} a ∈ Fin ( τ ) ⟺ d τ (( a − ε ) + ) < ∞ for every ε > 0. (7) To verify the forward direction, approximate a a a b b b Fin ( τ ) \operatorname{Fin}(\tau) Fin ( τ ) ∥ a − b ∥ < ε / 2 \lVert a-b\rVert<\varepsilon/2 ∥ a − b ∥ < ε /2
d τ ( ( a − ε ) + ) ≤ d τ ( ( b − ε / 2 ) + ) ≤ 2 ε τ ( b ) < ∞ . d_{\tau}((a-\varepsilon)_{+})\le d_{\tau}((b-\varepsilon/2)_{+})\le\frac{2}{\varepsilon}\tau(b)<\infty. d τ (( a − ε ) + ) ≤ d τ (( b − ε /2 ) + ) ≤ ε 2 τ ( b ) < ∞. The finite-trace positives are dense in this ideal: finite sums of terms x ∗ b x x^{*}bx x ∗ b x τ ( b ) < ∞ \tau(b)<\infty τ ( b ) < ∞ τ ( ( a − ε ) + ) ≤ ∥ a ∥ d τ ( ( a − ε ) + ) \tau((a-\varepsilon)_{+})\le\lVert a\rVert d_{\tau}((a-\varepsilon)_{+}) τ (( a − ε ) + ) ≤ ∥ a ∥ d τ (( a − ε ) + ) a a a (7) .
Proposition 2.2 (Transport of ideals). The maps F , G F,G F , G P P P Q Q Q Prim ( P ) = Prim ( Q ) = X \operatorname{Prim}(P)=\operatorname{Prim}(Q)=X Prim ( P ) = Prim ( Q ) = X U ⊆ X U\subseteq X U ⊆ X P ( U ) P(U) P ( U ) Q ( U ) Q(U) Q ( U )
G ( Cu ( P ( U ) ) ) = Cu ( Q ( U ) ) , G(\operatorname{Cu}(P(U)))=\operatorname{Cu}(Q(U)), G ( Cu ( P ( U ))) = Cu ( Q ( U )) , and
Fin ( τ ) = Q ( U ) ⟺ Fin ( F τ ) = P ( U ) , Ker ( τ ) = Q ( U ) ⟺ Ker ( F τ ) = P ( U ) . \operatorname{Fin}(\tau)=Q(U)\Longleftrightarrow\operatorname{Fin}(F\tau)=P(U), \qquad\operatorname{Ker}(\tau)=Q(U)\Longleftrightarrow\operatorname{Ker}(F\tau)=P(U). Fin ( τ ) = Q ( U ) ⟺ Fin ( F τ ) = P ( U ) , Ker ( τ ) = Q ( U ) ⟺ Ker ( F τ ) = P ( U ) . Proof. The additive idempotents of the trace cone are exactly the ideal weights τ I \tau_I τ I τ + τ = τ \tau+\tau=\tau τ + τ = τ
I ⊆ J ⟺ τ I + τ J = τ I . I\subseteq J \Longleftrightarrow\tau_I+\tau_J=\tau_I. I ⊆ J ⟺ τ I + τ J = τ I . Thus F F F F ( τ I ) = τ J F(\tau_I)=\tau_J F ( τ I ) = τ J
[ a ] ∈ Cu ( J ) ⟺ d F τ I ( a ) = 0 ⟺ d τ I ( G [ a ] ) = 0 ⟺ G [ a ] ∈ Cu ( I ) . [a]\in\operatorname{Cu}(J)\Longleftrightarrow d_{F\tau_I}(a)=0\Longleftrightarrow d_{\tau_I}(G[a])=0\Longleftrightarrow G[a]\in\operatorname{Cu}(I). [ a ] ∈ Cu ( J ) ⟺ d F τ I ( a ) = 0 ⟺ d τ I ( G [ a ]) = 0 ⟺ G [ a ] ∈ Cu ( I ) . This proves compatibility with G G G
The topology recovers the two other ideals through
lim r ↓ 0 r τ = τ F i n ( τ ) , lim r ↑ ∞ r τ = τ K e r ( τ ) . (8) \lim_{r\downarrow0} r\tau= \tau_{\mathrm{Fin}}(\tau), \qquad\lim_{r\uparrow\infty} r\tau= \tau_{\mathrm{Ker}}(\tau). \tag*{(8)} r ↓ 0 lim r τ = τ Fin ( τ ) , r ↑ ∞ lim r τ = τ Ker ( τ ) . (8) These are [14 ] , (3.4) . They can also be checked directly in (1) : all positive cuts of an element of Fin ( τ ) \operatorname{Fin}(\tau) Fin ( τ ) (7) , whereas an element outside that ideal has infinite trace; for the second limit use whether its trace is zero or positive. Continuity and positive homogeneity of F F F
An ideal-lattice isomorphism preserves prime ideals and their hull–kernel topology. In a separable C ∗ C^* C ∗ [11 ] , Corollary 1], which gives the asserted homeomorphism. Finally, the zero algebra is distinguished by the cone: every nonzero algebra has the nonzero weight τ { 0 } \tau_{\{0\}} τ { 0 } P , Q ≠ 0 P,Q\ne0 P , Q = 0
For a positive element c ∈ Q ( U ) c\in Q(U) c ∈ Q ( U ) c c c compactly Cuntz supported in Q ( U ) Q(U) Q ( U )
[ c ] ≤ [ ( b − δ ) + ] for some b ∈ Q ( U ) + , δ > 0. (9) [c]\leq[(b-\delta)_+]\quad\text{for some }b\in Q(U)_+,\ \delta>0. \tag*{(9)} [ c ] ≤ [( b − δ ) + ] for some b ∈ Q ( U ) + , δ > 0. (9) Equivalently, [ c ] [c] [ c ] Q ( U ) Q(U) Q ( U ) c c c
The next lemma permits one controller to be chosen before an approximation sequence is constructed. Its finite multiplicity may depend on the element being controlled. Later we will compensate for that multiplicity by choosing a sufficiently late coordinate.
Lemma 2.3 (Countable supports and common controllers). The open-set lattice of X X X U U U V ⋐ U V\Subset U V ⋐ U V ⋐ U V\Subset U V ⋐ U b V , U ∈ Q ( U ) + b_{V,U}\in Q(U)_+ b V , U ∈ Q ( U ) +
Q ( V ) ⊆ Ideal Q ( b V , U ) . Q(V)\subseteq\operatorname{Ideal}_Q(b_{V,U}). Q ( V ) ⊆ Ideal Q ( b V , U ) . Every compactly Cuntz supported c ∈ Q ( V ) + c\in Q(V)_+ c ∈ Q ( V ) +
[ c ] ≤ N [ b V , U ] for some finite N ∈ N . (10) [c]\leq N[b_{V,U}]\quad\text{for some finite }N\in\mathbb{N}. \tag*{(10)} [ c ] ≤ N [ b V , U ] for some finite N ∈ N . (10) The element b V , U b_{V,U} b V , U c c c N N N
Proof. Choose a norm-dense sequence ( a j ) (a_j) ( a j ) Q Q Q ( a j − r ) + (a_j-r)_+ ( a j − r ) + r r r U U U x ∈ U x\in U x ∈ U a ∈ Q ( U ) + a\in Q(U)_+ a ∈ Q ( U ) + x x x a j a_j a j Q ( U ) Q(U) Q ( U ) x x x ( a j − r ) + (a_j-r)_+ ( a j − r ) + ( a j − r / 2 ) + (a_j-r/2)_+ ( a j − r /2 ) + U U U
Let h h h Q ( U ) Q(U) Q ( U ) ( h − 1 / k ) + (h-1/k)_+ ( h − 1/ k ) + Q ( U ) Q(U) Q ( U ) k k k Q ( V ) ⊆ Ideal Q ( ( h − 1 / k ) + ) Q(V)\subseteq\operatorname{Ideal}_Q((h-1/k)_+) Q ( V ) ⊆ Ideal Q (( h − 1/ k ) + ) b V , U b_{V,U} b V , U c c c (9) inside Q ( V ) Q(V) Q ( V ) [ c ] ≪ [ z ] [c]\ll[z] [ c ] ≪ [ z ] z ∈ Q ( V ) + z\in Q(V)_+ z ∈ Q ( V ) + [ b V , U ] [b_{V,U}] [ b V , U ] sup N N [ b V , U ] \sup_N N[b_{V,U}] sup N N [ b V , U ] [ z ] ≤ sup N N [ b V , U ] [z]\leq\sup_N N[b_{V,U}] [ z ] ≤ sup N N [ b V , U ] (10) .
For the domains used below,
S Y = C 0 ( Y ) ⊗ P , S Y ( U ) = C 0 ( Y ) ⊗ P ( U ) . S_Y = C_0(Y) \otimes P,\qquad S_Y(U) = C_0(Y) \otimes P(U). S Y = C 0 ( Y ) ⊗ P , S Y ( U ) = C 0 ( Y ) ⊗ P ( U ) . Their primitive spaces are Y × X Y \times X Y × X s ∈ S Y s \in S_Y s ∈ S Y U s U_s U s X X X Ideal S Y ( s ) \operatorname{Ideal}_{S_Y}(s) Ideal S Y ( s )
s ∈ S Y ( U ) ⟺ U s ⊆ U . s \in S_Y(U) \quad\Longleftrightarrow\quad U_s \subseteq U. s ∈ S Y ( U ) ⟺ U s ⊆ U . Thus every element has a smallest coarse label. This is the lower-semicontinuity condition on the domain ideal assignment used in absorption. The fine labels on S Y S_Y S Y Y × X Y \times X Y × X
Folding and rational averaging
There are two matrix operations in the proof. Stabilization adds internal matrix corners, whose ranks add with no normalization. Convex averaging instead uses a separate UHF factor, whose normalized trace supplies the convex coefficients. The following identifications allow both operations to have range Q Q Q
Lemma 2.4 (Trace-preserving folding). Finite matrix amplifications, including amplifications whose sizes vary with the coordinate, may be folded into Q Q Q Q Q Q
If h 1 , … , h l h_1,\ldots,h_l h 1 , … , h l Q Q Q λ i > 0 \lambda_i > 0 λ i > 0 ∑ i λ i = 1 \sum_i \lambda_i = 1 ∑ i λ i = 1 h h h Q Q Q s ≥ 0 s \ge0 s ≥ 0
τ ( h ( s ) ) = ∑ i λ i τ ( h i ( s ) ) , τ ( h ( s ) 2 ) = ∑ i λ i τ ( h i ( s ) 2 ) , (11) \tau(h(s)) = \sum_i \lambda_i \tau(h_i(s)),\qquad
\tau(h(s)^2) = \sum_i \lambda_i \tau(h_i(s)^2), \tag*{(11)} τ ( h ( s )) = i ∑ λ i τ ( h i ( s )) , τ ( h ( s ) 2 ) = i ∑ λ i τ ( h i ( s ) 2 ) , (11) d τ ( ( h ( s ) − ε ) + ) = ∑ i λ i d τ ( ( h i ( s ) − ε ) + ) . (12) d_\tau((h(s)-\varepsilon)_+) = \sum_i \lambda_i d_\tau((h_i(s)-\varepsilon)_+). \tag*{(12)} d τ (( h ( s ) − ε ) + ) = i ∑ λ i d τ (( h i ( s ) − ε ) + ) . (12) The traces on the right include the unnormalized internal matrices. All equalities include zero and infinite values.
Proof. Use (6) once to exhibit Q = C ⊗ Q ⊗ K Q = \mathbb{C} \otimes Q \otimes\mathcal{K} Q = C ⊗ Q ⊗ K U k : C k ⊗ ℓ 2 → ℓ 2 U_k : \mathbb{C}^k \otimes\ell^2 \to\ell^2 U k : C k ⊗ ℓ 2 → ℓ 2 id C k ⊗ Q ⊗ Ad ( U k ) \operatorname{id}_{\mathbb{C}^k \otimes Q} \otimes\operatorname{Ad}(U_k) id C k ⊗ Q ⊗ Ad ( U k ) E ⊂ Q E \subset Q E ⊂ Q e e e ∥ a − e a e ∥ \lVert a-eae\rVert ∥ a − e a e ∥ a ∈ E a \in E a ∈ E B ( ℓ 2 ) B(\ell^2) B ( ℓ 2 ) e ℓ 2 e\ell^2 e ℓ 2 e ℓ 2 e\ell^2 e ℓ 2 E E E
Write λ i = l i / L \lambda_i = l_i/L λ i = l i / L e i e_i e i l i / L l_i/L l i / L
h ( s ) = ∑ i h i ( s ) ⊗ e i . h(s) = \sum_i h_i(s) \otimes e_i. h ( s ) = i ∑ h i ( s ) ⊗ e i . Orthogonality makes its norm the maximum of the block norms, and functional calculus is blockwise for functions vanishing at zero. This proves complete positivity, contractivity, and ((11) –(12) ) before the final UHF identification.
Choose that identification by interleaving tensor factors of a fixed infinite tensor-product realization of Q Q Q τ ~ ( a ⊗ e ) = tr ( e ) τ ( a ) \widetilde{\tau}(a \otimes e) = \operatorname{tr}(e)\tau(a) τ ( a ⊗ e ) = tr ( e ) τ ( a ) τ ( a ) = τ ~ ( a ⊗ 1 ) \tau(a) = \widetilde{\tau}(a \otimes1) τ ( a ) = τ ( a ⊗ 1 ) tr \operatorname{tr} tr Q Q Q a 1 / 2 u a^{1/2}u a 1/2 u
We always distinguish the internal matrix directions from the normalized candidate-averaging direction. In particular, an internal sector sum is not divided by its number of sectors. All subsequent padding and foldings use Lemma 2.4 with the existing constant and finite test elements included among the elements to be aligned.
Sequence models and local error ideals
Put D = Q ∞ = ℓ ∞ ( Q ) / c 0 ( Q ) D = Q_\infty= \ell^\infty(Q)/c_0(Q) D = Q ∞ = ℓ ∞ ( Q ) / c 0 ( Q ) Q Q Q
Regularized limits and their model data
Proposition 3.1 (Regularized limits). Let ω \omega ω N \mathbb{N} N τ n ∈ T ( Q ) \tau_n \in T(Q) τ n ∈ T ( Q ) ( x n ) (x_n) ( x n )
ρ ( [ ( x n ) ] ) = sup ε > 0 lim n → ω τ n ( ( x n − ε ) + ) (13) \rho([(x_n)]) = \sup_{\varepsilon>0}\lim_{n\to\omega}\tau_n((x_n-\varepsilon)_+) \tag*{(13)} ρ ([( x n )]) = ε > 0 sup n → ω lim τ n (( x n − ε ) + ) (13) defines a lower-semicontinuous tracial weight on D D D
d ρ ( [ ( x n ) ] ) = sup ε > 0 lim n → ω d τ n ( ( x n − ε ) + ) . (14) d_\rho([(x_n)]) = \sup_{\varepsilon>0}\lim_{n\to\omega}d_{\tau_n}((x_n-\varepsilon)_+). \tag*{(14)} d ρ ([( x n )]) = ε > 0 sup n → ω lim d τ n (( x n − ε ) + ) . (14) If σ = lim n → ω τ n \sigma= \lim_{n\to\omega}\tau_n σ = lim n → ω τ n T ( Q ) T(Q) T ( Q ) ρ ∣ Q = σ \rho|_Q = \sigma ρ ∣ Q = σ
Proof. Pull τ n \tau_n τ n ℓ ∞ ( Q ) \ell^\infty(Q) ℓ ∞ ( Q ) n n n ω \omega ω ρ ~ \widetilde{\rho} ρ x x x
lim ω τ n ( ( x n − ε ) + ) ≤ ρ ~ ( x ) , ρ ~ ( ( x − ε ) + ) ≤ lim ω τ n ( ( x n − ε ) + ) . \lim_{\omega}\tau_n((x_n-\varepsilon)_+) \le\widetilde{\rho}(x), \qquad\widetilde{\rho}((x-\varepsilon)_+) \le\lim_{\omega}\tau_n((x_n-\varepsilon)_+). ω lim τ n (( x n − ε ) + ) ≤ ρ ( x ) , ρ (( x − ε ) + ) ≤ ω lim τ n (( x n − ε ) + ) . Taking suprema over ε \varepsilon ε c 0 ( Q ) + c_0(Q)_+ c 0 ( Q ) + ρ ~ \widetilde{\rho} ρ ρ \rho ρ ρ ∣ Q = σ \rho|_Q = \sigma ρ ∣ Q = σ
For completeness, rescale the representatives so that 0 ≤ x n ≤ 1 0 \le x_n \le1 0 ≤ x n ≤ 1 L L L k k k t > 0 t > 0 t > 0
τ n ( ( x n 1 / k − t ) + ) ≤ d τ n ( ( x n − t k ) + ) . \tau_n((x_n^{1/k}-t)_+) \le d_{\tau_n}((x_n-t^k)_+). τ n (( x n 1/ k − t ) + ) ≤ d τ n (( x n − t k ) + ) . Regularization and then the supremum over k k k d ρ ( x ) ≤ L d_{\rho}(x) \le L d ρ ( x ) ≤ L g ε g_{\varepsilon} g ε [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 0 , ε / 2 ] [0,\varepsilon/2] [ 0 , ε /2 ] [ ε , 1 ] [\varepsilon,1] [ ε , 1 ] 0 < t < 1 0 < t < 1 0 < t < 1
τ n ( ( g ε ( x n ) − t ) + ) ≥ ( 1 − t ) d τ n ( ( x n − ε ) + ) . \tau_n((g_{\varepsilon}(x_n)-t)_+) \ge(1-t)d_{\tau_n}((x_n-\varepsilon)_+). τ n (( g ε ( x n ) − t ) + ) ≥ ( 1 − t ) d τ n (( x n − ε ) + ) . The support of g ε ( x ) g_{\varepsilon}(x) g ε ( x ) x x x d ρ ( x ) ≥ ρ ( g ε ( x ) ) d_{\rho}(x) \ge\rho(g_{\varepsilon}(x)) d ρ ( x ) ≥ ρ ( g ε ( x )) t ↓ 0 t \downarrow0 t ↓ 0 d ρ ( x ) ≥ lim ω d τ n ( ( x n − ε ) + ) d_{\rho}(x) \ge\lim_{\omega}d_{\tau_n}((x_n-\varepsilon)_+) d ρ ( x ) ≥ lim ω d τ n (( x n − ε ) + ) ε \varepsilon ε
We call the weights in Proposition 3.1 regularized limits . The coordinate weights need not be bounded, normalized, or densely finite. Passing to a subsequence is included by taking an ultrafilter concentrated on its indices.
Let Y Y Y R \mathbb{R} R m m m R \mathbb{R} R S = S Y S = S_Y S = S Y ν ∈ T ( P ) \nu\in T(P) ν ∈ T ( P ) m ⊗ ν m \otimes\nu m ⊗ ν C 0 ( Y ) ⊗ Fin ( ν ) C_0(Y) \otimes\operatorname{Fin}(\nu) C 0 ( Y ) ⊗ Fin ( ν )
d m ⊗ ν ( f ⊗ a ) = m ( { f > 0 } ) d ν ( a ) ( f ∈ C 0 ( Y ) + , a ∈ P + ) . (15) d_{m \otimes\nu}(f \otimes a) = m(\{f > 0\})d_{\nu}(a) \qquad(f \in C_0(Y)_+,\, a \in P_+). \tag*{(15)} d m ⊗ ν ( f ⊗ a ) = m ({ f > 0 }) d ν ( a ) ( f ∈ C 0 ( Y ) + , a ∈ P + ) . (15) Here a zero factor makes the product zero, including the expression 0 ⋅ ∞ 0 \cdot\infty 0 ⋅ ∞ m m m
Definition 3.2 (Model). A model for ( Y , m , F ) (Y,m,F) ( Y , m , F ) p : S Y → D p : S_Y \to D p : S Y → D
ρ ∘ p = m ⊗ F ( σ ) , σ = ρ ∣ Q , \rho\circ p = m \otimes F(\sigma), \qquad\sigma= \rho|_Q, ρ ∘ p = m ⊗ F ( σ ) , σ = ρ ∣ Q , for every regularized limit ρ \rho ρ
We fix a countable collection of positive elementary tensors for testing this identity. In each basis ideal P ( V ) P(V) P ( V ) C 0 ( Y ) C_0(Y) C 0 ( Y )
Lemma 3.3 (Elementary determination). A trace θ ∈ T ( S Y ) \theta\in T(S_Y) θ ∈ T ( S Y ) m ⊗ ν m \otimes\nu m ⊗ ν
Proof. We first extend the rank identities to all elementary tensors, then identify the ideal where the weights are densely finite, and finally compare their bounded restrictions inside that ideal.
Extension of the tests. Suppose that positive contractions f j , a i f_j,a_i f j , a i f , a f,a f , a η \eta η
( f − 3 η ) + ⊗ ( a − 3 η ) + ≾ ( f j − 2 η ) + ⊗ ( a i − 2 η ) + ≾ f ⊗ a . (f-3\eta)_+ \otimes(a-3\eta)_+ \precsim(f_j-2\eta)_+ \otimes(a_i-2\eta)_+ \precsim f \otimes a. ( f − 3 η ) + ⊗ ( a − 3 η ) + ≾ ( f j − 2 η ) + ⊗ ( a i − 2 η ) + ≾ f ⊗ a . The middle tensors belong to the test collection when η \eta η f f f a a a θ \theta θ m ⊗ ν m \otimes\nu m ⊗ ν η ↓ 0 \eta\downarrow0 η ↓ 0 (15) for every elementary tensor with compact function support. Exhaustion gives the remaining elementary tensors.
Identification of the finite ideal. Set I = Fin ( ν ) I = \operatorname{Fin}(\nu) I = Fin ( ν ) θ \theta θ C 0 ( Y ) ⊗ I C_0(Y) \otimes I C 0 ( Y ) ⊗ I I I I θ \theta θ C 0 ( Y ) ⊗ I C_0(Y) \otimes I C 0 ( Y ) ⊗ I V × U V \times U V × U Fin ( θ ) \operatorname{Fin}(\theta) Fin ( θ ) f ∈ C c ( V ) + f \in C_c(V)_+ f ∈ C c ( V ) + a ∈ P ( U ) + a \in P(U)_+ a ∈ P ( U ) + f 0 ∈ C c ( V ) + f_0 \in C_c(V)_+ f 0 ∈ C c ( V ) + supp ( f ) \operatorname{supp}(f) supp ( f ) b ∈ P ( U ) + b \in P(U)_+ b ∈ P ( U ) + δ > 0 \delta> 0 δ > 0 a ≾ ( b − δ ) + a \precsim(b-\delta)_+ a ≾ ( b − δ ) +
f ⊗ a ≾ ( f 0 ⊗ b − δ / 2 ) + . f \otimes a \precsim(f_0 \otimes b-\delta/2)_+. f ⊗ a ≾ ( f 0 ⊗ b − δ /2 ) + . The larger tensor belongs to Fin ( θ ) \operatorname{Fin}(\theta) Fin ( θ ) f ⊗ a f \otimes a f ⊗ a θ \theta θ (7) . Formula (15) and 0 < m ( f > 0 ) < ∞ 0 < m(f > 0) < \infty 0 < m ( f > 0 ) < ∞ d ν ( a ) < ∞ d_\nu(a) < \infty d ν ( a ) < ∞ P ( U ) P(U) P ( U ) P ( U ) ⊆ I P(U) \subseteq I P ( U ) ⊆ I
Equality on the finite ideal. It remains to identify the densely finite traces on this common ideal; there is nothing to prove if I = 0 I = 0 I = 0 h ∈ I h \in I h ∈ I e n = ( h − 1 / n ) + e_n = (h-1/n)_+ e n = ( h − 1/ n ) + f n f_n f n Y Y Y
H n = Her ( f n ⊗ e n ) = C 0 ( { f n > 0 } ) ⊗ Her ( e n ) H_n = \operatorname{Her}(f_n \otimes e_n) = C_0(\{f_n > 0\}) \otimes\operatorname{Her}(e_n) H n = Her ( f n ⊗ e n ) = C 0 ({ f n > 0 }) ⊗ Her ( e n ) increase with dense union in C 0 ( Y ) ⊗ I C_0(Y) \otimes I C 0 ( Y ) ⊗ I H n H_n H n m ( f n > 0 ) d ν ( e n ) < ∞ m(f_n > 0)d_\nu(e_n) < \infty m ( f n > 0 ) d ν ( e n ) < ∞ H n ∗ ∗ H_n^{**} H n ∗∗ H n H_n H n
θ ( f ⊗ a ) = ∫ 0 ∥ f ∥ ∫ 0 ∥ a ∥ d θ ( ( f − t ) + ⊗ ( a − u ) + ) d u d t = ∫ 0 ∥ f ∥ ∫ 0 ∥ a ∥ m ( f > t ) d ν ( ( a − u ) + ) d u d t = ( ∫ Y f d m ) ν ( a ) . \begin{aligned}
\theta(f \otimes a) &= \int_0^{\lVert f\rVert}\int_0^{\lVert a\rVert} d_\theta((f-t)_+ \otimes(a-u)_+) \,du\,dt \\
&= \int_0^{\lVert f\rVert}\int_0^{\lVert a\rVert} m(f > t)d_\nu((a-u)_+) \,du\,dt = \left(\int_Y f\,dm\right)\nu(a).
\end{aligned} θ ( f ⊗ a ) = ∫ 0 ∥ f ∥ ∫ 0 ∥ a ∥ d θ (( f − t ) + ⊗ ( a − u ) + ) d u d t = ∫ 0 ∥ f ∥ ∫ 0 ∥ a ∥ m ( f > t ) d ν (( a − u ) + ) d u d t = ( ∫ Y f d m ) ν ( a ) . The first integrand is the value of the normal extension on the product of the two spectral projections. Tonelli’s theorem applies to these nonnegative functions. Boundedness and algebraic tensor density now give equality of the two traces on H n H_n H n ( v n ) (v_n) ( v n ) v n v_n v n H j H_j H j x ≥ 0 x \ge0 x ≥ 0 v n x v n v_nxv_n v n x v n
θ ( v n x v n ) = θ ( x 1 / 2 v n 2 x 1 / 2 ) ≤ θ ( x ) , v n x v n ⟶ x . \theta(v_nxv_n) = \theta(x^{1/2}v_n^2x^{1/2}) \le\theta(x), \qquad v_nxv_n \longrightarrow x. θ ( v n x v n ) = θ ( x 1/2 v n 2 x 1/2 ) ≤ θ ( x ) , v n x v n ⟶ x . Lower semicontinuity gives θ ( x ) = lim n θ ( v n x v n ) \theta(x) = \lim_n \theta(v_nxv_n) θ ( x ) = lim n θ ( v n x v n ) m ⊗ ν m \otimes\nu m ⊗ ν
The following elementary estimate is the source of the uniformity needed when rows of models are combined.
Lemma 3.4 (Bounded witnesses into a fixed cut). Let a , b a,b a , b C ∗ C^* C ∗ a ≾ ( b − δ ) + a \precsim(b-\delta)_+ a ≾ ( b − δ ) + δ > 0 \delta> 0 δ > 0 η > 0 \eta> 0 η > 0 w w w
∥ w ∗ b w − a ∥ < η , ∥ w ∥ 2 ≤ ∥ a ∥ + η δ . (16) \lVert w^*bw-a\rVert< \eta, \qquad\lVert w\rVert^2 \le\frac{\lVert a\rVert+\eta}{\delta}. \tag*{(16)} ∥ w ∗ b w − a ∥ < η , ∥ w ∥ 2 ≤ δ ∥ a ∥ + η . (16) The bound is independent of the matrix size.
Proof. Choose v v v ∥ v ∗ ( b − δ ) + v − a ∥ < η \lVert v^*(b-\delta)_+v-a\rVert<\eta ∥ v ∗ ( b − δ ) + v − a ∥ < η
g δ ( t ) = { ( t − δ ) + / t , t > 0 , 0 , t = 0 , w = g δ ( b ) v . g_\delta(t)=
\begin{cases}
\sqrt{(t-\delta)_+/t}, & t>0,\\
0, & t=0,
\end{cases}
\qquad w=g_\delta(b)v. g δ ( t ) = { ( t − δ ) + / t , 0 , t > 0 , t = 0 , w = g δ ( b ) v . Then w ∗ b w = v ∗ ( b − δ ) + v w^*bw=v^*(b-\delta)_+v w ∗ b w = v ∗ ( b − δ ) + v g δ ( b ) 2 ≤ δ − 1 ( b − δ ) + g_\delta(b)^2\le\delta^{-1}(b-\delta)_+ g δ ( b ) 2 ≤ δ − 1 ( b − δ ) + (16) .
Lemma 3.5 (Strict comparison from regularized limits). Let x , y ∈ D + x,y\in D_+ x , y ∈ D + 0 < α < 1 0<\alpha<1 0 < α < 1
d ρ ( x ) ≤ α d ρ ( y ) for every regularized limit ρ , (17) d_\rho(x)\le\alpha d_\rho(y)\qquad\text{for every regularized limit }\rho, \tag*{(17)} d ρ ( x ) ≤ α d ρ ( y ) for every regularized limit ρ , (17) then x ≾ y x\precsim y x ≾ y
Proof. Choose bounded positive representatives x n , y n x_n,y_n x n , y n ε > 0 \varepsilon>0 ε > 0 ( x − 2 ε ) + ≾̸ y (x-2\varepsilon)_+\not\precsim y ( x − 2 ε ) + ≾ y δ > 0 \delta>0 δ > 0
( x n − ε ) + ≾̸ ( y n − δ ) + . (18) (x_n-\varepsilon)_+\not\precsim(y_n-\delta)_+. \tag*{(18)} ( x n − ε ) + ≾ ( y n − δ ) + . (18) Otherwise Lemma 3.4, applied for each approximation accuracy, supplies bounded sequences of witnesses showing ( x − ε ) + ≾ y (x-\varepsilon)_+\precsim y ( x − ε ) + ≾ y
Choose increasing indices n k n_k n k (18) holds for δ = 1 / k \delta=1/k δ = 1/ k Cu ( Q ) \operatorname{Cu}(Q) Cu ( Q ) (5) , gives a trace whose rank on the source is strictly larger than its rank on the target. The smaller rank is finite. If the larger rank is finite, divide the trace by that rank; if it is infinite, divide by one plus the smaller rank. Thus we may choose τ k \tau_k τ k
d τ k ( ( x n k − ε ) + ) ≥ 1 , d τ k ( ( y n k − 1 / k ) + ) ≤ 1. d_{\tau_k}\bigl((x_{n_k}-\varepsilon)_+\bigr)\ge1,\qquad d_{\tau_k}\bigl((y_{n_k}-1/k)_+\bigr)\le1. d τ k ( ( x n k − ε ) + ) ≥ 1 , d τ k ( ( y n k − 1/ k ) + ) ≤ 1. Take a free ultrafilter on these indices and the corresponding regularized limit. Formula (14) gives d ρ ( x ) ≥ 1 d_\rho(x)\ge1 d ρ ( x ) ≥ 1 t > 0 t>0 t > 0 ( y n k − t ) + (y_{n_k}-t)_+ ( y n k − t ) + ( y n k − 1 / k ) + (y_{n_k}-1/k)_+ ( y n k − 1/ k ) + d ρ ( y ) ≤ 1 d_\rho(y)\le1 d ρ ( y ) ≤ 1 (17) . All comparisons used to select τ k \tau_k τ k Q Q Q
Proposition 3.6 (Sandwich comparisons)
Let p : S Y → D p:S_Y\to D p : S Y → D s = f ⊗ a ≥ 0 s=f\otimes a\ge0 s = f ⊗ a ≥ 0 0 < m 0 : = m ( f > 0 ) < ∞ 0<m_0:=m(f>0)<\infty 0 < m 0 := m ( f > 0 ) < ∞ r > 0 r>0 r > 0 c r ∈ Q + c_r\in Q_+ c r ∈ Q + r G [ a ] rG[a] r G [ a ]
c r ≾ p ( s ) ( 0 < r < m 0 ) , p ( s ) ≾ c r ( r > m 0 ) . c_r\precsim p(s)\quad(0<r<m_0),\qquad p(s)\precsim c_r\quad(r>m_0). c r ≾ p ( s ) ( 0 < r < m 0 ) , p ( s ) ≾ c r ( r > m 0 ) . The following versions have fixed target cuts.
If η , δ > 0 \eta,\delta>0 η , δ > 0 0 < r < m ( f > η ) 0<r<m(f>\eta) 0 < r < m ( f > η ) c c c r G [ ( a − δ ) + ] rG[(a-\delta)_+] r G [( a − δ ) + ] 0 < κ < η δ 0<\kappa<\eta\delta 0 < κ < η δ
c ≾ ( p ( f ⊗ a ) − κ ) + . c\precsim\bigl(p(f\otimes a)-\kappa\bigr)_+. c ≾ ( p ( f ⊗ a ) − κ ) + . If r ′ > m 0 r'>m_0 r ′ > m 0 ε > 0 \varepsilon>0 ε > 0 b b b r ′ G [ a ] r'G[a] r ′ G [ a ] ( b − κ ) + (b-\kappa)_+ ( b − κ ) +
( p ( f ⊗ a ) − ε ) + ≾ ( b − κ ) + . (19) \bigl(p(f\otimes a)-\varepsilon\bigr)_+\precsim(b-\kappa)_+. \tag*{(19)} ( p ( f ⊗ a ) − ε ) + ≾ ( b − κ ) + . (19) The choices in these two displays can be made independently of the model p p p
Proof. For a regularized limit ρ \rho ρ σ = ρ ∣ Q \sigma=\rho|_Q σ = ρ ∣ Q t = d F σ ( a ) t=d_{F_\sigma}(a) t = d F σ ( a ) c r c_r c r p ( s ) p(s) p ( s ) r t rt r t m 0 t m_0t m 0 t (3.8) . This reasoning includes t = 0 t=0 t = 0 t = ∞ t=\infty t = ∞
Apply the lower comparison to s ′ = ( f − η ) + ⊗ ( a − δ ) + s'=(f-\eta)_+\otimes(a-\delta)_+ s ′ = ( f − η ) + ⊗ ( a − δ ) + f ⊗ a > η δ f\otimes a>\eta\delta f ⊗ a > η δ s ′ ≾ ( f ⊗ a − κ ) + s'\precsim(f\otimes a-\kappa)_+ s ′ ≾ ( f ⊗ a − κ ) + 0 < κ < η δ 0<\kappa<\eta\delta 0 < κ < η δ p p p (3.9) .
For the upper comparison, choose m 0 < r < r ′ m_0<r<r' m 0 < r < r ′ δ > 0 \delta>0 δ > 0 ∥ f ∥ δ < ε / 2 \|f\|_\delta<\varepsilon/2 ∥ f ∥ δ < ε /2 f ⊗ a f\otimes a f ⊗ a f ⊗ ( a − δ ) + f\otimes(a-\delta)_+ f ⊗ ( a − δ ) +
( p ( f ⊗ a ) − ε ) + ≾ p ( f ⊗ ( a − δ ) + ) ≾ c , [ c ] = r G [ ( a − δ ) + ] . (p(f\otimes a)-\varepsilon)_+\precsim p(f\otimes(a-\delta)_+)\precsim c,\qquad[c]=rG[(a-\delta)_+]. ( p ( f ⊗ a ) − ε ) + ≾ p ( f ⊗ ( a − δ ) + ) ≾ c , [ c ] = r G [( a − δ ) + ] . Since positive scalar multiplication preserves compact containment,
[ c ] ≪ r G [ a ] ≤ r ′ G [ a ] = [ b ] . [c]\ll rG[a]\le r'G[a]=[b]. [ c ] ≪ r G [ a ] ≤ r ′ G [ a ] = [ b ] . There is therefore κ > 0 \kappa>0 κ > 0 [ c ] ≤ [ ( b − κ ) + ] [c]\le[(b-\kappa)_+] [ c ] ≤ [( b − κ ) + ] p p p
Conversely, apply any regularized rank to the fixed-cut comparisons. Let δ , η ↓ 0 \delta,\eta\downarrow0 δ , η ↓ 0 m 0 m_0 m 0 (15) for the pullback trace on every compact test. Supremum of cuts handles zero or infinite ranks, and compact exhaustion handles infinite-measure supports. Lemma 3.3 now gives (3.4) . □ \square □
Corollary 3.7 (Constant ideals and all traces). For a model p p p s ∈ S s\in S s ∈ S
Ideal D ( p ( s ) ) = Ideal D ( Q ( U s ) ) . (20) \operatorname{Ideal}_D(p(s))=\operatorname{Ideal}_D(Q(U_s)). \tag*{(20)} Ideal D ( p ( s )) = Ideal D ( Q ( U s )) . (20) Given countably many models p i : S → D p_i:S\to D p i : S → D D 0 ⊆ D D_0\subseteq D D 0 ⊆ D Q Q Q γ ∈ T ( D 0 ) \gamma\in T(D_0) γ ∈ T ( D 0 )
γ ∘ p i = m ⊗ F ( γ ∣ Q ) for every i . (21) \gamma\circ p_i=m\otimes F(\gamma|_Q)\qquad\text{for every }i. \tag*{(21)} γ ∘ p i = m ⊗ F ( γ ∣ Q ) for every i . (21) It may also contain any prescribed countable subset of D D D
Proof. For a nonzero elementary tensor of finite function support, the upper sandwiches put its image in the ideal generated by the corresponding constants. The lower trimmed sandwiches recover the ideals of all cuts of a a a Q ( U a ) Q(U_a) Q ( U a ) s s s s s s S S S U s U_s U s Ideal D ( Q ( U s ) ) ⊆ Ideal D ( p ( s ) ) \operatorname{Ideal}_D(Q(U_s))\subseteq\operatorname{Ideal}_D(p(s)) Ideal D ( Q ( U s )) ⊆ Ideal D ( p ( s )) s s s S ( U s ) S(U_s) S ( U s )
For the second assertion, start with the separable algebra generated by the stated ranges, constants, and extra elements. For every countable test, every rational cut and scalar in Proposition 3.6, and every model, adjoin the matrix entries of comparison witnesses at accuracies tending to zero. The resulting algebra D 0 D_0 D 0 C U ( D 0 ) CU(D_0) C U ( D 0 ) C U ( D ) CU(D) C U ( D ) γ ∈ T ( D 0 ) \gamma\in T(D_0) γ ∈ T ( D 0 ) σ = γ ∣ Q \sigma=\gamma|_Q σ = γ ∣ Q r G [ a ] rG[a] r G [ a ] r d F σ ( a ) r d_{F_\sigma}(a) r d F σ ( a ) d σ d_\sigma d σ γ ∘ p i \gamma\circ p_i γ ∘ p i γ \gamma γ (21) . $\square`
The ideals of locally small elements
Definition 3.8 (Local error ideal). For an open U ⊆ X U \subseteq X U ⊆ X x = [ ( x n ) ] ∈ D x=[(x_n)] \in D x = [( x n )] ∈ D J ( U ) J(U) J ( U ) ε > 0 \varepsilon> 0 ε > 0 c c c Q ( U ) Q(U) Q ( U ) r n > 0 r_n > 0 r n > 0 N N N
d τ ( ( ∣ x n ∣ − ε ) + ) ≤ r n d τ ( c ) ( n ≥ N , τ ∈ T ( Q ) ) . (22) d_\tau\bigl((|x_n|-\varepsilon)_+\bigr) \le r_n d_\tau(c) \qquad(n \ge N,\ \tau\in T(Q)). \tag*{(22)} d τ ( ( ∣ x n ∣ − ε ) + ) ≤ r n d τ ( c ) ( n ≥ N , τ ∈ T ( Q )) . (22) The controller c c c N N N x x x U U U ε \varepsilon ε
The scalar r n r_n r n (22) is unambiguously infinity when d τ ( c ) = ∞ d_\tau(c)=\infty d τ ( c ) = ∞
For example, fix a nonzero positive contraction a a a Q ( U ) Q(U) Q ( U ) e n e_n e n t n → 0 t_n \to0 t n → 0 x n = a ⊗ e n x_n=a\otimes e_n x n = a ⊗ e n
∥ x n ∥ = ∥ a ∥ , d τ ( ( x n − ε ) + ) = t n d τ ( ( a − ε ) + ) ≤ t n d τ ( a ) . \|x_n\|=\|a\|,\qquad d_\tau\bigl((x_n-\varepsilon)_+\bigr)=t_n d_\tau\bigl((a-\varepsilon)_+\bigr)\le t_n d_\tau(a). ∥ x n ∥ = ∥ a ∥ , d τ ( ( x n − ε ) + ) = t n d τ ( ( a − ε ) + ) ≤ t n d τ ( a ) . Thus [ ( x n ) ] [(x_n)] [( x n )] J ( U ) J(U) J ( U )
Proposition 3.9 (Properties of the local error ideals). The definition of J ( U ) J(U) J ( U ) D D D
J ( U ) ⊆ Ideal D ( Q ( U ) ) . (23) J(U) \subseteq\operatorname{Ideal}_D(Q(U)). \tag*{(23)} J ( U ) ⊆ Ideal D ( Q ( U )) . (23) If ρ \rho ρ σ = ρ ∣ Q \sigma=\rho|_Q σ = ρ ∣ Q Q ( U ) Q(U) Q ( U )
ρ ∣ J ( U ) = 0. \rho|_{J(U)}=0. ρ ∣ J ( U ) = 0. All assertions use the same conventions in finite matrix algebras.
Proof. We recall the cut estimates that will be used. For bounded operators in a C ∗ C^* C ∗
d τ ( ( ∣ x + y ∣ − ε ) + ) ≤ d τ ( ( ∣ x ∣ − ε / 4 ) + ) + d τ ( ( ∣ y ∣ − ε / 4 ) + ) , (24) d_\tau\bigl((|x+y|-\varepsilon)_+\bigr)\le d_\tau\bigl((|x|-\varepsilon/4)_+\bigr)+d_\tau\bigl((|y|-\varepsilon/4)_+\bigr), \tag*{(24)} d τ ( ( ∣ x + y ∣ − ε ) + ) ≤ d τ ( ( ∣ x ∣ − ε /4 ) + ) + d τ ( ( ∣ y ∣ − ε /4 ) + ) , (24) d τ ( ( ∣ b x ∣ − ε ) + ) , d τ ( ( ∣ x b ∣ − ε ) + ) ≤ d τ ( ( ∣ x ∣ − ε / ( 2 M ) ) + ) ( ∥ b ∥ ≤ M , M > 0 ) . (25) d_\tau\bigl((|bx|-\varepsilon)_+\bigr),\ d_\tau\bigl((|xb|-\varepsilon)_+\bigr)\le d_\tau\bigl((|x|-\varepsilon/(2M))_+\bigr)\qquad(\|b\|\le M,\ M>0). \tag*{(25)} d τ ( ( ∣ b x ∣ − ε ) + ) , d τ ( ( ∣ x b ∣ − ε ) + ) ≤ d τ ( ( ∣ x ∣ − ε / ( 2 M ) ) + ) ( ∥ b ∥ ≤ M , M > 0 ) . (25) Also d τ ( ( ∣ x ∗ ∣ − t ) + ) = d τ ( ( ∣ x ∣ − t ) + ) d_\tau\bigl((|x^*|-t)_+\bigr)=d_\tau\bigl((|x|-t)_+\bigr) d τ ( ( ∣ x ∗ ∣ − t ) + ) = d τ ( ( ∣ x ∣ − t ) + )
∥ x − y ∥ < ε / 4 ⟹ d τ ( ( ∣ x ∣ − ε ) + ) ≤ d τ ( ( ∣ y ∣ − ε / 2 ) + ) . (26) \|x-y\|<\varepsilon/4\Longrightarrow d_\tau\bigl((|x|-\varepsilon)_+\bigr)\le d_\tau\bigl((|y|-\varepsilon/2)_+\bigr). \tag*{(26)} ∥ x − y ∥ < ε /4 ⟹ d τ ( ( ∣ x ∣ − ε ) + ) ≤ d τ ( ( ∣ y ∣ − ε /2 ) + ) . (26) Here is an explicit cut reduction. Set
x t = x f t ( ∣ x ∣ ) , f t ( s ) = { ( s − t ) + / s , s > 0 , 0 , s = 0. x_t=xf_t(|x|),\qquad f_t(s)=
\begin{cases}
(s-t)_+/s, & s>0,\\
0, & s=0.
\end{cases} x t = x f t ( ∣ x ∣ ) , f t ( s ) = { ( s − t ) + / s , 0 , s > 0 , s = 0. Then ∥ x − x t ∥ ≤ t \|x-x_t\|\le t ∥ x − x t ∥ ≤ t ∣ x t ∣ = ( ∣ x ∣ − t ) + |x_t|=(|x|-t)_+ ∣ x t ∣ = ( ∣ x ∣ − t ) + ∥ z − w ∥ < ε ⇒ ( ∣ z ∣ − ε ) + ≾ ∣ w ∣ \|z-w\|<\varepsilon\Rightarrow(|z|-\varepsilon)_+\precsim|w| ∥ z − w ∥ < ε ⇒ ( ∣ z ∣ − ε ) + ≾ ∣ w ∣ w = x ε / 4 + y ε / 4 w=x_{\varepsilon/4}+y_{\varepsilon/4} w = x ε /4 + y ε /4 ∣ w ∣ ≾ ∣ x ε / 4 ∣ ⊕ ∣ y ε / 4 ∣ |w|\precsim|x_{\varepsilon/4}|\oplus|y_{\varepsilon/4}| ∣ w ∣ ≾ ∣ x ε /4 ∣ ⊕ ∣ y ε /4 ∣ w = b x ε / ( 2 M ) w=bx_{\varepsilon/(2M)} w = b x ε / ( 2 M ) ∣ x ε / ( 2 M ) ∣ |x_{\varepsilon/(2M)}| ∣ x ε / ( 2 M ) ∣ (26) take w = y ε / 2 w=y_{\varepsilon/2} w = y ε /2 ∥ x − w ∥ < 3 ε / 4 \|x-w\|<3\varepsilon/4 ∥ x − w ∥ < 3 ε /4 [ v ∗ v ] = [ v v ∗ ] [v^*v]=[vv^*] [ v ∗ v ] = [ v v ∗ ] x , y ∈ J ( U ) x,y \in J(U) x , y ∈ J ( U ) ε / 4 \varepsilon/4 ε /4 Q ( U ) Q(U) Q ( U ) x + y x+y x + y (24) . Equation (25) proves closure under multiplication on either side by an arbitrary bounded sequence. The equality for adjoints proves self-adjointness. Equation (26) proves independence of the representative. For norm closure, first fix ε \varepsilon ε ε / 8 \varepsilon/8 ε /8 ε / 4 \varepsilon/4 ε /4 ε / 2 \varepsilon/2 ε /2 U U U
To prove (23) , fix a cut of ∣ x ∣ |x| ∣ x ∣ c ≾ ( b − δ ) + c \precsim(b-\delta)_+ c ≾ ( b − δ ) + Q ( U ) Q(U) Q ( U ) r n ≤ 1 r_n \le1 r n ≤ 1 Cu ( Q ) \operatorname{Cu}(Q) Cu ( Q ) ( ∣ x n ∣ − ε ) + ≾ c ≾ ( b − δ ) + (|x_n|-\varepsilon)_+ \precsim c \precsim(b-\delta)_+ ( ∣ x n ∣ − ε ) + ≾ c ≾ ( b − δ ) + b b b ∣ x ∣ |x| ∣ x ∣ Ideal D ( Q ( U ) ) \operatorname{Ideal}_D(Q(U)) Ideal D ( Q ( U )) x x x
For vanishing, fix ε \varepsilon ε
d τ n ( c ) ≤ 4 δ τ n ( ( b − 3 δ / 4 ) + ) . (27) d_{\tau_n}(c) \le\frac{4}{\delta}\tau_n((b-3\delta/4)_+). \tag*{(27)} d τ n ( c ) ≤ δ 4 τ n (( b − 3 δ /4 ) + ) . (27) Indeed, on the spectral region b > δ b>\delta b > δ ( b − 3 δ / 4 ) + (b-3\delta/4)_+ ( b − 3 δ /4 ) + δ / 4 \delta/4 δ /4 h = ( b − δ / 2 ) + h=(b-\delta/2)_+ h = ( b − δ /2 ) + Q ( U ) Q(U) Q ( U ) (7) give σ ( h ) < ∞ \sigma(h)<\infty σ ( h ) < ∞
lim ω τ n ( ( b − 3 δ / 4 ) + ) = lim ω τ n ( ( h − δ / 4 ) + ) ≤ σ ( h ) . \lim_{\omega}\tau_n((b-3\delta/4)_+) = \lim_{\omega}\tau_n((h-\delta/4)_+) \le\sigma(h). ω lim τ n (( b − 3 δ /4 ) + ) = ω lim τ n (( h − δ /4 ) + ) ≤ σ ( h ) . The ranks in (27) are therefore bounded on an ω \omega ω r n → 0 r_n \to0 r n → 0 (22) and using (14) yields d ρ ( ∣ x ∣ ) = 0 d_{\rho}(|x|)=0 d ρ ( ∣ x ∣ ) = 0 ρ \rho ρ J ( U ) + J(U)_+ J ( U ) +
A diagonalization with fixed controllers
The diagonalizations used below retain two different kinds of control. A fixed positive controller bounds a local error’s rank, while a fixed positive target cut bounds the norm of a comparison witness. Both must be chosen before advancing through the approximation stages. We call each stage’s sequence of coordinate approximations a row. Its finite matrix size may depend on the row and on the coordinate; all matrices are folded by Lemma 2.4.
Lemma 3.10 (Local diagonalization). Let A \mathcal{A} A Q + i Q ∗ \mathbb{Q}+i\mathbb{Q}^* Q + i Q ∗ S = S Y S=S_Y S = S Y j j j L j : A → D L_j:\mathcal{A}\to D L j : A → D L j , n ( a ) L_{j,n}(a) L j , n ( a ) a ∈ A a\in\mathcal{A} a ∈ A
lim sup j → ∞ ∥ L j ( a ) ∥ ≤ ∥ a ∥ , \limsup_{j\to\infty}\lVert L_j(a)\rVert\le\lVert a\rVert, j → ∞ lim sup ∥ L j ( a )∥ ≤ ∥ a ∥ , and that the quotient errors in every fixed rational linearity, involution, and multiplicativity relation tend to zero as j → ∞ j\to\infty j → ∞ j ( n ) → ∞ j(n)\to\infty j ( n ) → ∞
L ( a ) = [ ( L j ( n ) , n ( a ) ) ] L(a)=[(L_{j(n),n}(a))] L ( a ) = [( L j ( n ) , n ( a ))] extends to a contractive *-homomorphism L : S → D L:S\to D L : S → D
The same diagonal choice can retain the following countable data.
(i) For each l l l V l ⊆ U l V_l\subseteq U_l V l ⊆ U l e l ( j ) ∈ J ( V l ) e_l^{(j)}\in J(V_l) e l ( j ) ∈ J ( V l ) j j j e l , n ( j ) e_{l,n}^{(j)} e l , n ( j ) sup j ∥ e l ( j ) ∥ < ∞ \sup_j\lVert e_l^{(j)}\rVert<\infty sup j ∥ e l ( j ) ∥ < ∞
e l = [ ( e l , n ( j ( n ) ) ) ] e_l=[(e_{l,n}^{(j(n))})] e l = [( e l , n ( j ( n )) )]
belongs to J ( U l ) J(U_l) J ( U l ) a ( j ) , b ( j ) ∈ D + a^{(j)}, b^{(j)} \in D_+ a ( j ) , b ( j ) ∈ D + a ( j ) ≾ ( b ( j ) − δ ) + a^{(j)} \precsim(b^{(j)}-\delta)_+ a ( j ) ≾ ( b ( j ) − δ ) + δ > 0 \delta>0 δ > 0 j j j a , b a,b a , b a ≾ ( b − δ ′ ) + a \precsim(b-\delta')_+ a ≾ ( b − δ ′ ) + 0 < δ ′ < δ 0<\delta'<\delta 0 < δ ′ < δ
For local errors arising from maps on S S S V ⋐ U V\Subset U V ⋐ U U U U S ( U ) S(U) S ( U )
In particular, a diagonal of models is again a model when their data agree on the fixed compact tests and their enlarged versions. This also applies to models on increasing intervals with unchanged measure on these tests. Every subsequence of a model retains its model data.
Proof. The common controller and the coordinate choice. Enumerate all algebraic relations and positive rational cut tests. Reindex the rows so that row j j j j j j l l l ε l \varepsilon_l ε l b l = b V l , U l b_l=b_{V_l,U_l} b l = b V l , U l
In row j j j l l l
d τ ( ( ∣ e l , n ( j ) ∣ − ε l ) + ) ≤ r l , n ( j ) d τ ( c l ( j ) ) , r l , n ( j ) ⟶ 0 , d_\tau\left((|e_{l,n}^{(j)}|-\varepsilon_l)_+\right)\le r_{l,n}^{(j)}d_\tau(c_l^{(j)}),\qquad r_{l,n}^{(j)}\longrightarrow0, d τ ( ( ∣ e l , n ( j ) ∣ − ε l ) + ) ≤ r l , n ( j ) d τ ( c l ( j ) ) , r l , n ( j ) ⟶ 0 , uniformly for τ ∈ T ( Q ) \tau\in T(Q) τ ∈ T ( Q ) n n n c l ( j ) c_l^{(j)} c l ( j ) Q ( V l ) Q(V_l) Q ( V l ) N j , l N_{j,l} N j , l
[ c l ( j ) ] ≤ N j , l [ b l ] . [c_l^{(j)}]\le N_{j,l}[b_l]. [ c l ( j ) ] ≤ N j , l [ b l ] . This integer absorbs internal matrix sizes, repeated blocks, and any other finite multiplicity at that stage. No bound on it as j → ∞ j\to\infty j → ∞
Fix a sequence η j ↓ 0 \eta_j\downarrow0 η j ↓ 0 N j , l N_{j,l} N j , l l ≤ j l\le j l ≤ j n j ≥ j n_j\ge j n j ≥ j
N j , l r l , n ( j ) ≤ η j ( n ≥ n j , l ≤ j ) . (28) N_{j,l}r_{l,n}^{(j)}\le\eta_j\qquad(n\ge n_j,\ l\le j). \tag*{(28)} N j , l r l , n ( j ) ≤ η j ( n ≥ n j , l ≤ j ) . (28) At the same threshold impose the first j j j η j \eta_j η j
For n j ≤ n < n j + 1 n_j\le n<n_{j+1} n j ≤ n < n j + 1 j ( n ) = j j(n)=j j ( n ) = j j j j l l l
d τ ( ( ∣ e l , n ∣ − ε l ) + ) ≤ η j ( n ) d τ ( b l ) , j ( n ) ⟶ ∞ . d_\tau\left((|e_{l,n}|-\varepsilon_l)_+\right)\le\eta_{j(n)}d_\tau(b_l),\qquad j(n)\longrightarrow\infty. d τ ( ( ∣ e l , n ∣ − ε l ) + ) ≤ η j ( n ) d τ ( b l ) , j ( n ) ⟶ ∞. The controller on the right is fixed, so this is precisely the required J ( U l ) J(U_l) J ( U l ) ∥ L ( a ) ∥ ≤ ∥ a ∥ \|L(a)\|\le\|a\| ∥ L ( a ) ∥ ≤ ∥ a ∥ S S S
Extension of local errors. For an arbitrary open U U U S ( V ) S(V) S ( V ) V ⋐ U ′ ⊆ U V \Subset U' \subseteq U V ⋐ U ′ ⊆ U S ( U ) S(U) S ( U ) S ( U ) S(U) S ( U ) V i ⋐ U i ′ V_i \Subset U_i' V i ⋐ U i ′ eq:2 . Thus the union of the scheduled localized dense sets is dense in S ( U ) S(U) S ( U ) J ( U ′ ) ⊆ J ( U ) J(U') \subseteq J(U) J ( U ′ ) ⊆ J ( U ) J ( U ) J(U) J ( U )
Comparison witnesses with a fixed target cut. Suppose a row comparison has uniformly bounded source a ( j ) a^{(j)} a ( j ) b ( j ) b^{(j)} b ( j ) ( b ( j ) − δ ) + (b^{(j)}-\delta)_+ ( b ( j ) − δ ) + δ > 0 \delta> 0 δ > 0 j j j ( b ( j ) − δ ′ ) + (b^{(j)}-\delta')_+ ( b ( j ) − δ ′ ) + δ − δ ′ \delta-\delta' δ − δ ′ δ − δ ′ \delta-\delta' δ − δ ′ n j n_j n j
For rows of models, the lower bounds (3.9) use target thresholds κ < η δ \kappa< \eta\delta κ < η δ
Remark 3.11. When the resulting quotient map is a homomorphism with nuclear separable domain, the completely positive lifting theorem provides c.p.c. coordinate lifts [4 ] , Theorem 3.10. Those lifts represent the same quotient values. Consequently all fixed-cut comparisons and all memberships in J ( U ) J(U) J ( U )
Small stabilizers
We construct homomorphisms whose values belong to the local error ideals, but are large enough to absorb a prescribed countable family of errors. All matrix comparisons in this section are stable comparisons. The identifications of Lemma 2.4 are understood; in particular, a rational UHF fraction changes ranks, not operator norms.
Theorem 4.1 (Small stabilizers). Let S = C 0 ( Y ) ⊗ P S = C_0(Y) \otimes P S = C 0 ( Y ) ⊗ P and prescribe countably many pairs
x l ∈ S + , e l ∈ J ( U x l ) , l ∈ N . x_l \in S_+,\qquad e_l \in J(U_{x_l}),\qquad l \in\mathbb{N}. x l ∈ S + , e l ∈ J ( U x l ) , l ∈ N . There is a homomorphism θ : S → D \theta:S \to D θ : S → D satisfying
θ ( S ( U ) ) ⊆ J ( U ) , e l ∈ Ideal D ( θ ( x l ) ) ( U ⊆ X open , l ∈ N ) . (29) \theta(S(U)) \subseteq J(U),\qquad e_l \in\operatorname{Ideal}_D(\theta(x_l))\quad(U \subseteq X\ \text{open},\ l \in\mathbb{N}). \tag*{(29)} θ ( S ( U )) ⊆ J ( U ) , e l ∈ Ideal D ( θ ( x l )) ( U ⊆ X open , l ∈ N ) . (29) in either of the following situations.
(i) An existing model χ : S → D \chi:S\to D χ : S → D θ \theta θ χ \chi χ
θ ( S ( U x ) ) ⊆ Ideal D ( θ ( x ) ) ( x ∈ S + ) . (30) \theta(S(U_x))\subseteq\operatorname{Ideal}_D(\theta(x)) \qquad(x\in S_+). \tag*{(30)} θ ( S ( U x )) ⊆ Ideal D ( θ ( x )) ( x ∈ S + ) . (30) For every fixed N N N N N N χ \chi χ
(ii) Y = ( 0 , 1 ] Y=(0,1] Y = ( 0 , 1 ]
In both cases, for every l l l ε > 0 \varepsilon>0 ε > 0 δ l , ε > 0 \delta_{l,\varepsilon}>0 δ l , ε > 0
[ ( ∣ e l ∣ − ε ) + ] ≤ [ ( θ ( x l ) − δ l , ε ) + ] in Cu ( D ) . (31) [(\lvert e_l\rvert-\varepsilon)_+]\leq[(\theta(x_l)-\delta_{l,\varepsilon})_+]\quad\text{in }\operatorname{Cu}(D). \tag*{(31)} [(∣ e l ∣ − ε ) + ] ≤ [( θ ( x l ) − δ l , ε ) + ] in Cu ( D ) . (31) The construction permits the prescribed errors and the stabilizer to occupy separate matrix corners.
The cut in (4.3) will remain fixed while the accuracy, matrix sizes, and finite lists change. We first describe the countable tests and the comparison convention used to ensure this.
Fixed tests and comparison bounds
Choose a countable compact-containment basis for the open ideals as in Lemma 2.3. For every pair V ⋐ U V\Subset U V ⋐ U S ( V ) S(V) S ( V )
( a i , ε i , V i , U i ) , a i ∈ S ( V i ) + , V i ⋐ U i , ε i > 0. (32) (a_i,\varepsilon_i,V_i,U_i),\qquad a_i\in S(V_i)_+,\qquad V_i\Subset U_i,\qquad\varepsilon_i>0. \tag*{(32)} ( a i , ε i , V i , U i ) , a i ∈ S ( V i ) + , V i ⋐ U i , ε i > 0. (32) For each test fix a compactly Cuntz-supported positive b i ∈ Q ( U i ) b_i\in Q(U_i) b i ∈ Q ( U i ) Q ( V i ) Q(V_i) Q ( V i )
d τ ( ( θ n ( a i ) − ε i ) + ) ≤ s i , n d τ ( b i ) , s i , n ⟶ 0 , (33) d_\tau((\theta_n(a_i)-\varepsilon_i)_+)\leq s_{i,n}d_\tau(b_i),\qquad s_{i,n}\longrightarrow0, \tag*{(33)} d τ (( θ n ( a i ) − ε i ) + ) ≤ s i , n d τ ( b i ) , s i , n ⟶ 0 , (33) uniformly in τ \tau τ S ( V i ) S(V_i) S ( V i ) S ( V i ) S(V_i) S ( V i ) V i ⋐ U V_i\Subset U V i ⋐ U S ( U ) S(U) S ( U )
Choose bounded representatives e l , n e_{l,n} e l , n q = ( l , η ) q=(l,\eta) q = ( l , η )
A q , n = ( ∣ e l , n ∣ − η ) + , d τ ( A q , n ) ≤ r q ( n ) d τ ( c q ) , r q ( n ) > 0 , r q ( n ) ⟶ 0. (34) A_{q,n}=(\lvert e_{l,n}\rvert-\eta)_+,\qquad d_\tau(A_{q,n})\leq r_q(n)d_\tau(c_q),\qquad r_q(n)>0,\qquad r_q(n)\longrightarrow0. \tag*{(34)} A q , n = (∣ e l , n ∣ − η ) + , d τ ( A q , n ) ≤ r q ( n ) d τ ( c q ) , r q ( n ) > 0 , r q ( n ) ⟶ 0. (34) Here c q c_q c q Q ( U x l ) Q(U_{x_l}) Q ( U x l ) τ ∈ T ( Q ) \tau\in T(Q) τ ∈ T ( Q ) r q r_q r q
We will use the fixed-cut comparison bound from Lemma 3.4. Its elementary form is worth recording. If A ≾ ( B − δ ) + A\precsim(B-\delta)_+ A ≾ ( B − δ ) + v v v v ∗ ( B − δ ) + v v^*(B-\delta)_+v v ∗ ( B − δ ) + v A A A
w = g δ ( B ) v , g δ ( t ) = ( t − δ ) + t , g δ ( 0 ) = 0. w=g_\delta(B)v,\qquad g_\delta(t)=\sqrt{\frac{(t-\delta)_+}{t}},\qquad g_\delta(0)=0. w = g δ ( B ) v , g δ ( t ) = t ( t − δ ) + , g δ ( 0 ) = 0. Then
w ∗ B w = v ∗ ( B − δ ) + v , ∥ w ∥ 2 ≤ ∥ A ∥ + ∥ v ∗ ( B − δ ) + v − A ∥ δ . (35) w^*Bw=v^*(B-\delta)_+v,\qquad\lVert w\rVert^2\leq\frac{\lVert A\rVert+\lVert v^*(B-\delta)_+v-A\rVert}{\delta}. \tag*{(35)} w ∗ B w = v ∗ ( B − δ ) + v , ∥ w ∥ 2 ≤ δ ∥ A ∥ + ∥ v ∗ ( B − δ ) + v − A ∥ . (35) Thus coordinate comparisons into a fixed positive cut have bounded witnesses into the uncut target. This conclusion is independent of matrix size.
Fractions of an existing model
Proof of Theorem 4.1(i). Take c.p.c. lifts χ n : S → Q \chi_n:S\to Q χ n : S → Q χ \chi χ [4 ] , Theorem 3.10. The constant-ideal assertion of Corollary 3.7 gives
Ideal D ( χ ( x l ) ) = Ideal D ( Q ( U x l ) ) . \operatorname{Ideal}_{D}(\chi(x_l))=\operatorname{Ideal}_{D}(Q(Ux_l)). Ideal D ( χ ( x l )) = Ideal D ( Q ( U x l )) . Compact support of the controller in (34) , with an extra cut before passing to coordinates, consequently gives an integer M q M_q M q d q > 0 d_q>0 d q > 0
d τ ( c q ) ≤ M q d τ ( ( χ n ( x l ) − d q ) + ) eventually, for every τ . d_\tau(c_q)\leq M_qd_\tau((\chi_n(x_l)-d_q)_+)\quad\text{eventually, for every }\tau. d τ ( c q ) ≤ M q d τ (( χ n ( x l ) − d q ) + ) eventually, for every τ . Here is the cut detail. Write [ c q ] ≤ [ ( z − γ ) + ] [c_q]\leq[(z-\gamma)_+] [ c q ] ≤ [( z − γ ) + ] z ∈ Q ( U x l ) + z\in Q(Ux_l)_+ z ∈ Q ( U x l ) + ( z − γ / 2 ) + (z-\gamma/2)_+ ( z − γ /2 ) + χ ( x l ) \chi(x_l) χ ( x l ) χ ( x l ) \chi(x_l) χ ( x l ) ( z − γ ) + (z-\gamma)_+ ( z − γ ) + (4.8) at every late coordinate. Both its multiplicity and target tolerance depend only on the request, not on the coordinate.
For an upper test, choose a strictly positive h i ∈ Q ( V i ) h_i\in Q(V_i) h i ∈ Q ( V i ) χ ( a i ) ∈ Ideal D ( Q ( V i ) ) = Ideal D ( h i ) \chi(a_i)\in\operatorname{Ideal}_{D}(Q(V_i))=\operatorname{Ideal}_{D}(h_i) χ ( a i ) ∈ Ideal D ( Q ( V i )) = Ideal D ( h i ) χ ( a i ) \chi(a_i) χ ( a i ) N i N_i N i κ i > 0 \kappa_i>0 κ i > 0
[ ( χ ( a i ) − ε i / 2 ) + ] ≤ N i [ ( h i − κ i ) + ] . [(\chi(a_i)-\varepsilon_i/2)_+]\leq N_i[(h_i-\kappa_i)_+]. [( χ ( a i ) − ε i /2 ) + ] ≤ N i [( h i − κ i ) + ] . Let c i ∈ Q ( V i ) + c_i\in Q(V_i)_+ c i ∈ Q ( V i ) + N i N_i N i Q ( V i ) Q(V_i) Q ( V i ) [ c i ] ≤ L i [ b i ] [c_i]\leq L_i[b_i] [ c i ] ≤ L i [ b i ] L i L_i L i
d τ ( ( χ n ( a i ) − ε i ) + ) ≤ L i d τ ( b i ) eventually, for every τ . (36) d_\tau((\chi_n(a_i)-\varepsilon_i)_+)\leq L_i d_\tau(b_i)\quad\text{eventually, for every }\tau. \tag*{(36)} d τ (( χ n ( a i ) − ε i ) + ) ≤ L i d τ ( b i ) eventually, for every τ . (36) Thus both the controller b i b_i b i
At stage j j j t j t_j t j
t j ≤ j − 1 , t j L i ≤ j − 1 ( i ≤ j ) . t_j\leq j^{-1},\qquad t_jL_i\leq j^{-1}\quad(i\leq j). t j ≤ j − 1 , t j L i ≤ j − 1 ( i ≤ j ) . Only after choosing t j t_j t j N j N_j N j j j j
M q r q ( n ) ≤ t j ( q ≤ j , n ≥ N j ) . M_qr_q(n)\leq t_j\quad(q\leq j,n\geq N_j). M q r q ( n ) ≤ t j ( q ≤ j , n ≥ N j ) . For N j ≤ n < N j + 1 N_j\leq n<N_{j+1} N j ≤ n < N j + 1 p n p_n p n t j t_j t j
θ n ( a ) = χ n ( a ) ⊗ p n . \theta_n(a)=\chi_n(a)\otimes p_n. θ n ( a ) = χ n ( a ) ⊗ p n . After the fixed trace-preserving identifications these are c.p.c. maps into Q Q Q χ n \chi_n χ n θ : S → D \theta:S\to D θ : S → D (36) gives (33) with s i , n ≤ j − 1 s_{i,n}\leq j^{-1} s i , n ≤ j − 1 j j j
d τ ( A q , n ) ≤ r q ( n ) d τ ( c q ) ≤ M q r q ( n ) d τ ( ( χ n ( x l ) − d q ) + ) ≤ d τ ( ( θ n ( x l ) − d q ) + ) . (37) \begin{aligned}
d_\tau(A_{q,n})&\leq r_q(n)d_\tau(c_q)\\
&\leq M_qr_q(n)d_\tau((\chi_n(x_l)-d_q)_+)\\
&\leq d_\tau((\theta_n(x_l)-d_q)_+). \tag*{(37)}
\end{aligned} d τ ( A q , n ) ≤ r q ( n ) d τ ( c q ) ≤ M q r q ( n ) d τ (( χ n ( x l ) − d q ) + ) ≤ d τ (( θ n ( x l ) − d q ) + ) . (37) All inequalities use extended nonnegative arithmetic; the scalar factors are strictly positive. Pointwise order in Cu ( Q ) \operatorname{Cu}(Q) Cu ( Q ) (35) to the target ( θ n ( x l ) − d q / 2 ) + (\theta_n(x_l)-d_q/2)_+ ( θ n ( x l ) − d q /2 ) + d q / 2 d_q/2 d q /2 D D D (31) for the enumerated tolerances, and hence for every tolerance.
It remains to check (30) . If y ∈ S ( U x ) + y \in S(U_x)_+ y ∈ S ( U x ) + χ ( y ) ∈ Ideal D ( χ ( x ) ) \chi(y) \in\operatorname{Ideal}_D(\chi(x)) χ ( y ) ∈ Ideal D ( χ ( x )) χ ( x ) \chi(x) χ ( x ) p n p_n p n (35) give ( θ ( y ) − ε ) + ∈ Ideal D ( θ ( x ) ) (\theta(y)-\varepsilon)_+ \in\operatorname{Ideal}_D(\theta(x)) ( θ ( y ) − ε ) + ∈ Ideal D ( θ ( x )) ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 (30) . Finally N t j ≤ 1 Nt_j \leq1 N t j ≤ 1 N N N N N N
Cone blocks with a fixed lower signal
Without a model to cut down, we must first build finite matrix maps whose values detect each requested controller. The construction has three ingredients: a fixed quotient-norm threshold for the controlling input, finitely many positive support elements with plateaux, and finite-dimensional approximations of the corresponding quotient cones. The plateaux convert a matrix norm lower bound into a Cuntz comparison at the fixed threshold. Stage-dependent matrix sizes will change only the rank multiplicities.
We use the following finite approximation form of cone quasidiagonality: for a separable C ∗ C^* C ∗ R 0 R_0 R 0 C 0 ( ( 0 , 1 ] ) ⊗ R 0 C_0((0,1]) \otimes R_0 C 0 (( 0 , 1 ]) ⊗ R 0 [35 ] ; a direct proof in precisely this form is [33 ] , Proposition 33.
The ideal-covering strategy in the finite cone stage is inspired by Bosa, Gabe, Sims, and White; compare [2 ] , Lemma 3.5. Their target is O ∞ \mathcal{O}_\infty O ∞
Lemma 4.2 (A finite cone stage). Let S = C 0 ( ( 0 , 1 ] ) ⊗ P S = C_0((0,1]) \otimes P S = C 0 (( 0 , 1 ]) ⊗ P (32) , finitely many lower requests (34) , and a finite multiplicativity test set. For each lower request one can fix a number h q > 0 h_q > 0 h q > 0 Φ : S → M k ( Q ) \Phi: S \to M_k(Q) Φ : S → M k ( Q ) L i L_i L i M q ≥ 1 M_q \geq1 M q ≥ 1
d τ ( Φ ( a i ) ) ≤ L i d τ ( b i ) , d τ ( c q ) ≤ M q d τ ( ( Φ ( x l ) − 4 h q ) + ) ( q = ( l , η ) ) , \begin{aligned}
d_\tau\bigl(\Phi(a_i)\bigr) &\leq L_i d_\tau(b_i),\\
d_\tau(c_q) &\leq M_q d_\tau\bigl((\Phi(x_l)-4h_q)_+\bigr) \qquad(q=(l,\eta)),
\end{aligned} d τ ( Φ ( a i ) ) d τ ( c q ) ≤ L i d τ ( b i ) , ≤ M q d τ ( ( Φ ( x l ) − 4 h q ) + ) ( q = ( l , η )) , for every τ ∈ T ( Q ) \tau\in T(Q) τ ∈ T ( Q ) τ \tau τ
Proof. We first choose the lower threshold, once for each request. Write
[ c q ] ≤ [ ( z q − γ q ) + ] , z q ∈ Q ( U x l ) + , γ q > 0. [c_q] \leq[(z_q-\gamma_q)_+], \qquad z_q \in Q(U_{x_l})_+, \quad\gamma_q>0. [ c q ] ≤ [( z q − γ q ) + ] , z q ∈ Q ( U x l ) + , γ q > 0. For z ∈ X z \in X z ∈ X ( x l ) z (x_l)_z ( x l ) z C 0 ( ( 0 , 1 ] ) ⊗ ( P / P z ) C_0((0,1]) \otimes(P/P_z) C 0 (( 0 , 1 ]) ⊗ ( P / P z ) P z P_z P z z z z
O q ( h ) = { z : ∥ ( x l ) z ∥ > 8 h } O_q(h)=\{z:\|(x_l)_z\|>8h\} O q ( h ) = { z : ∥ ( x l ) z ∥ > 8 h } are open and increase to U x l U_{x_l} U x l h ↓ 0 h \downarrow0 h ↓ 0 h q > 0 h_q>0 h q > 0
z q ′ = ( z q − γ q / 2 ) + ∈ Q ( O q ( h q ) ) , [ c q ] ≪ [ z q ′ ] . z'_q=(z_q-\gamma_q/2)_+ \in Q(O_q(h_q)), \qquad[c_q]\ll[z'_q]. z q ′ = ( z q − γ q /2 ) + ∈ Q ( O q ( h q )) , [ c q ] ≪ [ z q ′ ] . Keep h q h_q h q
For a point z ∈ O q ( h q ) z \in O_q(h_q) z ∈ O q ( h q )
N z ⊆ O q ( h q ) ∩ ⋂ { i : ( a i ) z ≠ 0 } supp X ( b i ) . N_z \subseteq O_q(h_q) \cap\bigcap_{\{i:(a_i)_z\ne0\}} \operatorname{supp}_X(b_i). N z ⊆ O q ( h q ) ∩ { i : ( a i ) z = 0 } ⋂ supp X ( b i ) . It exists because ( a i ) z ≠ 0 (a_i)_z \ne0 ( a i ) z = 0 z ∈ V i z \in V_i z ∈ V i supp X ( b i ) \operatorname{supp}_X(b_i) supp X ( b i ) V i V_i V i Q ( N z ) Q(N_z) Q ( N z ) z z z h z , k z h_z,k_z h z , k z
h z k z = k z , ( k z ) z ≠ 0 , h z compactly Cuntz-supported in Q ( N z ) . h_zk_z=k_z,\qquad(k_z)_z\ne0,\qquad h_z\text{ compactly Cuntz-supported in }Q(N_z). h z k z = k z , ( k z ) z = 0 , h z compactly Cuntz-supported in Q ( N z ) . For example, first take a nonzero positive cut, make h z h_z h z k z k_z k z k z k_z k z O q ( h q ) O_q(h_q) O q ( h q ) (4.13) , finitely many of them suffice for
[ c q ] ≤ M q ∑ t = 1 ℓ q [ k q t ] (38) [c_q]\le M_q\sum_{t=1}^{\ell_q}[k_{qt}] \tag*{(38)} [ c q ] ≤ M q t = 1 ∑ ℓ q [ k q t ] (38) for some integer M q M_q M q k z k_z k z z q ′ z'_q z q ′ [ c q ] ≪ [ z q ′ ] [c_q]\ll[z'_q] [ c q ] ≪ [ z q ′ ] O q ( h q ) O_q(h_q) O q ( h q )
For each selected center z z z S z = C 0 ( ( 0 , 1 ] ) ⊗ ( P / P z ) S_z=C_0((0,1])\otimes(P/P_z) S z = C 0 (( 0 , 1 ]) ⊗ ( P / P z ) α t \alpha_t α t 0 ≤ t ≤ 1 0\le t\le1 0 ≤ t ≤ 1
( α t f ) ( r ) = f ( t r ) , α 0 = 0. (\alpha_t f)(r)=f(tr),\qquad\alpha_0=0. ( α t f ) ( r ) = f ( t r ) , α 0 = 0. The scaled images of a finite test set form norm-compact families. Choose finite nets in those families and their products. Cone quasidiagonality, followed by contractivity to pass from the nets to the families, gives a c.p.c. map ϕ z : S z → M d z \phi_z:S_z\to M_{d_z} ϕ z : S z → M d z
∥ ϕ z ( ( x l ) z ) ∥ > 7 h q . \|\phi_z((x_l)_z)\|>7h_q. ∥ ϕ z (( x l ) z ) ∥ > 7 h q . Define a matrix-valued cone function by
t ⟼ ϕ z ( α t ( s z ) ) . t\longmapsto\phi_z(\alpha_t(s_z)). t ⟼ ϕ z ( α t ( s z )) . It is norm-continuous and zero at t = 0 t=0 t = 0 h z h_z h z ψ z : S → M d z ( Q ) \psi_z:S\to M_{d_z}(Q) ψ z : S → M d z ( Q ) M d z ( Her ( h z ) ) M_{d_z}(\operatorname{Her}(h_z)) M d z ( Her ( h z ))
For the lower estimate, choose a unit vector v ∈ C d z v\in\mathbb{C}^{d_z} v ∈ C d z ϕ z ( ( x l ) z ) \phi_z((x_l)_z) ϕ z (( x l ) z ) λ > 7 h q \lambda>7h_q λ > 7 h q (4.14) implies that evaluation on k z 1 / 2 k_z^{1/2} k z 1/2
w = ( λ − 4 h q ) − 1 / 2 v ⊗ k z 1 / 2 w=(\lambda-4h_q)^{-1/2}v\otimes k_z^{1/2} w = ( λ − 4 h q ) − 1/2 v ⊗ k z 1/2 satisfies
w ∗ ( ψ z ( x l ) − 4 h q ) + w = k z , ∥ w ∥ ≤ ( 3 h q ) − 1 / 2 . (39) w^*(\psi_z(x_l)-4h_q)_+w=k_z,\qquad\|w\|\le(3h_q)^{-1/2}. \tag*{(39)} w ∗ ( ψ z ( x l ) − 4 h q ) + w = k z , ∥ w ∥ ≤ ( 3 h q ) − 1/2 . (39) Combining these comparisons with (38) gives the second inequality of (4.12) for the direct sum of all the selected blocks.
For an upper test, a block is zero if ( a i ) z = 0 (a_i)_z=0 ( a i ) z = 0 h z h_z h z Ideal Q ( b i ) \operatorname{Ideal}_Q(b_i) Ideal Q ( b i ) b i b_i b i 1 d z ⊗ h z 1_{d_z}\otimes h_z 1 d z ⊗ h z d z d τ ( h z ) d_zd_\tau(h_z) d z d τ ( h z ) (4.12) . The direct sum is contractive, since its norm is the maximum of the block norms. Its internal matrix trace is the sum of the block traces. This proves the lemma.
The finite stage now has a lower spectral cut independent of the accuracy. To turn these maps into a locally small homomorphism, we choose their UHF fractions next and delay the coordinates until the prescribed errors are smaller than those fractions.
Proof of Theorem 4.1(ii). At stage j j j j j j j − 1 j^{-1} j − 1 Φ j \Phi_j Φ j L i j L_{ij} L ij M q j M_{qj} M q j t j t_j t j
t j ≤ j − 1 , t j L i j ≤ j − 1 ( i ≤ j ) , β q j : = t j / M q j > 0. (40) t_j \le j^{-1}, \qquad t_j L_{ij} \le j^{-1}\quad(i \le j), \qquad\beta_{qj} := t_j/M_{qj} > 0. \tag*{(40)} t j ≤ j − 1 , t j L ij ≤ j − 1 ( i ≤ j ) , β q j := t j / M q j > 0. (40) After all the stage maps and fractions have been chosen, choose increasing coordinate thresholds N j N_j N j
r q ( n ) ≤ β q j ( q ≤ j , n ≥ N j ) . (41) r_q(n) \le\beta_{qj}\quad(q \le j, n \ge N_j). \tag*{(41)} r q ( n ) ≤ β q j ( q ≤ j , n ≥ N j ) . (41) On N j ≤ n < N j + 1 N_j \le n < N_{j+1} N j ≤ n < N j + 1 Φ j \Phi_j Φ j t j t_j t j θ n : S → Q \theta_n:S \to Q θ n : S → Q θ \theta θ
For a fixed upper test, (4.12) and (4.18) give (4.5) eventually with bound j − 1 j^{-1} j − 1 θ ( S ( U ) ) ⊆ J ( U ) \theta(S(U)) \subseteq J(U) θ ( S ( U )) ⊆ J ( U ) q = ( l , η ) q=(l,\eta) q = ( l , η )
d τ ( A q , n ) ≤ r q ( n ) d τ ( c q ) ≤ d τ ( ( θ n ( x l ) − 4 h q ) + ) (42) d_\tau(A_{q,n}) \le r_q(n)d_\tau(c_q) \le d_\tau\bigl((\theta_n(x_l)-4h_q)_+\bigr) \tag*{(42)} d τ ( A q , n ) ≤ r q ( n ) d τ ( c q ) ≤ d τ ( ( θ n ( x l ) − 4 h q ) + ) (42) eventually at every coordinate. Pointwise trace order in Cu( Q ) (Q) ( Q ) B q , n = ( θ n ( x l ) − 2 h q ) + B_{q,n}=(\theta_n(x_l)-2h_q)_+ B q , n = ( θ n ( x l ) − 2 h q ) + n − 1 n^{-1} n − 1 2 h q 2h_q 2 h q
sup n ∥ A q , n ∥ + 1 2 h q , \frac{\sup_n \|A_{q,n}\|+1}{2h_q}, 2 h q sup n ∥ A q , n ∥ + 1 , independently of the stage, its centers, and its matrix size. They prove
[ ( ∣ e l ∣ − η ) + ] ≤ [ ( θ ( x l ) − 2 h q ) + ] in Cu ( D ) . [(|e_l|-\eta)_+] \le[(\theta(x_l)-2h_q)_+] \quad\text{in Cu}(D). [( ∣ e l ∣ − η ) + ] ≤ [( θ ( x l ) − 2 h q ) + ] in Cu ( D ) . This establishes (4.3) and then (4.1). A very small β q j \beta_{qj} β q j N j N_j N j h q h_q h q
Coefficient families
Corollary 4.3 (Simultaneous coefficient domination). Suppose that one of the hypotheses of Theorem 4.1 holds. Let E j E_j E j
E j : S × Z j , 1 × ⋯ × Z j , r j ⟶ D , E_j:S \times Z_{j,1} \times\cdots\times Z_{j,r_j} \longrightarrow D, E j : S × Z j , 1 × ⋯ × Z j , r j ⟶ D , where the parameter spaces are separable normed spaces. Assume that
E j ( S ( U ) , Z j , 1 , … , Z j , r j ) ⊆ J ( U ) for every open U . E_j(S(U),Z_{j,1},\ldots,Z_{j,r_j}) \subseteq J(U) \quad\text{for every open } U. E j ( S ( U ) , Z j , 1 , … , Z j , r j ) ⊆ J ( U ) for every open U . Then the stabilizer may also be chosen so that, for every x ∈ S + x \in S_+ x ∈ S +
E j ( S ( U x ) , Z j , 1 , … , Z j , r j ) ⊆ Ideal D ( θ ( x ) ) . (43) E_j(S(U_x),Z_{j,1},\ldots,Z_{j,r_j}) \subseteq\operatorname{Ideal}_D(\theta(x)). \tag*{(43)} E j ( S ( U x ) , Z j , 1 , … , Z j , r j ) ⊆ Ideal D ( θ ( x )) . (43) The same conclusion holds for norm-continuous separably parametrized families of such maps, with local norm bounds, and for any choice of the localized multilinear input.
Proof. Choose a countable family B ⊆ S + \mathcal{B} \subseteq S_{+} B ⊆ S + x ∈ S + x \in S_{+} x ∈ S + B ∩ Ideal S ( x ) \mathcal{B} \cap\operatorname{Ideal}_{S}(x) B ∩ Ideal S ( x ) Ideal S ( x ) \operatorname{Ideal}_{S}(x) Ideal S ( x )
U x = ⋃ y ∈ B ∩ Ideal S ( x ) U y . (44) U_{x} = \bigcup_{y \in\mathcal{B} \cap\operatorname{Ideal}_{S}(x)} U_{y}. \tag*{(44)} U x = y ∈ B ∩ Ideal S ( x ) ⋃ U y . (44) For each y ∈ B y \in\mathcal{B} y ∈ B E j ( a , z 1 , … , z r j ) E_{j}(a,z_{1},\ldots,z_{r_{j}}) E j ( a , z 1 , … , z r j ) a a a S ( U y ) S(U_{y}) S ( U y ) y y y thm:4.1 applies. Continuity gives the desired containment for every input in S ( U y ) S(U_{y}) S ( U y )
Now fix x x x y ∈ B ∩ Ideal S ( x ) y \in\mathcal{B} \cap\operatorname{Ideal}_{S}(x) y ∈ B ∩ Ideal S ( x ) θ ( y ) ∈ Ideal D ( θ ( x ) ) \theta(y) \in\operatorname{Ideal}_{D}(\theta(x)) θ ( y ) ∈ Ideal D ( θ ( x )) (44) , S ( U x ) S(U_{x}) S ( U x ) S ( U y ) S(U_{y}) S ( U y ) (43) . Dense parameter sets and local norm continuity handle the asserted continuous families. Notice that the argument uses y ∈ Ideal S ( x ) y \in\operatorname{Ideal}_{S}(x) y ∈ Ideal S ( x ) U y ⊆ U x U_{y} \subseteq U_{x} U y ⊆ U x
Remark 4.4. Two forms of cor:4.3 will be used. If I I I d : S → D d:S \to D d : S → D d ( S ( U ) ) I ⊆ J ( U ) d(S(U))I \subseteq J(U) d ( S ( U )) I ⊆ J ( U ) E ( a , b ) = d ( a ) b E(a,b)=d(a)b E ( a , b ) = d ( a ) b b ∈ I b \in I b ∈ I
Alternatively, let H ⊆ D H \subseteq D H ⊆ D σ \sigma σ d : S → M ( H ) d:S \to M(H) d : S → M ( H ) J ( U ) J(U) J ( U ) ( u m ) (u_{m}) ( u m ) H H H u m d ( a ) u m u_{m}d(a)u_{m} u m d ( a ) u m h , k ∈ H h,k \in H h , k ∈ H
h d ( a ) k = lim m → ∞ h u m d ( a ) u m k . hd(a)k = \lim_{m \to\infty} h u_{m}d(a)u_{m} k. h d ( a ) k = m → ∞ lim h u m d ( a ) u m k . Thus all multiplier coefficients are controlled. The countability needed here comes from σ \sigma σ H H H
Norm uniqueness of sequence models
We prove uniqueness for the models of Definition 3.2, allowing any of the spaces Y Y Y m m m
S = S Y = C 0 ( Y ) ⊗ P , S ( U ) = C 0 ( Y ) ⊗ P ( U ) , U ⊆ X open . S = S_{Y} = C_{0}(Y) \otimes P,\qquad S(U) = C_{0}(Y) \otimes P(U),\qquad U \subseteq X\ \text{open}. S = S Y = C 0 ( Y ) ⊗ P , S ( U ) = C 0 ( Y ) ⊗ P ( U ) , U ⊆ X open . The same statements apply if the prescribed measure is multiplied by a strictly positive constant. A coordinate unitary will always mean a unitary in M ( Q ) M(Q) M ( Q )
Theorem 5.1 (Norm uniqueness). Let p , q : S Y → Q ∞ p,q:S_{Y} \to Q_{\infty} p , q : S Y → Q ∞ p n , q n : S Y → Q p_{n},q_{n}:S_{Y} \to Q p n , q n : S Y → Q u n ∈ U ( M ( Q ) ) u_{n} \in\mathcal{U}(M(Q)) u n ∈ U ( M ( Q ))
lim n → ∞ ∥ u n p n ( a ) u n ∗ − q n ( a ) ∥ = 0 , a ∈ S Y . \lim_{n \to\infty} \left\|u_{n}p_{n}(a)u_{n}^{*}-q_{n}(a)\right\|=0,\qquad a \in S_{Y}. n → ∞ lim ∥ u n p n ( a ) u n ∗ − q n ( a ) ∥ = 0 , a ∈ S Y . There are two steps. Equal trace data first give, after conjugacy and rational averaging, a difference in every local error ideal J ( U ) J(U) J ( U )
Compact sets of weights and tracial alignment
Lemma 5.2 (Controlled trace evaluations). Let C C C C ∗ C^* C ∗ b ∈ ( C ⊗ K ) + b \in(C \otimes\mathcal{K})_+ b ∈ ( C ⊗ K ) +
K b = { γ ∈ T ( C ) : d γ ( b ) ≤ 1 } K_b=\{\gamma\in T(C):d_\gamma(b)\leq1\} K b = { γ ∈ T ( C ) : d γ ( b ) ≤ 1 } is compact and convex. If x ∈ ( C ⊗ K ) + x\in(C\otimes\mathcal{K})_+ x ∈ ( C ⊗ K ) + sup γ ∈ K b d γ ( x ) ≤ L < ∞ \sup_{\gamma\in K_b}d_\gamma(x)\leq L<\infty sup γ ∈ K b d γ ( x ) ≤ L < ∞ γ ↦ γ ( x ) \gamma\mapsto\gamma(x) γ ↦ γ ( x ) K b K_b K b L L L x x x
Consequently this applies separately to all the first and square evaluations of a fixed completely positive candidate whose tested values have bounded rank on K b K_b K b
γ ⟼ γ ( ∣ u x 1 u ∗ − x 2 ∣ 2 ) , u ∈ U ( C ∼ ) , \gamma\longmapsto\gamma(|ux_1u^*-x_2|^2),\qquad u\in\mathcal{U}(C^\sim), γ ⟼ γ ( ∣ u x 1 u ∗ − x 2 ∣ 2 ) , u ∈ U ( C ∼ ) , whenever the positive elements x 1 , x 2 x_1,x_2 x 1 , x 2 u u u
Proof. Rank evaluation is lower semicontinuous on the compact trace cone, so K b K_b K b η > 0 \eta>0 η > 0
0 ≤ γ ( x ) − γ ( ( x − η ) + ) ≤ η L , γ ∈ K b . (45) 0\leq\gamma(x)-\gamma((x-\eta)_+)\leq\eta L,\qquad\gamma\in K_b. \tag*{(45)} 0 ≤ γ ( x ) − γ (( x − η ) + ) ≤ η L , γ ∈ K b . (45) and γ ( x ) ≤ ∥ x ∥ L \gamma(x)\leq\lVert x\rVert L γ ( x ) ≤ ∥ x ∥ L γ i → γ \gamma_i\to\gamma γ i → γ K b K_b K b
γ ( x ) ≤ lim inf i γ i ( x ) ≤ lim sup i γ i ( x ) ≤ γ ( x ) + η L . \gamma(x)\leq\liminf_i\gamma_i(x)\leq\limsup_i\gamma_i(x)\leq\gamma(x)+\eta L. γ ( x ) ≤ i lim inf γ i ( x ) ≤ i lim sup γ i ( x ) ≤ γ ( x ) + η L . Letting η ↓ 0 \eta\downarrow0 η ↓ 0
d γ ( ∣ u x 1 u ∗ − x 2 ∣ 2 ) ≤ d γ ( x 1 ) + d γ ( x 2 ) , d_\gamma(|ux_1u^*-x_2|^2)\leq d_\gamma(x_1)+d_\gamma(x_2), d γ ( ∣ u x 1 u ∗ − x 2 ∣ 2 ) ≤ d γ ( x 1 ) + d γ ( x 2 ) , which proves the last assertion.
Lemma 5.3 (Barycenters on a trace cap). Let K b K_b K b μ \mu μ K b K_b K b γ ˉ ∈ K b \bar{\gamma}\in K_b γ ˉ ∈ K b
v ( γ ˉ ) = ∫ K b v d μ (46) v(\bar{\gamma})=\int_{K_b}v\,d\mu \tag*{(46)} v ( γ ˉ ) = ∫ K b v d μ (46) for every finite continuous affine real function v v v K b K_b K b M > 0 M>0 M > 0 M v ( γ ˉ ) Mv(\bar{\gamma}) M v ( γ ˉ ) γ ˉ \bar{\gamma} γ ˉ μ / M \mu/M μ / M
Proof. Approximate μ \mu μ μ j = ∑ k t j k δ γ j k \mu_j=\sum_k t_{jk}\delta_{\gamma_{jk}} μ j = ∑ k t j k δ γ j k β j = ∑ k t j k γ j k \beta_j=\sum_k t_{jk}\gamma_{jk} β j = ∑ k t j k γ j k K b K_b K b K b K_b K b γ ˉ \bar{\gamma} γ ˉ v v v
v ( γ ˉ ) = lim j v ( β j ) = lim j ∫ v d μ j = ∫ v d μ . v(\bar{\gamma})=\lim_jv(\beta_j)=\lim_j\int v\,d\mu_j=\int v\,d\mu. v ( γ ˉ ) = j lim v ( β j ) = j lim ∫ v d μ j = ∫ v d μ . The same subnet works for all these functions. Neither uniqueness of γ ˉ \bar{\gamma} γ ˉ
Lemma 5.4 (Alignment for finitely many weights). Let C C C S S S p , q : S → C p,q:S\to C p , q : S → C T ( C ) T(C) T ( C ) ( s i , γ i ) (s_i,\gamma_i) ( s i , γ i ) s i ∈ S + s_i\in S_+ s i ∈ S +
d γ i ( p ( s i ) ) + d γ i ( q ( s i ) ) < ∞ , d_{\gamma_i}(p(s_i))+d_{\gamma_i}(q(s_i))<\infty, d γ i ( p ( s i )) + d γ i ( q ( s i )) < ∞ , and any δ > 0 \delta>0 δ > 0 u ∈ U ( M 2 ( C ) ∼ ) u\in\mathcal{U}(M_2(C)^{\sim}) u ∈ U ( M 2 ( C ) ∼ )
( γ i ⊗ Tr 2 ) ( ∣ u ( p ( s i ) ⊕ 0 ) u ∗ − ( q ( s i ) ⊕ 0 ) ∣ 2 ) < δ for every i . (47) (\gamma_i\otimes\operatorname{Tr}_2)\left(\left|u(p(s_i)\oplus0)u^*-(q(s_i)\oplus0)\right|^2\right)<\delta\quad\text{for every }i. \tag*{(47)} ( γ i ⊗ Tr 2 ) ( ∣ u ( p ( s i ) ⊕ 0 ) u ∗ − ( q ( s i ) ⊕ 0 ) ∣ 2 ) < δ for every i . (47) The added matrix direction is internal and has unnormalized trace.
Proof. The equality of pullback weights implies equality of all their ranks. By [7 ] , Lemma 5.1, the support projections of p ( s ) p(s) p ( s ) q ( s ) q(s) q ( s ) C ∗ ∗ C^{**} C ∗∗ s ∈ S + s\in S_+ s ∈ S + C C C
The finite-rank assumption puts p ( s i ) p(s_i) p ( s i ) q ( s i ) q(s_i) q ( s i ) Fin ( γ i ) \operatorname{Fin}(\gamma_i) Fin ( γ i ) M 2 ( Fin ( γ i ) ) M_2(\operatorname{Fin}(\gamma_i)) M 2 ( Fin ( γ i )) M 2 ( C ) ∼ M_2(C)^{\sim} M 2 ( C ) ∼ π i \pi_i π i C C C Tr i \operatorname{Tr}_i Tr i M i = π i ( C ) ′ ′ M_i=\pi_i(C)^{\prime\prime} M i = π i ( C ) ′′ π = π 0 ⊕ π 1 ⊕ ⋯ ⊕ π r \pi=\pi_0\oplus\pi_1\oplus\cdots\oplus\pi_r π = π 0 ⊕ π 1 ⊕ ⋯ ⊕ π r π 0 \pi_0 π 0 M = π ( C ) ′ ′ M=\pi(C)^{\prime\prime} M = π ( C ) ′′ M M M i i i q i : M → M i q_i:M\to M_i q i : M → M i M ( 1 − z i ) M(1-z_i) M ( 1 − z i ) z i z_i z i q i q_i q i M z i Mz_i M z i M i M_i M i
γ ^ i = Tr i ∘ q i . \widehat{\gamma}_i=\operatorname{Tr}_i\circ q_i. γ i = Tr i ∘ q i . This is the transported normal semifinite trace on M z i Mz_i M z i M ( 1 − z i ) M(1-z_i) M ( 1 − z i ) z i z_i z i γ i \gamma_i γ i γ ^ i π = γ i \widehat{\gamma}_i\pi=\gamma_i γ i π = γ i
We apply von Neumann uniqueness only after a common zero padding and unitization. Let S † S^\dagger S † M 2 ( M ) M_2(M) M 2 ( M )
p ^ ( a + λ 1 ) = ( π p ( a ) ⊕ 0 ) + λ 1 M 2 ( M ) , q ^ ( a + λ 1 ) = ( π q ( a ) ⊕ 0 ) + λ 1 M 2 ( M ) . \widehat{p}(a+\lambda1)=(\pi p(a)\oplus0)+\lambda1_{M_2(M)},\qquad
\widehat{q}(a+\lambda1)=(\pi q(a)\oplus0)+\lambda1_{M_2(M)}. p ( a + λ 1 ) = ( π p ( a ) ⊕ 0 ) + λ 1 M 2 ( M ) , q ( a + λ 1 ) = ( π q ( a ) ⊕ 0 ) + λ 1 M 2 ( M ) . These maps have equal support ranks on every positive element of S † S^\dagger S † M M M M 2 ( M ) M_2(M) M 2 ( M ) π p \pi p π p π q \pi q π q C C C
The source S † S^\dagger S † [7 ] , Proposition 2.1, Corollary 3.10, and Section 4. For completeness, the properly infinite factor argument needs only the partial-isometry approximation of [7 ] , Lemma 3.6 and the following unital completion. Including 1 1 1 v v v e = v ∗ v e=v^*v e = v ∗ v 1 1 1 e 0 ≤ e e_0\leq e e 0 ≤ e e e e 1 − e 0 ∼ 1 − v e 0 v ∗ ∼ 1 1-e_0\sim1-ve_0v^*\sim1 1 − e 0 ∼ 1 − v e 0 v ∗ ∼ 1 e 0 = e e_0=e e 0 = e e e e v e 0 ve_0 v e 0 u u u ( u − v ) e 0 = 0 (u-v)e_0=0 ( u − v ) e 0 = 0
∣ ρ ( u ∗ x u − v ∗ x v ) ∣ ≤ 4 ∥ x ∥ ρ ( 1 − e 0 ) 1 / 2 . \left|\rho\left(u^*xu-v^*xv\right)\right| \le4\lVert x\rVert\rho(1-e_0)^{1/2}. ∣ ρ ( u ∗ xu − v ∗ xv ) ∣ ≤ 4 ∥ x ∥ ρ ( 1 − e 0 ) 1/2 . Thus the partial-isometry conjugacies can be replaced by unitary conjugacies in the weak star topology; [7 ] makes them strong star. Section 4 then assembles these unital factor conclusions by central decomposition; separability lets one impose the support comparisons on one countable dense set of positive cuts before selecting the factorwise unitaries. We obtain w j ∈ M 2 ( M ) w_j\in M_2(M) w j ∈ M 2 ( M )
w j ( π p ( a ) ⊕ 0 ) w j ∗ ⟶ π q ( a ) ⊕ 0 strong star , a ∈ S . w_j\left(\pi p(a)\oplus0\right)w_j^* \longrightarrow\pi q(a)\oplus0 \quad\text{strong star}, \qquad a\in S. w j ( π p ( a ) ⊕ 0 ) w j ∗ ⟶ π q ( a ) ⊕ 0 strong star , a ∈ S . The unital extensions are needed here: support ranks of nonunital maps alone do not compare the complements of their support projections.
For clarity, the passage to L 2 L^2 L 2 x j = w j ( π p ( s i ) ⊕ 0 ) w j ∗ x_j=w_j(\pi p(s_i)\oplus0)w_j^* x j = w j ( π p ( s i ) ⊕ 0 ) w j ∗ x = π q ( s i ) ⊕ 0 x=\pi q(s_i)\oplus0 x = π q ( s i ) ⊕ 0 T i = γ i ^ ⊗ Tr 2 T_i=\widehat{\gamma_i}\otimes\operatorname{Tr}_2 T i = γ i ⊗ Tr 2 T i T_i T i
∥ x j ∥ 2 , T i 2 = γ i ( p ( s i 2 ) ) = γ i ( q ( s i 2 ) ) = ∥ x ∥ 2 , T i 2 . \lVert x_j\rVert_{2,T_i}^2=\gamma_i(p(s_i^2))=\gamma_i(q(s_i^2))=\lVert x\rVert_{2,T_i}^2. ∥ x j ∥ 2 , T i 2 = γ i ( p ( s i 2 )) = γ i ( q ( s i 2 )) = ∥ x ∥ 2 , T i 2 . For a bounded test y ∈ L 1 ( T i ) ∩ L 2 ( T i ) y\in L^1(T_i)\cap L^2(T_i) y ∈ L 1 ( T i ) ∩ L 2 ( T i ) x j − x x_j-x x j − x t ↦ T i ( y ∗ t ) t\mapsto T_i(y^*t) t ↦ T i ( y ∗ t ) L 2 ( T i ) L^2(T_i) L 2 ( T i ) L 2 L^2 L 2 L 2 L^2 L 2 ∥ x j − x ∥ 2 , T i → 0 \lVert x_j-x\rVert_{2,T_i}\to0 ∥ x j − x ∥ 2 , T i → 0
Choose one w j w_j w j M 2 ( π ( C ) ) ∼ M_2(\pi(C))^{\sim} M 2 ( π ( C ) ) ∼ M 2 ( C ) ∼ M_2(C)^{\sim} M 2 ( C ) ∼ w j w_j w j L 2 L^2 L 2 L 2 L^2 L 2 M 2 ( C ) ∼ M_2(C)^{\sim} M 2 ( C ) ∼ (47) .
Lemma 5.5 (Tracial alignment). If p , q : S → D p,q:S\to D p , q : S → D p ~ , q ~ : S → D \widetilde p,\widetilde q:S\to D p , q : S → D
( p ~ − q ~ ) ( S ( U ) ) ⊆ J ( U ) ( U ⊆ X open ) . (48) (\widetilde p-\widetilde q)(S(U))\subseteq J(U) \qquad(U\subseteq X\ \text{open}). \tag*{(48)} ( p − q ) ( S ( U )) ⊆ J ( U ) ( U ⊆ X open ) . (48) Each new model is coordinate approximately unitarily equivalent to its respective original model. The operations may be performed in one changing block while a prescribed spare block is carried along unchanged, apart from its common amplification and folding.
Proof. By Corollary 3.7, choose a separable subalgebra D 0 D_0 D 0 p p p q q q D 0 D_0 D 0
Choose a countable local positive test family s i s_i s i Y Y Y s ˉ i \bar{s}_i s ˉ i κ i > 0 \kappa_i>0 κ i > 0 s i ≾ ( s ˉ i − κ i ) + s_i\precsim(\bar{s}_i-\kappa_i)_+ s i ≾ ( s ˉ i − κ i ) + p ( s i ) ≾ ( p ( s ˉ i ) − κ i ) + p(s_i)\precsim(p(\bar{s}_i)-\kappa_i)_+ p ( s i ) ≾ ( p ( s ˉ i ) − κ i ) + q q q s i s_i s i b i ∈ Q ( U i ) + b_i\in Q(U_i)_+ b i ∈ Q ( U i ) + L i L_i L i
d γ ( p ( s i ) ) + d γ ( q ( s i ) ) ≤ L i d γ ( b i ) , γ ∈ T ( D 0 ) . (49) d_\gamma(p(s_i))+d_\gamma(q(s_i))\le L_i d_\gamma(b_i), \qquad\gamma\in T(D_0). \tag*{(49)} d γ ( p ( s i )) + d γ ( q ( s i )) ≤ L i d γ ( b i ) , γ ∈ T ( D 0 ) . (49) for a finite L i L_i L i D 0 D_0 D 0 b i b_i b i
Fix a finite initial list and δ > 0 \delta> 0 δ > 0 u ∈ U ( M 2 ( D 0 ) ∼ ) u \in\mathcal{U}(M_2(D_0)^{\sim}) u ∈ U ( M 2 ( D 0 ) ∼ )
f u , i ( γ ) = ( γ ⊗ Tr 2 ) ( ∣ u ( p ( s i ) ⊕ 0 ) u ∗ − ( q ( s i ) ⊕ 0 ) ∣ 2 ) , γ ∈ K b i . f_{u,i}(\gamma) = (\gamma\otimes\operatorname{Tr}_2)(\lvert u(p(s_i) \oplus0)u^* - (q(s_i) \oplus0)\rvert^2), \qquad\gamma\in K_{b_i}. f u , i ( γ ) = ( γ ⊗ Tr 2 ) (∣ u ( p ( s i ) ⊕ 0 ) u ∗ − ( q ( s i ) ⊕ 0 ) ∣ 2 ) , γ ∈ K b i . Here Tr 2 \operatorname{Tr}_2 Tr 2 (49) unchanged. Continuity follows from Lemma 5.2. We claim that zero lies in the closed convex hull of the tuples ( f u , i ) i (f_{u,i})_i ( f u , i ) i ⨁ i C ( K b i , R ) \bigoplus_i C(K_{b_i}, \mathbb{R}) ⨁ i C ( K b i , R ) f u , i ≥ 0 f_{u,i} \ge0 f u , i ≥ 0
It follows that a finite convex combination satisfies
sup γ ∈ K b i ∑ k = 1 r t k f u k , i ( γ ) < δ for every tested i . (50) \sup_{\gamma\in K_{b_i}} \sum_{k=1}^{r} t_k f_{u_k,i}(\gamma) < \delta\qquad\text{for every tested } i. \tag*{(50)} γ ∈ K b i sup k = 1 ∑ r t k f u k , i ( γ ) < δ for every tested i . (50) The coefficients may be chosen positive and rational: the finite functions being averaged are bounded, so sufficiently small changes of the coefficients preserve the strict inequality. Write t k = m k / l t_k = m_k/l t k = m k / l B l = M l ( M 2 ( Q ) ) B_l = M_l(M_2(Q)) B l = M l ( M 2 ( Q )) τ ⊗ Tr 2 ⊗ tr l \tau\otimes\operatorname{Tr}_2 \otimes\operatorname{tr}_l τ ⊗ Tr 2 ⊗ tr l Ad ( u k ) ( p ⊕ 0 ) \operatorname{Ad}(u_k)(p \oplus0) Ad ( u k ) ( p ⊕ 0 ) q ⊕ 0 q \oplus0 q ⊕ 0 m k m_k m k k k k D 0 D_0 D 0
ι l : D 0 ⟶ ( B l ) ∞ , x ⟼ ( x ⊕ 0 ) ⊗ 1 l . \iota_l: D_0 \longrightarrow(B_l)_\infty,\qquad x \longmapsto(x \oplus0) \otimes1_l. ι l : D 0 ⟶ ( B l ) ∞ , x ⟼ ( x ⊕ 0 ) ⊗ 1 l . We record explicitly the resulting coordinate estimate. Lift the finitely many unitaries to unitary multiplier sequences, and use c.p.c. representatives of p , q p,q p , q z i , n z_{i,n} z i , n Q Q Q t > 0 t > 0 t > 0
d τ ( ( ∣ z i , n ∣ − t ) + ) ≤ 4 δ t 2 ( d τ ( b i ) ≤ 1 ) (51) d_\tau\bigl((\lvert z_{i,n}\rvert- t)_+\bigr) \le\frac{4\delta}{t^2} \qquad(d_\tau(b_i) \le1) \tag*{(51)} d τ ( (∣ z i , n ∣ − t ) + ) ≤ t 2 4 δ ( d τ ( b i ) ≤ 1 ) (51) for every sufficiently large n n n τ n \tau_n τ n τ n ⊗ Tr 2 ⊗ tr l \tau_n \otimes\operatorname{Tr}_2 \otimes\operatorname{tr}_l τ n ⊗ Tr 2 ⊗ tr l ρ l \rho_l ρ l γ = ρ l ∘ ι l \gamma= \rho_l \circ\iota_l γ = ρ l ∘ ι l D 0 D_0 D 0
d γ ( b i ) = sup η > 0 lim sup n → ∞ d τ n ( ( b i − η ) + ) ≤ 1. d_\gamma(b_i) = \sup_{\eta> 0} \limsup_{n \to\infty} d_{\tau_n}\bigl((b_i - \eta)_+\bigr) \le1. d γ ( b i ) = η > 0 sup n → ∞ lim sup d τ n ( ( b i − η ) + ) ≤ 1. Thus γ ∈ K b i \gamma\in K_{b_i} γ ∈ K b i e k , i = u k ( p ( s i ) ⊕ 0 ) u k ∗ − ( q ( s i ) ⊕ 0 ) ∈ M 2 ( D 0 ) e_{k,i} = u_k(p(s_i) \oplus0)u_k^* - (q(s_i) \oplus0) \in M_2(D_0) e k , i = u k ( p ( s i ) ⊕ 0 ) u k ∗ − ( q ( s i ) ⊕ 0 ) ∈ M 2 ( D 0 ) z i z_i z i
ρ l ( z i ∗ z i ) = ∑ k m k l ( γ ⊗ Tr 2 ) ( e k , i ∗ e k , i ) < δ . (52) \rho_l(z_i^*z_i) = \sum_k \frac{m_k}{l}(\gamma\otimes\operatorname{Tr}_2)(e_{k,i}^*e_{k,i}) < \delta. \tag*{(52)} ρ l ( z i ∗ z i ) = k ∑ l m k ( γ ⊗ Tr 2 ) ( e k , i ∗ e k , i ) < δ . (52) Indeed, restriction to the first internal corner gives γ \gamma γ M 2 ( D 0 ) M_2(D_0) M 2 ( D 0 ) γ ⊗ Tr 2 \gamma\otimes\operatorname{Tr}_2 γ ⊗ Tr 2 a ∈ M 2 ( D 0 ) a \in M_2(D_0) a ∈ M 2 ( D 0 ) ρ l ( a ⊗ e j j ) = ( γ ⊗ Tr 2 ) ( a ) / l \rho_l(a \otimes e_{jj}) = (\gamma\otimes\operatorname{Tr}_2)(a)/l ρ l ( a ⊗ e j j ) = ( γ ⊗ Tr 2 ) ( a ) / l l l l a ⊗ 1 L a \otimes1_{\mathcal{L}} a ⊗ 1 L
τ ( ( ∣ z i , n ∣ 2 − t 2 / 2 ) + ) ≥ t 2 2 d τ ( ( ∣ z i , n ∣ − t ) + ) . \tau\left(\left(\lvert z_{i,n}\rvert^2-t^2/2\right)_+\right) \ge\frac{t^2}{2}d_\tau\left((\lvert z_{i,n}\rvert-t)_+\right). τ ( ( ∣ z i , n ∣ 2 − t 2 /2 ) + ) ≥ 2 t 2 d τ ( (∣ z i , n ∣ − t ) + ) . Regularization and the violations of the absolute estimate therefore give a square trace at least 2 δ 2\delta 2 δ 0 < d τ ( b i ) < ∞ 0<d_\tau(b_i)<\infty 0 < d τ ( b i ) < ∞ τ / d τ ( b i ) \tau/d_\tau(b_i) τ / d τ ( b i ) d τ ( b i ) = 0 d_\tau(b_i)=0 d τ ( b i ) = 0 τ \tau τ d τ ( b i ) = ∞ d_\tau(b_i)=\infty d τ ( b i ) = ∞
d τ ( ( ∣ z i , n ∣ − t ) + ) ≤ 4 δ t 2 d τ ( b i ) for all τ ∈ T ( Q ) . (53) d_\tau\left((\lvert z_{i,n}\rvert-t)_+\right)\le\frac{4\delta}{t^2}d_\tau(b_i)\qquad\text{for all }\tau\in T(Q). \tag*{(53)} d τ ( (∣ z i , n ∣ − t ) + ) ≤ t 2 4 δ d τ ( b i ) for all τ ∈ T ( Q ) . (53) At stage j j j j j j 1 , 1 / 2 , … , 1 / j 1,1/2,\ldots,1/j 1 , 1/2 , … , 1/ j δ j < 2 − j / ( 4 j 2 ) \delta_j<2^{-j}/(4j^2) δ j < 2 − j / ( 4 j 2 ) J ( U i ) J(U_i) J ( U i ) b i b_i b i
For completeness, these operations preserve the asserted relation to the original maps. At each coordinate, a diagonal sum of conjugates of a map is conjugate, by the diagonal of the same unitaries, to its tensor-first copy in the averaging matrix factor. The coordinate folding can be chosen, using Lemma 2.4, to carry that copy within the stage tolerance of the original on every current input. The first-corner clause of Lemma 2.4 removes the common internal zero padding at the same time. Make that tolerance tend to zero during the same diagonal choice. Inner conjugacy preserves every coordinate weight, and vanishing norm changes preserve regularized values by cuts, so the models keep their data. The argument applies to a designated block of a direct sum, amplifying the spare by the same averaging factor. Its unused part can always be placed in a separate stable corner before folding.
Tracial alignment has placed the difference of two models in each local error ideal. To remove that difference in norm, we will dominate its coefficients by a small stabilizer and apply stable uniqueness on a hereditary support. We first state the absorption and compact perturbation results used in that step.
We specify the version of the ideal-related machinery used below. An X X X H H H U ↦ H ( U ) U\mapsto H(U) U ↦ H ( U )
E H = ℓ 2 ( N ) ⊗ H , K ( E H ) = H ⊗ K , K ( E H ) ( U ) = H ( U ) ⊗ K . E_H=\ell^2(\mathbb{N})\otimes H,\qquad\mathcal{K}(E_H)=H\otimes\mathcal{K},\qquad\mathcal{K}(E_H)(U)=H(U)\otimes\mathcal{K}. E H = ℓ 2 ( N ) ⊗ H , K ( E H ) = H ⊗ K , K ( E H ) ( U ) = H ( U ) ⊗ K . A representation ϕ : S → L ( E H ) \phi:S\to\mathcal{L}(E_H) ϕ : S → L ( E H )
⟨ ξ , ϕ ( a ) η ⟩ ∈ H ( U ) ( a ∈ S ( U ) , ξ , η ∈ E H ) . \langle\xi,\phi(a)\eta\rangle\in H(U)\qquad(a\in S(U),\ \xi,\eta\in E_H). ⟨ ξ , ϕ ( a ) η ⟩ ∈ H ( U ) ( a ∈ S ( U ) , ξ , η ∈ E H ) . Equivalently, ϕ ( S ( U ) ) E H ⊂ E H H ( U ) \phi(S(U))E_H\subset E_HH(U) ϕ ( S ( U )) E H ⊂ E H H ( U ) a ↦ ⟨ ξ , ϕ ( a ) ξ ⟩ a\mapsto\langle\xi,\phi(a)\xi\rangle a ↦ ⟨ ξ , ϕ ( a ) ξ ⟩ S S S [5 ] ; other coefficients follow by polarization. Thus weak equivariance here implies weak X X X
Theorem 5.6 (Ideal-related absorption). Let S S S X X X H H H θ : S → H ⊗ K \theta:S\to H\otimes\mathcal{K} θ : S → H ⊗ K X X X
Ideal H ⊗ K ( θ ( a ) ) = H ( U a ) ⊗ K ( a ∈ S + ) . \operatorname{Ideal}_{H\otimes\mathcal{K}}(\theta(a))=H(U_a)\otimes\mathcal{K}\qquad(a\in S_+). Ideal H ⊗ K ( θ ( a )) = H ( U a ) ⊗ K ( a ∈ S + ) . Then its infinite repeat is weakly X X X X X X
This is [16 ] , applied to H ⊗ K H\otimes\mathcal{K} H ⊗ K H H H H H H θ 0 : S → H \theta_0:S\to H θ 0 : S → H θ 0 ⊗ e 11 \theta_0\otimes e_{11} θ 0 ⊗ e 11 diag ( θ 0 , 0 , θ 0 , 0 , … ) \operatorname{diag}(\theta_0,0,\theta_0,0,\ldots) diag ( θ 0 , 0 , θ 0 , 0 , … )
For weakly equivariant representations ϕ , ψ : S → L ( E H ) \phi,\psi:S\to\mathcal{L}(E_H) ϕ , ψ : S → L ( E H ) [ ϕ , ψ ] X [\phi,\psi]_X [ ϕ , ψ ] X
( E H ⊕ E H , ( ϕ 0 0 ψ ) , ( 0 1 1 0 ) ) . \left(E_H\oplus E_H,
\begin{pmatrix}\phi&0\\0&\psi\end{pmatrix},
\begin{pmatrix}0&1\\1&0\end{pmatrix}\right). ( E H ⊕ E H , ( ϕ 0 0 ψ ) , ( 0 1 1 0 ) ) . The two summands have opposite gradings. The compact-difference condition supplies the compact commutators in this cycle. This notation refers to the pair, without assigning separate classes to the multiplier-valued maps.
Proposition 5.7 (Multiplier stable uniqueness). Suppose S S S X X X H H H θ : S → H ⊗ K \theta:S\to H\otimes\mathcal{K} θ : S → H ⊗ K X X X ϕ , ψ : S → L ( E H ) \phi,\psi:S\to\mathcal{L}(E_H) ϕ , ψ : S → L ( E H ) [ ϕ , ψ ] X = 0 [\phi,\psi]_X=0 [ ϕ , ψ ] X = 0 θ ∞ \theta^\infty θ ∞ F ⊂ S F\subset S F ⊂ S ε > 0 \varepsilon>0 ε > 0 v ∈ 1 + K ( E H ⊕ E H ) v\in1+\mathcal{K}(E_H\oplus E_H) v ∈ 1 + K ( E H ⊕ E H )
∥ v ( ϕ ⊕ θ ∞ ) ( a ) v ∗ − ( ψ ⊕ θ ∞ ) ( a ) ∥ < ε , a ∈ F . (54) \left\|v(\phi\oplus\theta^\infty)(a)v^*-(\psi\oplus\theta^\infty)(a)\right\|<\varepsilon,\qquad a\in F. \tag*{(54)} ∥ v ( ϕ ⊕ θ ∞ ) ( a ) v ∗ − ( ψ ⊕ θ ∞ ) ( a ) ∥ < ε , a ∈ F . (54) Proof. The multiplier-pair theorem [16 ] supplies a weakly X X X Θ \Theta Θ u u u ϕ ⊕ Θ \phi\oplus\Theta ϕ ⊕ Θ ψ ⊕ Θ \psi\oplus\Theta ψ ⊕ Θ δ \delta δ S S S σ = Θ ⊕ θ ∞ \sigma=\Theta\oplus\theta^\infty σ = Θ ⊕ θ ∞ V V V σ \sigma σ
∥ V ∗ σ ( a ) V − θ ∞ ( a ) ∥ < η , a ∈ F . \left\|V^*\sigma(a)V-\theta^\infty(a)\right\|<\eta,\qquad a\in F. ∥ V ∗ σ ( a ) V − θ ∞ ( a ) ∥ < η , a ∈ F . With W 0 = 1 ⊕ V W_0=1\oplus V W 0 = 1 ⊕ V
v = W 0 ∗ ( u ⊕ 1 ) W 0 . v=W_0^*(u\oplus1)W_0. v = W 0 ∗ ( u ⊕ 1 ) W 0 . It is a unitary, and v − 1 v-1 v − 1 (54) is at most δ + 2 η \delta+2\eta δ + 2 η δ + 2 η < ε \delta+2\eta<\varepsilon δ + 2 η < ε [10 ] ; no initial value of a continuous unitary path is needed here. □ \square □
Lemma 5.8 (The trivially labeled factor). For the coarse X X X S = C 0 ( Y ) ⊗ P S=C_0(Y)\otimes P S = C 0 ( Y ) ⊗ P X X X H H H
K K ( X ; S , H ) = 0. KK(X;S,H)=0. K K ( X ; S , H ) = 0. In particular every weakly equivariant multiplier pair with compact difference has zero class in (54) .
Proof. Since W \mathcal{W} W C ∗ C^* C ∗ [17 ] . Together with the transported labels of Proposition 2.2, this gives, after reordering the tensor factors, an X X X C = C 0 ( Y ) ⊗ A ⊗ K C=C_0(Y)\otimes A\otimes\mathcal{K} C = C 0 ( Y ) ⊗ A ⊗ K
S = W ⊗ C , S ( U ) = W ⊗ C ( U ) . S=\mathcal{W}\otimes C,\qquad S(U)=\mathcal{W}\otimes C(U). S = W ⊗ C , S ( U ) = W ⊗ C ( U ) . The identity of W \mathcal{W} W C C C c ∈ C ( U ) c\in C(U) c ∈ C ( U )
∑ i , j ⟨ ξ i , π ( w ) ξ j ⟩ ⊗ c i ∗ c c j ∈ W ⊗ C ( U ) . \sum_{i,j}\langle\xi_i,\pi(w)\xi_j\rangle\otimes c_i^*cc_j\in\mathcal{W}\otimes C(U). i , j ∑ ⟨ ξ i , π ( w ) ξ j ⟩ ⊗ c i ∗ c c j ∈ W ⊗ C ( U ) . The same formula applies throughout the homotopy. It follows that the identity cycle of S S S S S S H H H
The cycle relation and mixed product statements used here are [16 ] , Propositions 12.17 and 12.20 and Lemmas 12.18, 12.19, and 12.23. They require the first algebra to be separable and the remaining coefficient algebras to be sigma-unital. Thus they apply to this H H H S S S X X X
Lemma 5.9 (Hereditary support and local multiplication). Let I ⊆ C I\subseteq C I ⊆ C C ∗ C^* C ∗ e e e k ∈ C + k\in C_+ k ∈ C +
H = Her M 2 ( C ) ( e ⊕ k ) . H=\operatorname{Her}_{M_2(C)}(e\oplus k). H = Her M 2 ( C ) ( e ⊕ k ) . Then H H H r ∈ C r\in C r ∈ C I I I r ⊕ 0 r\oplus0 r ⊕ 0 H H H L L L M 2 ( C ) M_2(C) M 2 ( C )
( r I ) ⊕ 0 ⊆ L , ( I r ) ⊕ 0 ⊆ L , (rI)\oplus0\subseteq L,\qquad(Ir)\oplus0\subseteq L, ( r I ) ⊕ 0 ⊆ L , ( I r ) ⊕ 0 ⊆ L , then
( r ⊕ 0 ) H + H ( r ⊕ 0 ) ⊆ H ∩ L . (r\oplus0)H+H(r\oplus0)\subseteq H\cap L. ( r ⊕ 0 ) H + H ( r ⊕ 0 ) ⊆ H ∩ L . For x ∈ H x\in H x ∈ H H ∩ Ideal M 2 ( C ) ( x ) = Ideal H ( x ) H\cap\operatorname{Ideal}_{M_2(C)}(x)=\operatorname{Ideal}_H(x) H ∩ Ideal M 2 ( C ) ( x ) = Ideal H ( x )
Proof. The defining positive element is strictly positive in its hereditary algebra, so H H H r e 1 / 2 re^{1/2} r e 1/2 e 1 / 2 r e^{1/2}r e 1/2 r I I I ( e 1 / 2 ⊕ k 1 / 2 ) M 2 ( C ) ( e 1 / 2 ⊕ k 1 / 2 ) (e^{1/2}\oplus k^{1/2})M_2(C)(e^{1/2}\oplus k^{1/2}) ( e 1/2 ⊕ k 1/2 ) M 2 ( C ) ( e 1/2 ⊕ k 1/2 ) r ⊕ 0 r\oplus0 r ⊕ 0 H H H H H H L L L H H H x x x H H H
Lemma 5.10 (Finite truncation of a proper conjugacy). Let H H H p , q : S → M ( H ) p,q:S\to\mathcal{M}(H) p , q : S → M ( H ) H H H θ : S → H \theta:S\to H θ : S → H H H H
A 0 = p ⊕ θ ⊕ θ ⊕ ⋯ , B 0 = q ⊕ θ ⊕ θ ⊕ ⋯ . A_0=p\oplus\theta\oplus\theta\oplus\cdots,\qquad B_0=q\oplus\theta\oplus\theta\oplus\cdots. A 0 = p ⊕ θ ⊕ θ ⊕ ⋯ , B 0 = q ⊕ θ ⊕ θ ⊕ ⋯ . Identical zero summands may be inserted in these diagonals. If an identity-plus-compact unitary compares A 0 , B 0 A_0,B_0 A 0 , B 0 F \mathcal{F} F N , M ≥ 0 N, M \ge0 N , M ≥ 0 v N v_N v N N N N
max a ∈ F ∥ v N ( p ⊕ θ ⊕ N ⊕ 0 ⊕ M ) ( a ) v N ∗ − ( q ⊕ θ ⊕ N ⊕ 0 ⊕ M ) ( a ) ∥ < ε . \max_{a\in\mathcal{F}}\left\|v_N\left(p\oplus\theta^{\oplus N}\oplus0^{\oplus M}\right)(a)v_N^*-\left(q\oplus\theta^{\oplus N}\oplus0^{\oplus M}\right)(a)\right\|<\varepsilon. a ∈ F max v N ( p ⊕ θ ⊕ N ⊕ 0 ⊕ M ) ( a ) v N ∗ − ( q ⊕ θ ⊕ N ⊕ 0 ⊕ M ) ( a ) < ε . All resulting errors are compact. If the representations are weakly equivariant, their compact errors on S ( U ) S(U) S ( U ) H ( U ) H(U) H ( U )
Proof. Choose u = 1 + k u=1+k u = 1 + k ε / 2 \varepsilon/2 ε /2 R = max ( 1 , max a ∈ F ∥ a ∥ ) R=\max(1,\max_{a\in\mathcal{F}}\|a\|) R = max ( 1 , max a ∈ F ∥ a ∥ ) E N E_N E N
∥ k − E N k E N ∥ < η , 0 < η < min { 1 / 4 , ε / ( 8 R ) } . \|k-E_NkE_N\|<\eta,\qquad0<\eta<\min\{1/4,\varepsilon/(8R)\}. ∥ k − E N k E N ∥ < η , 0 < η < min { 1/4 , ε / ( 8 R )} . The operator z N = 1 + E N k E N z_N=1+E_NkE_N z N = 1 + E N k E N η \eta η ∣ z N ∣ |z_N| ∣ z N ∣ [ 1 − η , 1 + η ] [1-\eta,1+\eta] [ 1 − η , 1 + η ]
v N = z N ( z N ∗ z N ) − 1 / 2 , ∥ v N − z N ∥ ≤ η , ∥ v N − u ∥ ≤ 2 η . v_N=z_N(z_N^*z_N)^{-1/2},\qquad\|v_N-z_N\|\le\eta,\qquad\|v_N-u\|\le2\eta. v N = z N ( z N ∗ z N ) − 1/2 , ∥ v N − z N ∥ ≤ η , ∥ v N − u ∥ ≤ 2 η . It is the identity off E N E_N E N H H H E N → 1 E_N\to1 E N → 1 4 R η 4R\eta 4 R η
For localization, a compact operator which is weakly equivariant on an input in S ( U ) S(U) S ( U ) H ( U ) H(U) H ( U ) H ( U ) ⊗ K H(U)\otimes\mathcal{K} H ( U ) ⊗ K E H H ( U ) E_HH(U) E H H ( U )
Removing the locally small difference
Proposition 5.11 (Uniqueness with a spare model). Let p , q : S → D p,q:S\to D p , q : S → D ( p − q ) ( S ( U ) ) ⊆ J ( U ) (p-q)(S(U))\subseteq J(U) ( p − q ) ( S ( U )) ⊆ J ( U ) U U U χ \chi χ p ⊕ χ p\oplus\chi p ⊕ χ q ⊕ χ q\oplus\chi q ⊕ χ
Proof. Put C 1 = C ∗ ( p ( S ) , q ( S ) ) C_1=C^*(p(S),q(S)) C 1 = C ∗ ( p ( S ) , q ( S )) I I I C 1 C_1 C 1 p ( a ) − q ( a ) p(a)-q(a) p ( a ) − q ( a ) U U U
p ( S ( U ) ) I + q ( S ( U ) ) I + I p ( S ( U ) ) + I q ( S ( U ) ) ⊆ J ( U ) . p(S(U))I+q(S(U))I+Ip(S(U))+Iq(S(U))\subseteq J(U). p ( S ( U )) I + q ( S ( U )) I + I p ( S ( U )) + I q ( S ( U )) ⊆ J ( U ) . Indeed, modulo J ( U ) J(U) J ( U ) S ( U ) S(U) S ( U ) p ( a ) p(a) p ( a ) q ( a ) q(a) q ( a ) s ∈ S ( U ) s\in S(U) s ∈ S ( U )
p ( s ) ( p ( a ) − q ( a ) ) = p ( s a ) − q ( s a ) = 0 ( m o d J ( U ) ) , p(s)(p(a)-q(a))=p(sa)-q(sa)=0\pmod{J(U)}, p ( s ) ( p ( a ) − q ( a )) = p ( s a ) − q ( s a ) = 0 ( mod J ( U )) , and the other side is identical. It therefore annihilates the ideal generated by the differences, proving (5.14).
Use the existing-model version of Theorem 4.1, with the spare χ \chi χ θ \theta θ
p ( S ( U x ) ) I + q ( S ( U x ) ) I + I p ( S ( U x ) ) + I q ( S ( U x ) ) ⊆ Ideal ( θ ( x ) ) , x ∈ S + . (55) p(S(U_x))I+q(S(U_x))I+Ip(S(U_x))+Iq(S(U_x))\subseteq\operatorname{Ideal}(\theta(x)),\qquad x\in S_+. \tag*{(55)} p ( S ( U x )) I + q ( S ( U x )) I + I p ( S ( U x )) + I q ( S ( U x )) ⊆ Ideal ( θ ( x )) , x ∈ S + . (55) Apply Corollary 4.3 to the four bounded bilinear maps ( a , b ) ↦ p ( a ) b (a,b)\mapsto p(a)b ( a , b ) ↦ p ( a ) b q ( a ) b q(a)b q ( a ) b b p ( a ) bp(a) b p ( a ) b q ( a ) bq(a) b q ( a ) b ∈ I b\in I b ∈ I (5.14) gives their local hypotheses, and that corollary supplies the displayed inclusion for every positive x x x
θ ( S ( U x ) ) ⊆ Ideal ( θ ( x ) ) . (56) \theta(S(U_x))\subseteq\operatorname{Ideal}(\theta(x)). \tag*{(56)} θ ( S ( U x )) ⊆ Ideal ( θ ( x )) . (56) Moreover θ \theta θ χ \chi χ
Choose strictly positive elements e ∈ I e\in I e ∈ I k ∈ θ ( S ) k\in\theta(S) k ∈ θ ( S )
H = Her M 2 ( D ) ( e ⊕ k ) . H=\operatorname{Her}_{M_2(D)}(e\oplus k). H = Her M 2 ( D ) ( e ⊕ k ) . The maps p , q p,q p , q H H H H H H H H H
H ( U ) = Ideal H ( 0 ⊕ θ ( S ( U ) ) ) . (57) H(U)=\operatorname{Ideal}_H(0\oplus\theta(S(U))). \tag*{(57)} H ( U ) = Ideal H ( 0 ⊕ θ ( S ( U ))) . (57) No intersection property of this assignment is needed. From (55) , Lemma 5.9, and its ideal-intersection identity, values of the first-corner maps on S ( U x ) S(U_x) S ( U x ) H H H Ideal H ( 0 ⊕ θ ( x ) ) \operatorname{Ideal}_H(0\oplus\theta(x)) Ideal H ( 0 ⊕ θ ( x )) p , q p,q p , q (57) . By (56) , 0 ⊕ θ ( x ) 0\oplus\theta(x) 0 ⊕ θ ( x ) H ( U x ) H(U_x) H ( U x ) X X X H H H
The multiplier pair agrees modulo H H H I ⊕ 0 ⊆ H I\oplus0\subseteq H I ⊕ 0 ⊆ H H H H (5.11) is zero by Lemma 5.8. The coarse source action is lower semicontinuous, since the projection Y × X → X Y\times X\to X Y × X → X 0 ⊕ θ 0\oplus\theta 0 ⊕ θ θ \theta θ H H H H H H H H H F \mathcal{F} F ε > 0 \varepsilon>0 ε > 0 N N N
p ⊕ θ ⊕ N and q ⊕ θ ⊕ N p\oplus\theta^{\oplus N}\quad\text{and}\quad q\oplus\theta^{\oplus N} p ⊕ θ ⊕ N and q ⊕ θ ⊕ N within ε \varepsilon ε
These estimates hold for the actual ambient elements, as follows. If v = 1 + c v=1+c v = 1 + c A , B A,B A , B
v A ( a ) v ∗ − B ( a ) = ( A ( a ) − B ( a ) ) + c A ( a ) + A ( a ) c ∗ + c A ( a ) c ∗ . vA(a)v^*-B(a)=(A(a)-B(a))+cA(a)+A(a)c^*+cA(a)c^*. v A ( a ) v ∗ − B ( a ) = ( A ( a ) − B ( a )) + c A ( a ) + A ( a ) c ∗ + c A ( a ) c ∗ . Every term belongs to a finite matrix over H H H H H H H H H D D D M ( H ) \mathcal{M}(H) M ( H ) H H H
There is also no lifting obstruction for the finite unitary. Write it as 1 + c 1+c 1 + c D D D c n c_n c n Q Q Q 1 + c n 1+c_n 1 + c n n n n
v n = ( 1 + c n ) ( ( 1 + c n ) ∗ ( 1 + c n ) ) − 1 / 2 v_n=(1+c_n)\bigl((1+c_n)^*(1+c_n)\bigr)^{-1/2} v n = ( 1 + c n ) ( ( 1 + c n ) ∗ ( 1 + c n ) ) − 1/2 is a unitary in the corresponding matrix multiplier algebra and differs from 1 + c n 1+c_n 1 + c n
Finally fix this N N N θ n \theta_n θ n N N N χ n \chi_n χ n p ⊕ χ p \oplus\chi p ⊕ χ q ⊕ χ q \oplus\chi q ⊕ χ
Proof of Theorem 5.1. Split the tensor-first copy of each map along two equivalent UHF projections of trace 1 / 2 1/2 1/2 p 0 , p 1 p_0,p_1 p 0 , p 1 q 0 , q 1 q_0,q_1 q 0 , q 1 1 / 2 1/2 1/2
First compare p 0 ⊕ p 1 p_0 \oplus p_1 p 0 ⊕ p 1 q 0 ⊕ p 1 q_0 \oplus p_1 q 0 ⊕ p 1 p 1 p_1 p 1 J ( U ) J(U) J ( U ) q 0 ⊕ p 1 q_0 \oplus p_1 q 0 ⊕ p 1 q 0 ⊕ q 1 q_0 \oplus q_1 q 0 ⊕ q 1 q 0 q_0 q 0 u n u_n u n
Finally, any two c.p.c. lifts of a fixed model differ in norm by a sequence tending to zero on each input. The conclusion therefore holds for the lifts specified in the statement.
Tracial approximation on a cone
Throughout this section, Y = ( 0 , 1 ] Y=(0,1] Y = ( 0 , 1 ] m m m Y Y Y S = C 0 ( Y ) ⊗ P S=C_0(Y)\otimes P S = C 0 ( Y ) ⊗ P F : T ( Q ) → T ( P ) F:T(Q)\to T(P) F : T ( Q ) → T ( P ) G : Cu ( P ) → Cu ( Q ) G:\operatorname{Cu}(P)\to\operatorname{Cu}(Q) G : Cu ( P ) → Cu ( Q )
d τ ( G [ a ] ) = d F ( τ ) ( a ) . d_\tau(G[a])=d_{F(\tau)}(a). d τ ( G [ a ]) = d F ( τ ) ( a ) . We seek a completely positive map that is multiplicative modulo the local error ideals. It must carry two kinds of information. Its square moments will identify the prescribed trace wherever that trace is densely finite. The ideal generated by its image of a nonzero elementary tensor must also contain the corresponding constant ideal. This second condition will prevent the corrected map from turning an infinite part of a weight into a finite one.
Matrix traces are unnormalized. The rational averages used below have a separate, normalized UHF tensor factor; Lemma 2.4 identifies their ranges with Q Q Q
Choose a countable family E \mathcal{E} E s = f ⊗ a s=f\otimes a s = f ⊗ a C 0 ( Y ) C_0(Y) C 0 ( Y ) E \mathcal{E} E Y × X Y\times X Y × X
For a test s i = f i ⊗ a i s_i=f_i\otimes a_i s i = f i ⊗ a i U i U_i U i b i ∈ Q ( U i ) b_i\in Q(U_i) b i ∈ Q ( U i )
G [ a i ] ≤ [ b i ] . (58) G[a_i]\le[b_i]. \tag*{(58)} G [ a i ] ≤ [ b i ] . (58) The larger pre-cuts allow this choice inside Q ( U i ) Q(U_i) Q ( U i ) (19) .
Theorem 6.1. There is a completely positive contraction h : S → D = Q ∞ h:S\to D=Q_\infty h : S → D = Q ∞
For every open U ⊆ X U\subseteq X U ⊆ X s ∈ S ( U ) s\in S(U) s ∈ S ( U ) z ∈ S z\in S z ∈ S
h ( s z ) − h ( s ) h ( z ) , h ( z s ) − h ( z ) h ( s ) ∈ J ( U ) . (59) h(sz)-h(s)h(z),\quad h(zs)-h(z)h(s)\in J(U). \tag*{(59)} h ( sz ) − h ( s ) h ( z ) , h ( z s ) − h ( z ) h ( s ) ∈ J ( U ) . (59) If s = f ⊗ a ∈ E s=f\otimes a\in\mathcal{E} s = f ⊗ a ∈ E U U U ρ \rho ρ σ = ρ ∣ Q \sigma=\rho|_Q σ = ρ ∣ Q Q ( U ) Q(U) Q ( U )
ρ ( h ( s ) 2 ) = ( ∫ Y f 2 d m ) F ( σ ) ( a 2 ) . (60) \rho\bigl(h(s)^2\bigr)=\left(\int_Y f^2\,\mathrm{d}m\right)F(\sigma)(a^2). \tag*{(60)} ρ ( h ( s ) 2 ) = ( ∫ Y f 2 d m ) F ( σ ) ( a 2 ) . (60) Finally, for every f ⊗ a ∈ E f\otimes a\in\mathcal{E} f ⊗ a ∈ E f ≠ 0 f\ne0 f = 0
Q ( U a ) ⊆ Ideal D ( h ( f ⊗ a ) ) . (61) Q(U_a)\subseteq\operatorname{Ideal}_D\bigl(h(f\otimes a)\bigr). \tag*{(61)} Q ( U a ) ⊆ Ideal D ( h ( f ⊗ a ) ) . (61) The test family may be enlarged in advance by any countable family of the controlled elementary tests just described.
For a finite-stage completely positive contraction H : S → M k ( Q ) H:S\to M_k(Q) H : S → M k ( Q ) s s s H ( s ) H(s) H ( s ) H ( s 2 ) H(s^2) H ( s 2 ) H ( s ) 2 H(s)^2 H ( s ) 2
Trace approximation alone does not give the lower ideal inclusion. For that we also approximate positive expressions y v ∗ g ( H ( s ) ) v y yv^*g(H(s))vy y v ∗ g ( H ( s )) v y y 2 y^2 y 2 y y y Q Q Q v ∈ M k , 1 ( Q ) v\in M_{k,1}(Q) v ∈ M k , 1 ( Q ) g : [ 0 , ∞ ) → [ 0 , 1 ] g:[0,\infty)\to[0,1] g : [ 0 , ∞ ) → [ 0 , 1 ] Q Q Q
Realization in von Neumann algebras
We first record explicitly the trace-preserving embedding needed for extended weights. The embedding is not required to be unital. We write R R R I I 1 \mathrm{II}_1 II 1 R ∞ = R ⊗ B ( ℓ 2 ) R_\infty=R\otimes B(\ell^2) R ∞ = R ⊗ B ( ℓ 2 ) I I ∞ \mathrm{II}_\infty II ∞ [8 ] ; see also the distinct contemporary report [9 ] , Theorems 4.2.1 and 4.3.1.
Lemma 6.2 (Trace-preserving semifinite embedding). Let N N N T T T N N N R ∞ R_\infty R ∞ I I ∞ \mathrm{II}_\infty II ∞ T T T I I ∞ \mathrm{II}_\infty II ∞ Proof. The central decomposition of a semifinite hyperfinite algebra gives a countable direct sum of algebras
L ∞ ( X j , μ j 0 ) ⊗ N j , N j ∈ { M n , B ( ℓ 2 ) , R , R ∞ } . L^\infty(X_j,\mu_j^0)\otimes N_j,\qquad N_j\in\{M_n,B(\ell^2),R,R_\infty\}. L ∞ ( X j , μ j 0 ) ⊗ N j , N j ∈ { M n , B ( ℓ 2 ) , R , R ∞ } . Here the measure spaces are standard and the factor traces are the usual matrix trace or the standard finite or semifinite trace. This is the semifinite form of the hyperfinite central decomposition; see [19 ] , Section 6, p. 49, after (6.48) and (6.50)]. Relative to these traces and a finite measure μ j 0 \mu_j^0 μ j 0 T T T h j h_j h j 0 < h j < ∞ 0<h_j<\infty 0 < h j < ∞ μ j 0 \mu_j^0 μ j 0 μ j = h j μ j 0 \mu_j=h_j\mu_j^0 μ j = h j μ j 0 σ \sigma σ
There is a normal, trace-preserving injection
L ∞ ( X j , μ j ) ⟶ L ∞ ( [ 0 , ∞ ) , d t ) . L^\infty(X_j,\mu_j)\longrightarrow L^\infty([0,\infty),dt). L ∞ ( X j , μ j ) ⟶ L ∞ ([ 0 , ∞ ) , d t ) . Indeed, partition X j X_j X j L ∞ ( [ 0 , ∞ ) ) L^\infty([0,\infty)) L ∞ ([ 0 , ∞ )) L ∞ ( [ 0 , 1 ] ) ⊗ ℓ ∞ L^\infty([0,1])\otimes\ell^\infty L ∞ ([ 0 , 1 ]) ⊗ ℓ ∞ R ⊗ B ( ℓ 2 ) = R ∞ R\otimes B(\ell^2)=R_\infty R ⊗ B ( ℓ 2 ) = R ∞
Each standard factor N j N_j N j R ∞ R_\infty R ∞ 1 R ⊗ M n 1_R\otimes M_n 1 R ⊗ M n 1 R ⊗ B ( ℓ 2 ) 1_R\otimes B(\ell^2) 1 R ⊗ B ( ℓ 2 ) R ⊗ e 11 R\otimes e_{11} R ⊗ e 11 R ∞ ⊗ R ∞ ≅ R ∞ R_\infty\otimes R_\infty\cong R_\infty R ∞ ⊗ R ∞ ≅ R ∞ R ∞ ⊗ B ( ℓ 2 ) R_\infty\otimes B(\ell^2) R ∞ ⊗ B ( ℓ 2 )
Finally, in an infinite corner of an arbitrary factor of type I I ∞ \mathrm{II}_\infty II ∞ M 2 n M_{2^n} M 2 n R R R R ∞ R_\infty R ∞
Realization in a factor
Lemma 6.3 (Realization in a factor). Let π : Q → M \pi:Q\to M π : Q → M ι : S → M \iota:S\to M ι : S → M M M M T T T
d T ( ι ( f ⊗ a ) ) = m ( { f > 0 } ) d F ( T ∘ π ) ( a ) ( f , a ≥ 0 ) . (62) d_T(\iota(f\otimes a))=m(\{f>0\})d_{F(T\circ\pi)}(a)\qquad(f,a\geq0). \tag*{(62)} d T ( ι ( f ⊗ a )) = m ({ f > 0 }) d F ( T ∘ π ) ( a ) ( f , a ≥ 0 ) . (62) In a type III factor the corresponding support projection is nonzero exactly when the transported ideal data prescribe a nonzero support.
Proof. Stability of Q Q Q M M M M M M [1 ] , III.1.7.9–10 and III.1.3.7. Suppose first that M M M d T ∘ π d_{T\circ\pi} d T ∘ π Cu ( Q ) \operatorname{Cu}(Q) Cu ( Q ) 0 0 0 ∞ \infty ∞ Cu ( Q ) \operatorname{Cu}(Q) Cu ( Q ) (62) also takes only these two values. Its zero ideal on P P P ker π \ker\pi ker π S S S M M M
Suppose now that M M M I I ∞ \mathrm{II}_{\infty} II ∞
ν = m ⊗ F ( T ∘ π ) , I = Fin ( ν ) , K = Ker ( ν ) . \nu= m \otimes F(T \circ\pi), \qquad I = \operatorname{Fin}(\nu), \qquad K = \operatorname{Ker}(\nu). ν = m ⊗ F ( T ∘ π ) , I = Fin ( ν ) , K = Ker ( ν ) . Here Fin ( ν ) \operatorname{Fin}(\nu) Fin ( ν ) ν \nu ν K ⊆ I K \subseteq I K ⊆ I I / K I/K I / K N N N ν \nu ν I / K I/K I / K N N N [8 ] ; the nuclear-representation consequence is also stated in [9 ] , Theorem 4.1 and the following paragraph]. The algebra N N N I I I S S S S S S I I I M M M
In an orthogonal infinite corner, choose a copy of B ( ℓ 2 ) B(\ell^{2}) B ( ℓ 2 ) S / I S/I S / I I I I x ∈ S + x \in S_{+} x ∈ S + I I I ν ( x ) = ∞ \nu(x) = \infty ν ( x ) = ∞ I I I I I I ν \nu ν T ∘ l = ν T \circ l = \nu T ∘ l = ν K K K I = 0 I = 0 I = 0 I = S I = S I = S
Factorwise existence is enough only if the maps can be assembled measurably. We encode their rank requirements by partial isometries between support projections. These witnesses assemble along with the maps, and later allow each selected normal trace to evaluate the required ranks directly.
Lemma 6.4 (Measurable realization). Let π \pi π Q Q Q M = π ( Q ) ′ ′ M = \pi(Q)^{\prime\prime} M = π ( Q ) ′′ l : S → M l : S \to M l : S → M E \mathcal{E} E
Proof. Use the standard central decomposition of M M M M x M_x M x [1 ] , III.5.1.14–15. We explain the measurable selection in a form that does not choose a normalization of the factor traces. For each elementary test s j = f j ⊗ a j s_j = f_j \otimes a_j s j = f j ⊗ a j c j ∈ Q + c_j \in Q_{+} c j ∈ Q +
[ c j ] = m ( { f j > 0 } ) G [ a j ] . [c_j] = m(\{f_j > 0\})G[a_j]. [ c j ] = m ({ f j > 0 }) G [ a j ] . Realification and stability provide these representatives. Code a contractive homomorphism on a countable dense rational ∗ * ∗ S S S ∗ * ∗ ∗ * ∗ w j ∈ M x w_j \in M_x w j ∈ M x
w j ∗ w j = supp ( l x ( s j ) ) , w j w j ∗ = supp ( π x ( c j ) ) . (63) w_j^{*}w_j = \operatorname{supp}(l_x(s_j)), \qquad w_jw_j^{*} = \operatorname{supp}(\pi_x(c_j)). \tag*{(63)} w j ∗ w j = supp ( l x ( s j )) , w j w j ∗ = supp ( π x ( c j )) . (63) The support map is the strong limit of t ↦ t ( t + 1 / n ) − 1 t \mapsto t(t+1/n)^{-1} t ↦ t ( t + 1/ n ) − 1
Thus the augmented admissible-code relation is Borel in a countable product of standard measurable bounded operator fields. Its sections are nonempty by Lemma 6.3; equal factor dimensions give the partial isometries in (63) . Apply Jankov–von Neumann measurable uniformization to this augmented analytic relation, retaining both the homomorphism coordinates and all the contraction witnesses w j w_j w j [34 ] , Section 8, especially Lemma 8.6(1), and the proof of Proposition 3.2. The uniform operator bounds make the selected values, including the w j w_j w j l : S → M l:S\to M l : S → M (63) in M M M
Here is the finite-trace consequence of this construction that we will use. It specifies the normal trace associated to each selected weight and the finite constants through which the rank data are evaluated.
Lemma 6.5 (Realization for finitely many weights). Fix finitely many controlled tests s i = f i ⊗ a i s_i=f_i\otimes a_i s i = f i ⊗ a i τ \tau τ i i i d τ ( b i ) < ∞ d_\tau(b_i)<\infty d τ ( b i ) < ∞ π \pi π Q Q Q l l l τ ~ \widetilde{\tau} τ M = π ( Q ) ′ ′ M=\pi(Q)'' M = π ( Q ) ′′
τ ~ ( l ( s i k ) ) = ( ∫ f i k d m ) F ( τ ) ( a i k ) , k = 1 , 2. (64) \widetilde{\tau}(l(s_i^k))=\left(\int f_i^k\,\mathrm{d}m\right)F(\tau)(a_i^k),\qquad k=1,2. \tag*{(64)} τ ( l ( s i k )) = ( ∫ f i k d m ) F ( τ ) ( a i k ) , k = 1 , 2. (64) This assertion requires no commutation of F F F
Proof. Fix one selected τ \tau τ I = Fin ( τ ) I=\operatorname{Fin}(\tau) I = Fin ( τ ) π τ \pi_\tau π τ I / Ker ( τ ) I/\operatorname{Ker}(\tau) I / Ker ( τ ) N τ N_\tau N τ T τ T_\tau T τ π τ \pi_\tau π τ Q Q Q N τ N_\tau N τ ( u λ ) (u_\lambda) ( u λ ) I I I π τ ( q ) π τ ( u λ ) ∈ π τ ( I ) \pi_\tau(q)\pi_\tau(u_\lambda)\in\pi_\tau(I) π τ ( q ) π τ ( u λ ) ∈ π τ ( I ) π τ ( q ) \pi_\tau(q) π τ ( q ) π τ ( Q ) ′ ′ = N τ \pi_\tau(Q)''=N_\tau π τ ( Q ) ′′ = N τ
The projection onto this representation summand commutes with M = π ( Q ) ′ ′ M=\pi(Q)'' M = π ( Q ) ′′ θ τ : M → N τ \theta_\tau:M\to N_\tau θ τ : M → N τ ( 1 − z τ ) M (1-z_\tau)M ( 1 − z τ ) M z τ z_\tau z τ z τ M z_\tau M z τ M
τ ~ = T τ ∘ θ τ . \widetilde{\tau}=T_\tau\circ\theta_\tau. τ = T τ ∘ θ τ . This is a normal semifinite trace, zero on ( 1 − z τ ) M (1-z_\tau)M ( 1 − z τ ) M τ \tau τ π ( I ) + \pi(I)_+ π ( I ) + d ~ τ ( π ( c ) ) = d τ ( c ) \widetilde{d}_\tau(\pi(c))=d_\tau(c) d τ ( π ( c )) = d τ ( c ) c ∈ I + c\in I_+ c ∈ I + z τ z_\tau z τ τ \tau τ I I I
Include the rectangle tests ( f i − r ) + ⊗ ( a i − t ) + (f_i-r)_+\otimes(a_i-t)_+ ( f i − r ) + ⊗ ( a i − t ) + r , t ≥ 0 r,t\geq0 r , t ≥ 0 c r , t ∈ Q + c_{r,t}\in Q_+ c r , t ∈ Q +
[ c r , t ] = m ( { f i > r } ) G [ ( a i − t ) + ] . [c_{r,t}]=m(\{f_i>r\})G[(a_i-t)_+]. [ c r , t ] = m ({ f i > r }) G [( a i − t ) + ] . Their ranks are at most m ( Y ) d τ ( b i ) < ∞ m(Y)d_\tau(b_i)<\infty m ( Y ) d τ ( b i ) < ∞ c r , t ∈ I c_{r,t}\in I c r , t ∈ I (63) gives, in M M M l ( ( f i − r ) + ⊗ ( a i − t ) + ) l((f_i-r)_+\otimes(a_i-t)_+) l (( f i − r ) + ⊗ ( a i − t ) + ) supp ( π ( c r , t ) ) \operatorname{supp}(\pi(c_{r,t})) supp ( π ( c r , t )) τ ~ \widetilde{\tau} τ
τ ~ ( supp l ( ( f i − r ) + ⊗ ( a i − t ) + ) ) = m ( { f i > r } ) d F ( τ ) ( ( a i − t ) + ) . \widetilde{\tau}\bigl(\operatorname{supp}l((f_i-r)_+\otimes(a_i-t)_+)\bigr)=m(\{f_i>r\})d_{F(\tau)}((a_i-t)_+). τ ( supp l (( f i − r ) + ⊗ ( a i − t ) + ) ) = m ({ f i > r }) d F ( τ ) (( a i − t ) + ) . In particular, supp ( l ( s i ) ) \operatorname{supp}(l(s_i)) supp ( l ( s i )) τ ~ \widetilde{\tau} τ
In the support corner of l l l f j ⊗ 1 f_j\otimes1 f j ⊗ 1 1 ⊗ a i 1\otimes a_i 1 ⊗ a i l ( s i ) l(s_i) l ( s i ) f i f_i f i m m m a i a_i a i F ( τ ) F(\tau) F ( τ ) ( u , v ) ↦ u k v k (u,v)\mapsto u^{k}v^{k} ( u , v ) ↦ u k v k (64) by normal monotone convergence, for k = 1 , 2 k=1,2 k = 1 , 2 Fin ( τ ) \operatorname{Fin}(\tau) Fin ( τ )
Simultaneous finite approximation
The von Neumann algebra realizations attain the prescribed trace and support data. We now approximate finitely many of those data inside Q Q Q
Fix finitely many tests s i = f i ⊗ a i s_i=f_i\otimes a_i s i = f i ⊗ a i 1 ≤ i ≤ n 1\leq i\leq n 1 ≤ i ≤ n
K i = { τ ∈ T ( Q ) : d τ ( b i ) ≤ 1 } , t i ( k ) ( τ ) = ( ∫ f i k d m ) F ( τ ) ( a i k ) . (65) K_i=\{\tau\in T(Q):d_\tau(b_i)\leq1\},\qquad t_i^{(k)}(\tau)=\left(\int f_i^k\,\mathrm{d}m\right)F(\tau)(a_i^k). \tag*{(65)} K i = { τ ∈ T ( Q ) : d τ ( b i ) ≤ 1 } , t i ( k ) ( τ ) = ( ∫ f i k d m ) F ( τ ) ( a i k ) . (65) By Lemma 5.2, K i K_i K i t i ( k ) t_i^{(k)} t i ( k ) K i K_i K i
We impose finitely many lower comparisons, whose indices j j j δ j > 0 \delta_j>0 δ j > 0 r j > 0 r_j>0 r j > 0
r j < m ( { f j > δ j } ) , [ y j ] = r j G [ ( a j − δ j ) + ] , y j ∈ Q + . r_j<m(\{f_j>\delta_j\}),\qquad[y_j]=r_jG[(a_j-\delta_j)_+],\qquad y_j\in Q_+. r j < m ({ f j > δ j }) , [ y j ] = r j G [( a j − δ j ) + ] , y j ∈ Q + . Rescale representatives so that ∥ y j ∥ ≤ 1 \lVert y_j\rVert\leq1 ∥ y j ∥ ≤ 1 g j : [ 0 , ∞ ) → [ 0 , 1 ] g_j:[0,\infty)\to[0,1] g j : [ 0 , ∞ ) → [ 0 , 1 ]
0 < η j < θ j < δ j 2 , g j ∣ [ 0 , η j ] = 0 , g j ∣ [ θ j , ∞ ) = 1. 0<\eta_j<\theta_j<\delta_j^2,\qquad g_j|_{[0,\eta_j]}=0,\qquad g_j|_{[\theta_j,\infty)}=1. 0 < η j < θ j < δ j 2 , g j ∣ [ 0 , η j ] = 0 , g j ∣ [ θ j , ∞ ) = 1. These constants are fixed before any finite approximation or averaging.
A candidate consists of a completely positive contraction H : S → M k ( Q ) H:S\to M_k(Q) H : S → M k ( Q ) v j ∈ M k , 1 ( Q ) v_j\in M_{k,1}(Q) v j ∈ M k , 1 ( Q ) i i i C H , i C_{H,i} C H , i τ ∈ T ( Q ) \tau\in T(Q) τ ∈ T ( Q )
d τ ⊗ Tr ( H ( s i ) ) ≤ C H , i d τ ( b i ) , d τ ⊗ Tr ( H ( s i 2 ) ) ≤ C H , i d τ ( b i ) . (66) \begin{aligned}
d_{\tau\otimes\operatorname{Tr}}(H(s_i))&\leq C_{H,i}d_\tau(b_i),\\
d_{\tau\otimes\operatorname{Tr}}(H(s_i^2))&\leq C_{H,i}d_\tau(b_i). \tag*{(66)}
\end{aligned} d τ ⊗ Tr ( H ( s i )) d τ ⊗ Tr ( H ( s i 2 )) ≤ C H , i d τ ( b i ) , ≤ C H , i d τ ( b i ) . (66) The constants may depend on the candidate. Its vector in the real locally convex space
∏ i = 1 n C ( K i , R ) 3 × ∏ j Q s a \prod_{i=1}^{n}C(K_i,\mathbb{R})^3\times\prod_j Q_{\mathrm{sa}} i = 1 ∏ n C ( K i , R ) 3 × j ∏ Q sa is
( ( τ ( H ( s i ) ) , τ ( H ( s i 2 ) ) , τ ( H ( s i ) 2 ) ) i ; ( y j v j ∗ g j ( H ( s j ) ) v j y j ) j ) . \left((\tau(H(s_i)),\tau(H(s_i^2)),\tau(H(s_i)^2))_i;(y_jv_j^*g_j(H(s_j))v_jy_j)_j\right). ( ( τ ( H ( s i )) , τ ( H ( s i 2 )) , τ ( H ( s i ) 2 ) ) i ; ( y j v j ∗ g j ( H ( s j )) v j y j ) j ) . Here and below τ \tau τ τ ⊗ Tr \tau\otimes\operatorname{Tr} τ ⊗ Tr
Proposition 6.6 (Simultaneous finite approximation). The target vector
( ( t i ( 1 ) , t i ( 2 ) , t i ( 2 ) ) i ; ( y j 2 ) j ) \left((t_i^{(1)},t_i^{(2)},t_i^{(2)})_i;(y_j^2)_j\right) ( ( t i ( 1 ) , t i ( 2 ) , t i ( 2 ) ) i ; ( y j 2 ) j ) belongs to the closed convex hull of the candidate vectors.
Proof. Reduce a separator to finitely many weights. Suppose a continuous real linear functional strictly separates the target from this convex hull. On each C ( K i , R ) C(K_i,\mathbb{R}) C ( K i , R ) K i K_i K i
Realize the selected values exactly. Choose a faithful separably acting representation π \pi π Q Q Q M = π ( Q ) ′ ′ M=\pi(Q)^{\prime\prime} M = π ( Q ) ′′ Q Q Q l : S → M l:S\to M l : S → M M M M
Let q j q_j q j l ( ( f j − δ j ) + ⊗ ( a j − δ j ) + ) l((f_j-\delta_j)_+\otimes(a_j-\delta_j)_+) l (( f j − δ j ) + ⊗ ( a j − δ j ) + )
q j ≤ 1 ( δ j 2 , ∞ ) ( l ( s j ) ) . q_j\leq1_{(\delta_j^2,\infty)}(l(s_j)). q j ≤ 1 ( δ j 2 , ∞ ) ( l ( s j )) . The rank data and (6.11) give, in every factor, supp ( y j ) ≾ q j \operatorname{supp}(y_j)\precsim q_j supp ( y j ) ≾ q j v j ∈ M v_j\in M v j ∈ M supp ( y j ) \operatorname{supp}(y_j) supp ( y j ) q j q_j q j
y j v j ∗ g j ( l ( s j ) ) v j y j = y j 2 . y_jv_j^*g_j(l(s_j))v_jy_j=y_j^2. y j v j ∗ g j ( l ( s j )) v j y j = y j 2 . Approximate inside fixed supports. It remains to approximate these finite data by candidates satisfying (6.13) on the entire caps. A tested value must vanish exactly in a block where its controller vanishes; strong approximation alone would not retain this condition. We therefore split into central sectors and factor each sector map through the required source quotient before approximating it. Let z i z_i z i b i b_i b i M M M
z E = ∏ i ∈ E z i ∏ i ∉ E ( 1 − z i ) , E ⊆ { 1 , … , n } . z_E=\prod_{i\in E}z_i\prod_{i\notin E}(1-z_i),\qquad E\subseteq\{1,\ldots,n\}. z E = i ∈ E ∏ z i i ∈ / E ∏ ( 1 − z i ) , E ⊆ { 1 , … , n } . If i ∉ E i\notin E i ∈ / E G [ a i ] ≤ [ b i ] G[a_i]\leq[b_i] G [ a i ] ≤ [ b i ] z E l ( s i ) = 0 z_E l(s_i)=0 z E l ( s i ) = 0 J E S J_E^S J E S S S S s i s_i s i
I E = ⋂ i ∈ E Ideal Q ( b i ) , I ∅ = Q . I_E=\bigcap_{i\in E}\operatorname{Ideal}_Q(b_i),\qquad I_\varnothing=Q. I E = i ∈ E ⋂ Ideal Q ( b i ) , I ∅ = Q . The weak closure of I E I_E I E z E z_E z E e E ∈ I E e_E\in I_E e E ∈ I E
ψ E ( s ) = e E z E l ( s ) e E \psi_E(s)=e_Ez_E l(s)e_E ψ E ( s ) = e E z E l ( s ) e E take values in Her ( e E ) ′ ′ \operatorname{Her}(e_E)^{\prime\prime} Her ( e E ) ′′ S / J E S S/J_E^S S / J E S e E e_E e E
D 0 ( s ) = diag E ( z E l ( s ) ) , D e ( s ) = diag E ( e E z E l ( s ) e E ) . D_0(s)=\operatorname{diag}_E(z_E l(s)),\qquad D_e(s)=\operatorname{diag}_E(e_Ez_E l(s)e_E). D 0 ( s ) = diag E ( z E l ( s )) , D e ( s ) = diag E ( e E z E l ( s ) e E ) . Thus D 0 D_0 D 0 D e D_e D e e E e_E e E D e ( s ) → D 0 ( s ) D_e(s)\to D_0(s) D e ( s ) → D 0 ( s ) s s s i i i i ∉ E i \notin E i ∈ / E i ∈ E i \in E i ∈ E z E l ( s i ) z_E l(s_i) z E l ( s i ) z E l ( s i 2 ) z_E l(s_i^2) z E l ( s i 2 ) z E l ( s i ) z_E l(s_i) z E l ( s i ) (64) . Strongly converging bounded approximate units therefore make the compressions converge in the required L 1 L^1 L 1 L 2 L^2 L 2 L 1 L^1 L 1 L 2 L^2 L 2 z E l ( s i ) z_E l(s_i) z E l ( s i )
For the lower signal set V j = ( z E v j ) E V_j=(z_Ev_j)_E V j = ( z E v j ) E g j ( 0 ) = 0 g_j(0)=0 g j ( 0 ) = 0
y j V j ∗ g j ( D 0 ( s j ) ) V j y j = y j 2 . y_jV_j^*g_j(D_0(s_j))V_jy_j=y_j^2. y j V j ∗ g j ( D 0 ( s j )) V j y j = y j 2 . Bounded strong-* functional calculus gives g j ( D e ( s j ) ) → g j ( D 0 ( s j ) ) g_j(D_e(s_j))\to g_j(D_0(s_j)) g j ( D e ( s j )) → g j ( D 0 ( s j )) e E e_E e E D 0 D_0 D 0 D e D_e D e y j 2 y_j^2 y j 2
We give the remaining approximation explicitly. Nuclearity of the quotient R 0 = S / J E S R_0=S/J_E^S R 0 = S / J E S [6 ] ; see also [20 ] , Corollary 3.3. For nonunital R 0 R_0 R 0 R 0 R_0 R 0 β \beta β x ↦ e β ( x ) e x\mapsto e\beta(x)e x ↦ e β ( x ) e e ∈ R 0 e\in R_0 e ∈ R 0 e e e ∥ e a e − a ∥ < δ / 2 \lVert eae-a\rVert<\delta/2 ∥ e a e − a ∥ < δ /2 δ / 2 \delta/2 δ /2 δ \delta δ R 0 R_0 R 0 ψ E \psi_E ψ E M d M_d M d Her ( e E ) ′ ′ \operatorname{Her}(e_E)^{\prime\prime} Her ( e E ) ′′ Her ( e E ) \operatorname{Her}(e_E) Her ( e E ) β \beta β t = β ( 1 ) t=\beta(1) t = β ( 1 )
x ⟼ max ( 1 , t ) − 1 / 2 β ( x ) max ( 1 , t ) − 1 / 2 . x\longmapsto\max(1,t)^{-1/2}\beta(x)\max(1,t)^{-1/2}. x ⟼ max ( 1 , t ) − 1/2 β ( x ) max ( 1 , t ) − 1/2 . The functional calculus is in the unitization. This replacement is completely positive and contractive, stays in Her ( e E ) \operatorname{Her}(e_E) Her ( e E )
H E : S / J E S ⟶ Her ( e E ) ⊆ Q H_E:S/J_E^S\longrightarrow\operatorname{Her}(e_E)\subseteq Q H E : S / J E S ⟶ Her ( e E ) ⊆ Q approximating (6.17) on the required tests.
The support element e E e_E e E Her ( e E ) \operatorname{Her}(e_E) Her ( e E ) i ∈ E i\in E i ∈ E I E I_E I E N E , i N_{E,i} N E , i [ e E ] ≤ N E , i [ b i ] [e_E]\leq N_{E,i}[b_i] [ e E ] ≤ N E , i [ b i ] N E , i d τ ( b i ) N_{E,i}d_\tau(b_i) N E , i d τ ( b i ) τ \tau τ i ∉ E i\notin E i ∈ / E s i s_i s i s i 2 s_i^2 s i 2
Put the sector maps in separate matrix blocks: H = ⨁ E H E H=\bigoplus_E H_E H = ⨁ E H E i ∉ E i\notin E i ∈ / E s i s_i s i s i 2 s_i^2 s i 2
d τ ( H ( s i ) ) , d τ ( H ( s i 2 ) ) ≤ ( ∑ E ∋ i N E , i ) d τ ( b i ) . d_\tau(H(s_i)),d_\tau(H(s_i^2))\leq\left(\sum_{E\ni i}N_{E,i}\right)d_\tau(b_i). d τ ( H ( s i )) , d τ ( H ( s i 2 )) ≤ ( E ∋ i ∑ N E , i ) d τ ( b i ) . The maps H E H_E H E D e D_e D e D 0 D_0 D 0 V j V_j V j ∗ * ∗ Q Q Q ∗ * ∗ D e D_e D e e E e_E e E y j 2 y_j^2 y j 2 π \pi π
Thus candidate vectors approximate the value of every finite functional obtained from a supposed separator, arbitrarily closely to the target value. This contradicts strict separation and proves the proposition.
Averaging and the lower rank bound
The preceding proposition gives finite convex combinations with arbitrarily small uniform errors on the caps and arbitrarily small norm errors in the algebra coordinates. The convex coefficients can be made rational: only finitely many bounded vectors are involved in any one combination. If their common denominator is N N N N N N Q Q Q H a v H_{\mathrm{av}} H av k = 1 , 2 k=1,2 k = 1 , 2
τ ( H a v ( s i k ) ) = ∑ α λ α τ ( H α ( s i k ) ) , τ ( H a v ( s i ) 2 ) = ∑ α λ α τ ( H α ( s i ) 2 ) . (67) \begin{aligned}
\tau(H_{\mathrm{av}}(s_i^k)) &= \sum_{\alpha}\lambda_\alpha\tau(H_\alpha(s_i^k)),\\
\tau(H_{\mathrm{av}}(s_i)^2) &= \sum_{\alpha}\lambda_\alpha\tau(H_\alpha(s_i)^2). \tag*{(67)}
\end{aligned} τ ( H av ( s i k )) τ ( H av ( s i ) 2 ) = α ∑ λ α τ ( H α ( s i k )) , = α ∑ λ α τ ( H α ( s i ) 2 ) . (67) Only the α \alpha α Q Q Q
Lemma 6.7 (A lower rank bound preserved by averaging). Fix lower data y , g , η y,g,\eta y , g , η (6.11) –(6.12) . If a finite rational average satisfies
∥ ∑ α λ α y v α ∗ g ( H α ( s ) ) v α y − y 2 ∥ < ε 2 / 2 , \left\|\sum_{\alpha}\lambda_\alpha yv_\alpha^*g(H_\alpha(s))v_\alpha y-y^2\right\|<\varepsilon^2/2, α ∑ λ α y v α ∗ g ( H α ( s )) v α y − y 2 < ε 2 /2 , then for every extended trace on Q Q Q
d τ ( ( y − ε ) + ) ≤ 2 ∥ y ∥ 2 ε 2 d τ ( ( H a v ( s ) − η ) + ) . (68) d_\tau((y-\varepsilon)_+) \le\frac{2\lVert y\rVert^2}{\varepsilon^2}d_\tau((H_{\mathrm{av}}(s)-\eta)_+). \tag*{(68)} d τ (( y − ε ) + ) ≤ ε 2 2 ∥ y ∥ 2 d τ (( H av ( s ) − η ) + ) . (68) The constant and the cut η \eta η
Proof. Write the displayed average as T T T p = 1 ( ε , ∞ ) ( y ) p=1_{(\varepsilon,\infty)}(y) p = 1 ( ε , ∞ ) ( y ) p T p ≥ ( ε 2 / 2 ) p pTp\ge(\varepsilon^2/2)p pT p ≥ ( ε 2 /2 ) p v α v_\alpha v α 0 ≤ g ( t ) ≤ 1 ( η , ∞ ) ( t ) 0\le g(t)\le1_{(\eta,\infty)}(t) 0 ≤ g ( t ) ≤ 1 ( η , ∞ ) ( t )
ε 2 2 τ ( p ) ≤ ∥ y ∥ 2 ∑ α λ α d τ ( ( H α ( s ) − η ) + ) . \frac{\varepsilon^2}{2}\tau(p)\le\lVert y\rVert^2\sum_{\alpha}\lambda_\alpha d_\tau((H_\alpha(s)-\eta)_+). 2 ε 2 τ ( p ) ≤ ∥ y ∥ 2 α ∑ λ α d τ (( H α ( s ) − η ) + ) . The sum is exactly the rank on the right of (68) , by the normalized averaging convention. Here is a justification when τ \tau τ g ( H α ( s ) ) g(H_\alpha(s)) g ( H α ( s )) ( y 2 − ε 2 / 2 ) + ≾ T (y^{2}-\varepsilon^{2}/2)_{+} \precsim T ( y 2 − ε 2 /2 ) + ≾ T T T T g ( H α ( s ) ) g(H_{\alpha}(s)) g ( H α ( s )) p p p ( y 2 − ε 2 ) + (y^{2}-\varepsilon^{2})_{+} ( y 2 − ε 2 ) + p p p p p p
The diagonal choice
Proof of Theorem 6.1. Fix the comparisons before forming averages. Fix the test family, its larger pre-cuts, its local controllers, and all lower data before choosing the sequence. For each nonzero f ⊗ a ∈ E f \otimes a \in\mathcal{E} f ⊗ a ∈ E ( δ , r ) (\delta,r) ( δ , r )
δ ↓ 0 , 0 < r < m ( { f > δ } ) , \delta\downarrow0,\qquad0<r<m(\{f>\delta\}), δ ↓ 0 , 0 < r < m ({ f > δ }) , and positives y y y r G [ ( a − δ ) + ] rG[(a-\delta)_{+}] r G [( a − δ ) + ] y y y Q ( U a ) Q(U_a) Q ( U a ) a a a G G G g g g η \eta η ε \varepsilon ε
At stage n n n n n n n n n Q Q Q h n : S → Q h_n:S\to Q h n : S → Q γ n ↓ 0 \gamma_n\downarrow0 γ n ↓ 0 i ≤ n i\le n i ≤ n
∣ τ ( h n ( s i ) ) − t i ( 1 ) ( τ ) ∣ ≤ γ n d τ ( b i ) , ∣ τ ( h n ( s i 2 ) ) − t i ( 2 ) ( τ ) ∣ ≤ γ n d τ ( b i ) , ∣ τ ( h n ( s i ) 2 ) − t i ( 2 ) ( τ ) ∣ ≤ γ n d τ ( b i ) (69) \begin{aligned}
|\tau(h_n(s_i)) - t_i^{(1)}(\tau)| &\le\gamma_n d_\tau(b_i),\\
|\tau(h_n(s_i^2)) - t_i^{(2)}(\tau)| &\le\gamma_n d_\tau(b_i),\\
|\tau(h_n(s_i)^2) - t_i^{(2)}(\tau)| &\le\gamma_n d_\tau(b_i) \tag*{(69)}
\end{aligned} ∣ τ ( h n ( s i )) − t i ( 1 ) ( τ ) ∣ ∣ τ ( h n ( s i 2 )) − t i ( 2 ) ( τ ) ∣ ∣ τ ( h n ( s i ) 2 ) − t i ( 2 ) ( τ ) ∣ ≤ γ n d τ ( b i ) , ≤ γ n d τ ( b i ) , ≤ γ n d τ ( b i ) (69) whenever d τ ( b i ) < ∞ d_\tau(b_i)<\infty d τ ( b i ) < ∞ K i K_i K i τ \tau τ ε 2 / 2 \varepsilon^2/2 ε 2 /2 n n n
Localize the multiplicative defects. The coordinate maps define a completely positive contraction h : S → D h:S\to D h : S → D Δ n , i = h n ( s i 2 ) − h n ( s i ) 2 ≥ 0 \Delta_{n,i}=h_n(s_i^2)-h_n(s_i)^2\ge0 Δ n , i = h n ( s i 2 ) − h n ( s i ) 2 ≥ 0 t > 0 t>0 t > 0
d τ ( ( Δ n , i − t ) + ) ≤ 2 γ n t d τ ( b i ) . (70) d_\tau((\Delta_{n,i}-t)_{+})\le\frac{2\gamma_n}{t}d_\tau(b_i). \tag*{(70)} d τ (( Δ n , i − t ) + ) ≤ t 2 γ n d τ ( b i ) . (70) For zero controller rank this follows by the same rescaling argument; for infinite controller rank the upper bound is automatic. Thus h ( s i 2 ) − h ( s i ) 2 ∈ J ( U i ) h(s_i^2)-h(s_i)^2\in J(U_i) h ( s i 2 ) − h ( s i ) 2 ∈ J ( U i ) D / J ( U i ) D/J(U_i) D / J ( U i ) s i s_i s i s i s_i s i J ( U i ) J(U_i) J ( U i ) z ∈ S z\in S z ∈ S
For an arbitrary open U U U U U U S ( U ) S(U) S ( U ) J ( V ) ⊆ J ( U ) J(V)\subseteq J(U) J ( V ) ⊆ J ( U ) s ∈ S ( U ) s\in S(U) s ∈ S ( U )
Recover the square moments. Let ρ \rho ρ τ n \tau_n τ n σ = ρ ∣ Q \sigma=\rho|_Q σ = ρ ∣ Q Q ( U i ) Q(U_i) Q ( U i ) d τ n ( b i ) d_{\tau_n}(b_i) d τ n ( b i ) σ \sigma σ F F F
lim ω t i ( k ) ( τ n ) = t i ( k ) ( σ ) , k = 1 , 2. \lim_{\omega}t_i^{(k)}(\tau_n)=t_i^{(k)}(\sigma),\qquad k=1,2. ω lim t i ( k ) ( τ n ) = t i ( k ) ( σ ) , k = 1 , 2. Consequently (6.20) gives convergence of the uncut square traces to t i ( 2 ) ( σ ) t_i^{(2)}(\sigma) t i ( 2 ) ( σ ) lim ω τ n ( h n ( s i ) ) < ∞ \lim_{\omega}\tau_n(h_n(s_i))<\infty lim ω τ n ( h n ( s i )) < ∞
There is no loss of square mass on regularization. For x ≥ 0 x\geq0 x ≥ 0 t > 0 t>0 t > 0
0 ≤ τ ( x 2 ) − τ ( ( x 2 − t ) + ) ≤ t τ ( x ) (71) 0\leq\tau(x^2)-\tau((x^2-t)_+)\leq\sqrt{t}\tau(x) \tag*{(71)} 0 ≤ τ ( x 2 ) − τ (( x 2 − t ) + ) ≤ t τ ( x ) (71) whenever the right side is finite. Apply this with x = h n ( s i ) x=h_n(s_i) x = h n ( s i ) t ↓ 0 t\downarrow0 t ↓ 0 ρ \rho ρ (71) yield exactly (6.4).
Retain the lower ideal support. Fix one lower datum ( y , ε , η ) (y,\varepsilon,\eta) ( y , ε , η ) s = f ⊗ a s=f\otimes a s = f ⊗ a n n n
d τ ( ( y − ε ) + ) ≤ C d τ ( ( h n ( s ) − η ) + ) for every τ ∈ T ( Q ) , C = 2 ∥ y ∥ 2 / ε 2 . d_\tau((y-\varepsilon)_+)\leq C d_\tau((h_n(s)-\eta)_+)\quad\text{for every }\tau\in T(Q),\qquad C=2\lVert y\rVert^2/\varepsilon^2. d τ (( y − ε ) + ) ≤ C d τ (( h n ( s ) − η ) + ) for every τ ∈ T ( Q ) , C = 2 ∥ y ∥ 2 / ε 2 . The order determination in Theorem 2.1 implies
[ ( y − ε ) + ] ≤ ⌈ C ⌉ [ ( h n ( s ) − η ) + ] . [(y-\varepsilon)_+]\leq\lceil C\rceil[(h_n(s)-\eta)_+]. [( y − ε ) + ] ≤ ⌈ C ⌉ [( h n ( s ) − η ) + ] . If ( y − ε ) + = 0 (y-\varepsilon)_+=0 ( y − ε ) + = 0 C > 0 C>0 C > 0 N = ⌈ C ⌉ ≥ 1 N=\lceil C\rceil\geq1 N = ⌈ C ⌉ ≥ 1 w n w_n w n
∥ w n ∗ diag N ( h n ( s ) ) w n − ( y − ε ) + ∥ < 1 / n , ∥ w n ∥ 2 ≤ ∥ y ∥ 2 + 1 / n η . \lVert w_n^*\operatorname{diag}_N(h_n(s))w_n-(y-\varepsilon)_+\rVert<1/n,\qquad\lVert w_n\rVert^2\leq\frac{\lVert y\rVert^2+1/n}{\eta}. ∥ w n ∗ diag N ( h n ( s )) w n − ( y − ε ) + ∥ < 1/ n , ∥ w n ∥ 2 ≤ η ∥ y ∥ 2 + 1/ n . This bound is independent even of N N N ( y − ε ) + (y-\varepsilon)_+ ( y − ε ) + Ideal D ( h ( s ) ) \operatorname{Ideal}_D(h(s)) Ideal D ( h ( s )) y y y y y y Q ( U a ) Q(U_a) Q ( U a ) □ \square □
Correction of the cone models
Throughout this section,
S = C 0 ( ( 0 , 1 ] ) ⊗ P , D = Q ∞ . S=C_0((0,1])\otimes P,\qquad D=Q_\infty. S = C 0 (( 0 , 1 ]) ⊗ P , D = Q ∞ . We first correct the multiplicative defects of the map in Theorem 6.1. The correction takes place on a hereditary support containing the defects and a small cone representation. Its ideal estimates are stronger than equivariance for the projected labels. This strength is what permits the cone homotopy to be used with the finer labels of Prim ( S ) \operatorname{Prim}(S) Prim ( S )
The error support and its ideal estimates
Lemma 7.1 (A hereditary support for the defects). Let h : S → D h:S\to D h : S → D U ⊂ X U\subset X U ⊂ X
h ( s z ) − h ( s ) h ( z ) , h ( z s ) − h ( z ) h ( s ) ∈ J ( U ) ( s ∈ S ( U ) , z ∈ S ) . h(sz)-h(s)h(z),\qquad h(zs)-h(z)h(s)\in J(U)\qquad(s\in S(U),\ z\in S). h ( sz ) − h ( s ) h ( z ) , h ( z s ) − h ( z ) h ( s ) ∈ J ( U ) ( s ∈ S ( U ) , z ∈ S ) . Let E = C ∗ ( h ( S ) ) E=C^*(h(S)) E = C ∗ ( h ( S )) I ◃ E I\triangleleft E I ◃ E h ( a b ) − h ( a ) h ( b ) h(ab)-h(a)h(b) h ( ab ) − h ( a ) h ( b ) a , b ∈ S a,b\in S a , b ∈ S
h ( S ( U ) ) I + I h ( S ( U ) ) ⊂ J ( U ) . h(S(U))I+Ih(S(U))\subset J(U). h ( S ( U )) I + I h ( S ( U )) ⊂ J ( U ) . There is a locally small cone homomorphism θ : S → D \theta:S\to D θ : S → D h h h θ \theta θ σ \sigma σ H ⊂ M 2 ( D ) H\subset M_2(D) H ⊂ M 2 ( D ) h h h H H H H H H θ ( S ) ⊂ H \theta(S)\subset H θ ( S ) ⊂ H x ∈ S + x\in S_+ x ∈ S +
h ( S ( U x ) ) H + H h ( S ( U x ) ) ⊂ H x , H x : = Ideal H ( θ ( x ) ) . (72) h(S(U_x))H+Hh(S(U_x))\subset H_x,\qquad H_x:=\operatorname{Ideal}_H(\theta(x)). \tag*{(72)} h ( S ( U x )) H + H h ( S ( U x )) ⊂ H x , H x := Ideal H ( θ ( x )) . (72) Here θ ( S ( U ) ) ⊂ J ( U ) \theta(S(U))\subset J(U) θ ( S ( U )) ⊂ J ( U )
Proof. Fix U U U D / J ( U ) D/J(U) D / J ( U ) s ∈ S ( U ) s\in S(U) s ∈ S ( U ) a , b ∈ S a,b\in S a , b ∈ S
h ( s ) ‾ ( h ( a b ) ‾ − h ( a ) ‾ h ( b ) ‾ ) = h ( s a b ) ‾ − h ( s a ) ‾ h ( b ) ‾ = 0. \overline{h(s)}\bigl(\overline{h(ab)}-\overline{h(a)}\overline{h(b)}\bigr)=\overline{h(sab)}-\overline{h(sa)}\overline{h(b)}=0. h ( s ) ( h ( ab ) − h ( a ) h ( b ) ) = h ( s ab ) − h ( s a ) h ( b ) = 0. The analogous right product is zero. More generally a word in h ( S ) ‾ \overline{h(S)} h ( S ) h ( s ) ‾ \overline{h(s)} h ( s ) S ( U ) S(U) S ( U ) E E E (7.2) for the whole generated ideal I I I
The algebra E E E I I I ( s , b ) ↦ h ( s ) b (s,b)\mapsto h(s)b ( s , b ) ↦ h ( s ) b b ∈ I b\in I b ∈ I θ \theta θ x ∈ S + x\in S_+ x ∈ S +
h ( S ( U x ) ) I + I h ( S ( U x ) ) ⊂ Ideal M 2 ( D ) ( θ ( x ) ) . (73) h(S(U_x))I+Ih(S(U_x))\subset\operatorname{Ideal}_{M_2(D)}(\theta(x)). \tag*{(73)} h ( S ( U x )) I + I h ( S ( U x )) ⊂ Ideal M 2 ( D ) ( θ ( x )) . (73) The first-corner and second-corner placements are understood in this formula; the ideal on the right is the ambient matrix ideal.
Choose strictly positive elements e ∈ I + e\in I_+ e ∈ I + k ∈ θ ( S ) + k\in\theta(S)_+ k ∈ θ ( S ) +
H = Her M 2 ( D ) ( e ⊕ k ) . (74) H=\operatorname{Her}_{M_2(D)}(e\oplus k). \tag*{(74)} H = Her M 2 ( D ) ( e ⊕ k ) . (74) It is σ \sigma σ e ⊕ k e\oplus k e ⊕ k H H H H H H I ◃ E I\triangleleft E I ◃ E h ( a ) I + I h ( a ) ⊂ I h(a)I+Ih(a)\subset I h ( a ) I + I h ( a ) ⊂ I a ∈ S a\in S a ∈ S I I I (74) : on the dense linear span of products
f ( e ⊕ k ) z g ( e ⊕ k ) , z ∈ M 2 ( D ) , f(e\oplus k)z g(e\oplus k),\qquad z\in M_2(D), f ( e ⊕ k ) z g ( e ⊕ k ) , z ∈ M 2 ( D ) , left multiplication by h ( a ) h(a) h ( a ) I I I H H H I ⊂ H I\subset H I ⊂ H
For a ∈ S ( U x ) a\in S(U_x) a ∈ S ( U x ) (73) , puts h ( a ) H h(a)H h ( a ) H H h ( a ) Hh(a) H h ( a ) H ∩ Ideal M 2 ( D ) ( θ ( x ) ) H\cap\operatorname{Ideal}_{M_2(D)}(\theta(x)) H ∩ Ideal M 2 ( D ) ( θ ( x ))
H ∩ Ideal M 2 ( D ) ( θ ( x ) ) = Ideal H ( θ ( x ) ) . H\cap\operatorname{Ideal}_{M_2(D)}(\theta(x))=\operatorname{Ideal}_H(\theta(x)). H ∩ Ideal M 2 ( D ) ( θ ( x )) = Ideal H ( θ ( x )) . This proves (72) ; equivalently, one may apply Lemma 5.9. All matrix placements preserve the local smallness of θ \theta θ
We use the right Hilbert H H H L H ( E ) \mathcal{L}_{H}(\mathcal{E}) L H ( E ) K H ( E ) \mathcal{K}_{H}(\mathcal{E}) K H ( E ) H H H E \mathcal{E} E L ◃ H L \triangleleft H L ◃ H E L \mathcal{E}L E L { ξ b : ξ ∈ E , b ∈ L } \{\xi b:\xi\in\mathcal{E},b\in L\} { ξ b : ξ ∈ E , b ∈ L }
Lemma 7.2 (A dilation compatible with fine ideal labels). Let d : S → L H ( H ) = M ( H ) d:S\to\mathcal{L}_{H}(H)=M(H) d : S → L H ( H ) = M ( H ) h h h H H H E ′ \mathcal{E}' E ′
Π : S ⟶ L H ( H ⊕ E ′ ) , Π ( a ) = ( d ( a ) C ( a ) B ( a ) d ′ ( a ) ) , (75) \Pi:S\longrightarrow\mathcal{L}_{H}(H\oplus\mathcal{E}'),\qquad
\Pi(a)=
\begin{pmatrix}
d(a) & C(a)\\
B(a) & d'(a)
\end{pmatrix}, \tag*{(75)} Π : S ⟶ L H ( H ⊕ E ′ ) , Π ( a ) = ( d ( a ) B ( a ) C ( a ) d ′ ( a ) ) , (75) whose off-diagonal entries are compact. The maps d , d ′ d,d' d , d ′ H H H E ′ \mathcal{E}' E ′
Give S S S Prim ( S ) \operatorname{Prim}(S) Prim ( S ) H H H
H ( O ) = Ideal H ( θ ( S ( O ) ) ) ( O ⊂ Prim ( S ) open ) . (76) H(O)=\operatorname{Ideal}_{H}(\theta(S(O)))\qquad(O\subset\operatorname{Prim}(S)\ \text{open}). \tag*{(76)} H ( O ) = Ideal H ( θ ( S ( O ))) ( O ⊂ Prim ( S ) open ) . (76) Let α t \alpha_t α t
( α t a ) ( r ) = a ( t r ) , 0 ≤ t ≤ 1 , (\alpha_ta)(r)=a(tr),\qquad0\leq t\leq1, ( α t a ) ( r ) = a ( t r ) , 0 ≤ t ≤ 1 , where a ( 0 ) = 0 a(0)=0 a ( 0 ) = 0 Π t : = Π α t \Pi_t:=\Pi\alpha_t Π t := Π α t Prim ( S ) \operatorname{Prim}(S) Prim ( S ) (76) . The same is true of its diagonal compressions d t , d t ′ d_t,d'_t d t , d t ′
Proof. Use the Hilbert-module Stinespring construction in [23 ] , proof of Theorem 3, pp. 140–141, applied to the unital extension d + d^+ d + d d d S S S H H H V : H → E V:H\to\mathcal{E} V : H → E V ξ = 1 ⊗ ξ V\xi=1\otimes\xi V ξ = 1 ⊗ ξ V ∗ ( b ⊗ η ) = d + ( b ) η V^*(b\otimes\eta)=d^+(b)\eta V ∗ ( b ⊗ η ) = d + ( b ) η V ∗ Π ( a ) V = d ( a ) V^*\Pi(a)V=d(a) V ∗ Π ( a ) V = d ( a ) H H H E = V H ⊕ E ′ \mathcal{E}=VH\oplus\mathcal{E}' E = V H ⊕ E ′ V H VH V H H H H
For B ( a ) = ( 1 − V V ∗ ) Π ( a ) V B(a)=(1-VV^*)\Pi(a)V B ( a ) = ( 1 − V V ∗ ) Π ( a ) V
B ( a ) ∗ B ( a ) = d ( a ∗ a ) − d ( a ) ∗ d ( a ) ∈ H = K H ( H ) . (77) B(a)^*B(a)=d(a^*a)-d(a)^*d(a)\in H=\mathcal{K}_{H}(H). \tag*{(77)} B ( a ) ∗ B ( a ) = d ( a ∗ a ) − d ( a ) ∗ d ( a ) ∈ H = K H ( H ) . (77) An adjointable operator whose square modulus is compact is compact, by functional calculus in the compact-operator ideal. Thus B ( a ) B(a) B ( a ) C ( a ) = B ( a ∗ ) ∗ C(a)=B(a^*)^* C ( a ) = B ( a ∗ ) ∗ [23 ] , Theorem 2 of E ′ \mathcal{E}' E ′ V H VH V H
We check the ideal assertion before using the homotopy. For a ∈ S ( U x ) a\in S(U_x) a ∈ S ( U x )
⟨ b ⊗ ξ , Π ( a ) ( c ⊗ η ) ⟩ = ξ ∗ d ( b ∗ a c ) η ∈ H x , \langle b\otimes\xi,\Pi(a)(c\otimes\eta)\rangle=\xi^*d(b^*ac)\eta\in H_x, ⟨ b ⊗ ξ , Π ( a ) ( c ⊗ η )⟩ = ξ ∗ d ( b ∗ a c ) η ∈ H x , where b , c b,c b , c b ∗ a c ∈ S ( U x ) b^*ac\in S(U_x) b ∗ a c ∈ S ( U x ) (72) . It extends to all vectors. Passing to the quotient module by E H x \mathcal{E}H_x E H x Π ( a ) E ⊂ E H x \Pi(a)\mathcal{E}\subset\mathcal{E}H_x Π ( a ) E ⊂ E H x
Now take a ∈ S ( O ) a\in S(O) a ∈ S ( O ) x = ∣ a ∣ x=|a| x = ∣ a ∣
α t ( a ) ∈ S ( U x ) ( 0 ≤ t ≤ 1 ) . (78) \alpha_t(a)\in S(U_x)\qquad(0\leq t\leq1). \tag*{(78)} α t ( a ) ∈ S ( U x ) ( 0 ≤ t ≤ 1 ) . (78) Indeed a point quotient in the P P P a a a a a a x ∈ S ( O ) x\in S(O) x ∈ S ( O ) H x ⊂ H ( O ) H_x\subset H(O) H x ⊂ H ( O ) (7.9) applied to α t ( a ) \alpha_t(a) α t ( a ) Π t \Pi_t Π t
This argument does not assert that α t \alpha_t α t S ( O ) S(O) S ( O ) S ( U x ) S(U_x) S ( U x )
For later use, if U ⊂ X U \subset X U ⊂ X O U = ( 0 , 1 ] × U O_U=(0,1]\times U O U = ( 0 , 1 ] × U
H ( O U ) ⊂ H ∩ M 2 ( J ( U ) ) . H(O_U)\subset H\cap M_2(J(U)). H ( O U ) ⊂ H ∩ M 2 ( J ( U )) . This follows from the ideal property of J ( U ) J(U) J ( U ) θ \theta θ S ( O U ) S(O_U) S ( O U ) E H ( O U ) \mathcal{E}H(O_U) E H ( O U ) E H ( O U ) \mathcal{E}H(O_U) E H ( O U ) K H ( E H ( O U ) ) \mathcal{K}_H(\mathcal{E}H(O_U)) K H ( E H ( O U )) J ( U ) J(U) J ( U )
A finite correction on the original module
The dilation now respects every fine ideal label, including along cone scaling. We will compare nearby times in a finite subdivision of that homotopy. Absorption makes a finite chain of dilations available as a compact perturbation of a repeat of θ \theta θ
We then truncate only the compact perturbation. This produces a finite matrix correction with the original h h h
Lemma 7.3 (A finite correction retaining the original corner). Under the hypotheses of Lemma 7.1, let F \mathcal{F} F S S S η > 0 \eta>0 η > 0 N N N ∗ \ast ∗ L : S → M N + 1 ( M 2 ( D ) ) L:S\to M_{N+1}(M_2(D)) L : S → M N + 1 ( M 2 ( D )) K N : S → M N + 1 ( H ) K_N:S\to M_{N+1}(H) K N : S → M N + 1 ( H )
L ( a ) = h ( a ) ⊕ θ ( a ) ⊕ N + K N ( a ) , L(a)=h(a)\oplus\theta(a)^{\oplus N}+K_N(a), L ( a ) = h ( a ) ⊕ θ ( a ) ⊕ N + K N ( a ) , and, for a , b ∈ F a,b\in\mathcal{F} a , b ∈ F
∥ L ( a ) ∥ ≤ ∥ a ∥ + η , ∥ L ( a b ) − L ( a ) L ( b ) ∥ ≤ 3 η + 2 η 2 . (79) \lVert L(a)\rVert\le\lVert a\rVert+\eta,\qquad\lVert L(ab)-L(a)L(b)\rVert\le3\eta+2\eta^2. \tag*{(79)} ∥ L ( a )∥ ≤ ∥ a ∥ + η , ∥ L ( ab ) − L ( a ) L ( b )∥ ≤ 3 η + 2 η 2 . (79) For every open U ⊂ X U\subset X U ⊂ X
K N ( S ( U ) ) ⊂ M N + 1 ( H ∩ M 2 ( J ( U ) ) ) . K_N(S(U))\subset M_{N+1}(H\cap M_2(J(U))). K N ( S ( U )) ⊂ M N + 1 ( H ∩ M 2 ( J ( U ))) . The first summand in (7.12) is the original h h h
Proof. Absorb a finite chain of dilations. Include F ∗ \mathcal{F}^* F ∗ F \mathcal{F} F t ↦ α t ( a ) t\mapsto\alpha_t(a) t ↦ α t ( a )
0 = t 0 < t 1 < ⋯ < t k = 1 0=t_0<t_1<\cdots<t_k=1 0 = t 0 < t 1 < ⋯ < t k = 1 so that adjacent values differ in norm by less than η \eta η d i = d t i d_i=d_{t_i} d i = d t i d i ′ = d t i ′ d_i'=d_{t_i}' d i ′ = d t i ′ Π i = Π t i \Pi_i=\Pi_{t_i} Π i = Π t i H H H
For the absorption application put θ ^ ( a ) = θ ( a ) ⊗ e 11 ∈ H ⊗ K \hat{\theta}(a)=\theta(a)\otimes e_{11}\in H\otimes K θ ^ ( a ) = θ ( a ) ⊗ e 11 ∈ H ⊗ K H ( O ) ⊗ K H(O)\otimes K H ( O ) ⊗ K O a O_a O a a ∈ S + a\in S_+ a ∈ S + S ( O a ) = Ideal S ( a ) S(O_a)=\operatorname{Ideal}_S(a) S ( O a ) = Ideal S ( a )
Ideal H ( θ ( S ( O a ) ) ) = Ideal H ( θ ( a ) ) . \operatorname{Ideal}_H(\theta(S(O_a)))=\operatorname{Ideal}_H(\theta(a)). Ideal H ( θ ( S ( O a ))) = Ideal H ( θ ( a )) . Thus θ ^ \hat{\theta} θ ^ Θ \Theta Θ θ ^ \hat{\theta} θ ^ Π i \Pi_i Π i H H H
Θ ( a ) = diag ( θ ( a ) , 0 , θ ( a ) , 0 , … ) . \Theta(a)=\operatorname{diag}(\theta(a),0,\theta(a),0,\ldots). Θ ( a ) = diag ( θ ( a ) , 0 , θ ( a ) , 0 , … ) . Indeed each repeated copy of θ ^ \hat{\theta} θ ^ θ \theta θ H H H (7.15) by a fixed permutation. We retain every zero slot throughout absorption and finite compression. The bare diagonal repeat with all these zero slots removed need not absorb a zero representation. The hypotheses hold: the tightly labeled source is separable, exact and lower semicontinuous; H ⊗ K H\otimes K H ⊗ K σ \sigma σ
Θ ⊕ Π 1 ⊕ ⋯ ⊕ Π k \Theta\oplus\Pi_1\oplus\cdots\oplus\Pi_k Θ ⊕ Π 1 ⊕ ⋯ ⊕ Π k so that its difference from the original repeat is compact on every input. More precisely, absorption in [16 ] gives a repeat-side module unitary U U U
U ∗ ( Θ ⊕ ⨁ i Π i ) ( a ) U − Θ ( a ) ∈ K H ( H ∞ ) ( a ∈ S ) ; U^*(\Theta\oplus\bigoplus_i\Pi_i)(a)U-\Theta(a)\in K_H(H^\infty)\qquad(a\in S); U ∗ ( Θ ⊕ i ⨁ Π i ) ( a ) U − Θ ( a ) ∈ K H ( H ∞ ) ( a ∈ S ) ; the compact membership holds for every input for each chosen unitary, in addition to the finite-test norm approximation. Only this compact-membership conclusion from absorption is needed below. Its finite-test approximation error does not enter our bound: after the unitary identification we keep the absorbed representation itself. All these identifications act on the repeat side; the initial module carrying d = d k d=d_k d = d k
Shift the first summands along the chain. Denote the initial copy of H H H H 0 H_0 H 0 Π i \Pi_i Π i H i H_i H i E i \mathcal{E}_i E i Π i \Pi_i Π i
d k ∣ H 0 ⊕ ⨁ i = 1 k ( d i ∣ H i ⊕ d i ′ ∣ E i ) . (80) d_k|_{H_0}\oplus\bigoplus_{i=1}^{k}\left(d_i|_{H_i}\oplus d'_i|_{\mathcal{E}_i}\right). \tag*{(80)} d k ∣ H 0 ⊕ i = 1 ⨁ k ( d i ∣ H i ⊕ d i ′ ∣ E i ) . (80) Pair H 0 H_0 H 0 E k \mathcal{E}_k E k i < k i<k i < k H i + 1 H_{i+1} H i + 1 E i \mathcal{E}_i E i H 1 H_1 H 1 Π i \Pi_i Π i E i \mathcal{E}_i E i
T ′ = Θ ⊕ d 1 ⊕ ⨁ i = 1 k − 1 [ Π i + ( d i + 1 − d i 0 0 0 ) ] ⊕ Π k . T'=\Theta\oplus d_1\oplus\bigoplus_{i=1}^{k-1}\left[\Pi_i+\begin{pmatrix}d_{i+1}-d_i&0\\0&0\end{pmatrix}\right]\oplus\Pi_k. T ′ = Θ ⊕ d 1 ⊕ i = 1 ⨁ k − 1 [ Π i + ( d i + 1 − d i 0 0 0 ) ] ⊕ Π k . It is a compact perturbation of the map in (80) , including its repeat: the diagonal entries were only permuted, and the restored entries are compact. The representation
R ′ = Θ ⊕ 0 ⊕ ⨁ i = 1 k − 1 Π i ⊕ Π k R'=\Theta\oplus0\oplus\bigoplus_{i=1}^{k-1}\Pi_i\oplus\Pi_k R ′ = Θ ⊕ 0 ⊕ i = 1 ⨁ k − 1 Π i ⊕ Π k is within η \eta η T ′ T' T ′ ∥ d 1 ( a ) ∥ ≤ ∣ α 1 ( a ) ∣ < η \lVert d_1(a)\rVert\leq\lvert\alpha_1(a)\rvert<\eta ∥ d 1 ( a )∥ ≤ ∣ α 1 ( a )∣ < η α 0 = 0 \alpha_0=0 α 0 = 0 (7.17) has norm less than η \eta η d i + 1 ( a ) − d i ( a ) d_{i+1}(a)-d_i(a) d i + 1 ( a ) − d i ( a )
Pull both maps back through the block permutation and the absorption identification to the original module L = H ⊕ H ⊕ ∞ \mathcal{L}=H\oplus H^{\oplus\infty} L = H ⊕ H ⊕ ∞
Writing T 0 = d ⊕ Θ T_0=d\oplus\Theta T 0 = d ⊕ Θ
T = T 0 + K , K ( a ) ∈ K H ( L ) , ∥ T ( a ) − R ( a ) ∥ < η (81) T=T_0+K,\qquad K(a)\in\mathcal{K}_H(\mathcal{L}),\qquad\lVert T(a)-R(a)\rVert<\eta \tag*{(81)} T = T 0 + K , K ( a ) ∈ K H ( L ) , ∥ T ( a ) − R ( a )∥ < η (81) on the enlarged list, where R R R ∗ * ∗ K K K a , b ∈ F a,b\in\mathcal{F} a , b ∈ F R R R
∥ T ( a ) ∥ ≤ ∥ a ∥ + η , ∥ T ( a b ) − T ( a ) T ( b ) ∥ ≤ 3 η + η 2 . (82) \lVert T(a)\rVert\leq\lVert a\rVert+\eta,\qquad\lVert T(ab)-T(a)T(b)\rVert\leq3\eta+\eta^2. \tag*{(82)} ∥ T ( a )∥ ≤ ∥ a ∥ + η , ∥ T ( ab ) − T ( a ) T ( b )∥ ≤ 3 η + η 2 . (82) Compress the compact correction. Let E N E_N E N H H H N N N original diagonal repeat (7.15), including both active and zero slots. It commutes with T 0 ( a ) T_0(a) T 0 ( a ) a a a E N E_N E N N N N K ( a ) K(a) K ( a ) η \eta η
E N T ( a b ) E N − E N T ( a ) E N T ( b ) E N = E N ( T ( a b ) − T ( a ) T ( b ) ) E N + E N K ( a ) ( 1 − E N ) K ( b ) E N . (83) \begin{aligned}
E_NT(ab)E_N-E_NT(a)E_NT(b)E_N
&=E_N\bigl(T(ab)-T(a)T(b)\bigr)E_N\\
&\quad+E_NK(a)(1-E_N)K(b)E_N. \tag*{(83)}
\end{aligned} E N T ( ab ) E N − E N T ( a ) E N T ( b ) E N = E N ( T ( ab ) − T ( a ) T ( b ) ) E N + E N K ( a ) ( 1 − E N ) K ( b ) E N . (83) The second term has norm at most η 2 \eta^2 η 2
The entries of E N K ( a ) E N E_NK(a)E_N E N K ( a ) E N H H H h h h θ \theta θ Z N ( a ) Z_N(a) Z N ( a ) θ ( a ) \theta(a) θ ( a )
K N ( a ) = E N K ( a ) E N − Z N ( a ) . K_N(a)=E_NK(a)E_N-Z_N(a). K N ( a ) = E N K ( a ) E N − Z N ( a ) . Replacing each zero baseline slot by θ ( a ) \theta(a) θ ( a ) L L L Z N Z_N Z N ∗ * ∗ H H H J ( U ) J(U) J ( U ) θ \theta θ K N K_N K N T 0 T_0 T 0
Return to the ambient sequence algebra. To transfer the multiplier estimates, let p H p_H p H H H H M 2 ( D ) ∗ ∗ M_2(D)^{**} M 2 ( D ) ∗∗ h ( a ) h(a) h ( a ) p H p_H p H θ \theta θ p H p_H p H h h h I ⊂ H I\subset H I ⊂ H K K K
Passage to a corrected sequence homomorphism
Proposition 7.4 (Cone correction). Suppose h : S → D h:S\to D h : S → D λ : S → D \lambda:S\to D λ : S → D
( λ − h p a d ) ( S ( U ) ) ⊆ J ( U ) ( U ⊂ X open ) . (84) (\lambda-h^{\mathrm{pad}})(S(U))\subseteq J(U)\qquad(U\subset X\ \text{open}). \tag*{(84)} ( λ − h pad ) ( S ( U )) ⊆ J ( U ) ( U ⊂ X open ) . (84) The padding can be chosen to fix the constant copy of Q Q Q h h h
Proof. If the defect ideal is zero, h h h S S S h h h [4 ] ; fix one such lift. Fix a countable rational ∗ \ast ∗ S S S S ( V ) S(V) S ( V ) η j ↓ 0 \eta_j \downarrow0 η j ↓ 0 j j j j j j η j \eta_j η j L j L_j L j N j N_j N j ∥ L j ( a ) ∥ ≤ ∥ a ∥ + η j \lVert L_j(a)\rVert\le\lVert a\rVert+ \eta_j ∥ L j ( a )∥ ≤ ∥ a ∥ + η j a a a
Write h p a d , j = h ⊕ 0 ⊕ N j h^{\mathrm{pad},j}=h\oplus0^{\oplus N_j} h pad , j = h ⊕ 0 ⊕ N j L j L_j L j N j N_j N j θ \theta θ J ( V ) J(V) J ( V ) S ( V ) S(V) S ( V ) a a a
∥ L j ( a ) − h p a d , j ( a ) ∥ ≤ 2 ∥ a ∥ + η j \lVert L_j(a)-h^{\mathrm{pad},j}(a)\rVert\le2\lVert a\rVert+\eta_j ∥ L j ( a ) − h pad , j ( a )∥ ≤ 2 ∥ a ∥ + η j for all sufficiently large j j j L j L_j L j
d τ ( ( ∣ z j , n ∣ − ε ) + ) ≤ r j , n d τ ( c j ) , r j , n ⟶ 0 ( n → ∞ ) , (85) d_\tau\left((\lvert z_{j,n}\rvert-\varepsilon)_+\right)\le r_{j,n}d_\tau(c_j),\qquad r_{j,n}\longrightarrow0\quad(n\to\infty), \tag*{(85)} d τ ( (∣ z j , n ∣ − ε ) + ) ≤ r j , n d τ ( c j ) , r j , n ⟶ 0 ( n → ∞ ) , (85) where c j c_j c j Q ( V ) Q(V) Q ( V ) τ \tau τ
For each scheduled pair V ⋐ U V\Subset U V ⋐ U b V , U ∈ Q ( U ) b_{V,U}\in Q(U) b V , U ∈ Q ( U ) Q ( V ) Q(V) Q ( V ) [ c j ] ≤ M j [ b V , U ] [c_j]\le M_j[b_{V,U}] [ c j ] ≤ M j [ b V , U ] M j M_j M j j j j M j r j , n ≤ 1 / j M_jr_{j,n}\le1/j M j r j , n ≤ 1/ j M j M_j M j N j N_j N j j j j
Choose coordinate representatives linearly on each finite rational span, preserving the representatives already needed for its displayed algebraic tests. The preceding eventual estimates allow successive rows to be joined. Matrix folding in Lemma 2.4 is applied only after these choices. Its corner identification is adjusted to be within 1 / j 1/j 1/ j j j j Q Q Q Q Q Q h h h h p a d h^{\mathrm{pad}} h pad
The joined corrected representatives define a contractive, linear, ∗ \ast ∗ ∗ \ast ∗ λ : S → D \lambda:S\to D λ : S → D b V , U b_{V,U} b V , U h p a d h^{\mathrm{pad}} h pad J ( U ) J(U) J ( U ) S ( V ) S(V) S ( V ) S ( V ) S(V) S ( V ) V ⋐ U V\Subset U V ⋐ U S ( U ) S(U) S ( U ) J ( U ) J(U) J ( U ) U U U
For completeness, if ι n : Q → Q \iota_n:Q\to Q ι n : Q → Q ι : D → D \iota:D\to D ι : D → D ι ∣ Q = id Q \iota|_Q=\operatorname{id}_Q ι ∣ Q = id Q h p a d = ι ∘ h h^{\mathrm{pad}}=\iota\circ h h pad = ι ∘ h ι \iota ι τ n ι n \tau_n\iota_n τ n ι n σ \sigma σ Q Q Q h h h ι \iota ι
We record the trace determination used at the last step. A positive element compactly Cuntz supported in the densely finite ideal of a lower-semicontinuous trace has finite rank. Indeed that ideal is generated by finite-trace positives; their positive cuts have finite rank. Compact containment dominates the given element by finitely many such cuts. Conversely, if an ideal is generated by finite-rank positives, the trace is densely finite on it, since their hereditary algebras consist of finite-trace elements.
Lemma 7.5 (Determination on a finite ideal). Let V ⊂ X V \subset X V ⊂ X S ( V ) S(V) S ( V ) S ( V ) S(V) S ( V ) S ( V ) S(V) S ( V )
Proof. Write ω 1 , ω 2 \omega_1,\omega_2 ω 1 , ω 2 ω \omega ω s s s ε > 0 \varepsilon>0 ε > 0
d ω ( ( s − ε ) + ) ≤ ε − 2 ω ( s 2 ) . d_\omega((s-\varepsilon)_+) \le\varepsilon^{-2}\omega(s^2). d ω (( s − ε ) + ) ≤ ε − 2 ω ( s 2 ) . The preceding observation proves dense finiteness for both traces. Fix s s s g g g g ( t ) / t g(t)/t g ( t ) / t ω ( s 2 k ) \omega(s^{2k}) ω ( s 2 k ) g ( s 2 ) g(s^2) g ( s 2 ) ω ( s 2 ) \omega(s^2) ω ( s 2 ) s 2 s^2 s 2
The test supply contains cuts from dense sets of the two factors, so these squares approximate every positive elementary tensor in S ( V ) S(V) S ( V )
∥ a − b ∥ < δ ⟹ ( a − t − δ ) + ≾ ( b − t ) + ( t ≥ 0 ) \lVert a-b\rVert<\delta\quad\Longrightarrow\quad(a-t-\delta)_+ \precsim(b-t)_+ \qquad(t\ge0) ∥ a − b ∥ < δ ⟹ ( a − t − δ ) + ≾ ( b − t ) + ( t ≥ 0 ) transfers equality of all cut ranks to each such elementary limit, by allowing the cut slack to decrease to zero. Spectral integration now gives equality of trace values on all positive elementary tensors.
To pass from elementary tensors to the whole algebra, let L L L ( 0 , 1 ] (0,1] ( 0 , 1 ] H c = Her ( c ) ⊂ P ( V ) H_c=\operatorname{Her}(c)\subset P(V) H c = Her ( c ) ⊂ P ( V ) c c c P ( V ) P(V) P ( V ) f 0 ∈ C c ( ( 0 , 1 ] ) + f_0\in C_c((0,1])_+ f 0 ∈ C c (( 0 , 1 ] ) + L ‾ \overline{L} L c ≾ ( b − δ ) + c\precsim(b-\delta)_+ c ≾ ( b − δ ) + b ∈ P ( V ) + b\in P(V)_+ b ∈ P ( V ) + δ > 0 \delta>0 δ > 0
f 0 ⊗ c ≾ ( f 1 ⊗ b − δ / 2 ) + . f_0\otimes c\precsim(f_1\otimes b-\delta/2)_+. f 0 ⊗ c ≾ ( f 1 ⊗ b − δ /2 ) + . for a compactly supported f 1 f_1 f 1 supp ( f 0 ) \operatorname{supp}(f_0) supp ( f 0 ) S ( V ) S(V) S ( V ) f 0 ⊗ c f_0\otimes c f 0 ⊗ c C 0 ( L ) ⊗ H c C_0(L)\otimes H_c C 0 ( L ) ⊗ H c P P P P ( V ) P(V) P ( V ) S ( V ) S(V) S ( V )
Theorem 7.6 (Existence of cone models). *Let m m m ( 0 , 1 ] (0,1] ( 0 , 1 ] p : S → D p:S\to D p : S → D ρ \rho ρ σ = ρ ∣ Q \sigma=\rho|_Q σ = ρ ∣ Q
ρ ∘ p = m ⊗ F ( σ ) . \rho\circ p=m\otimes F(\sigma). ρ ∘ p = m ⊗ F ( σ ) . This includes weights whose densely finite ideal is proper or zero.
Proof. We first preserve the lower ideal signal through correction. For a fixed weight we will then identify its trace on the prescribed finite ideal, and use that signal to rule out any larger finite ideal.
Preserve the lower signal. Take h h h s = f ⊗ a s=f\otimes a s = f ⊗ a δ ( s ) = λ ( s ) − h p a d ( s ) \delta(s)=\lambda(s)-h^{\mathrm{pad}}(s) δ ( s ) = λ ( s ) − h pad ( s ) J ( U s ) J(U_s) J ( U s ) θ 1 : S → D \theta_1:S\to D θ 1 : S → D
δ ( s ) ∈ Ideal D ( θ 1 ( s ) ) (86) \delta(s)\in\operatorname{Ideal}_{D}(\theta_1(s)) \tag*{(86)} δ ( s ) ∈ Ideal D ( θ 1 ( s )) (86) for every such test. The countable coefficient form permits all the required cuts and localized families simultaneously. Set p = λ ⊕ θ 1 p=\lambda\oplus\theta_1 p = λ ⊕ θ 1
The ideal of p ( s ) p(s) p ( s ) λ ( s ) \lambda(s) λ ( s ) θ 1 ( s ) \theta_1(s) θ 1 ( s ) h p a d ( s ) h^{\mathrm{pad}}(s) h pad ( s )
Q ( U a ) ⊂ Ideal D ( p ( f ⊗ a ) ) ( f ≠ 0 ) (87) Q(U_a)\subset\operatorname{Ideal}_{D}(p(f\otimes a))\qquad(f\ne0) \tag*{(87)} Q ( U a ) ⊂ Ideal D ( p ( f ⊗ a )) ( f = 0 ) (87) for each controlled test. Also
( p − h p a d ) ( S ( U ) ) ⊂ J ( U ) , (p-h^{\mathrm{pad}})(S(U))\subset J(U), ( p − h pad ) ( S ( U )) ⊂ J ( U ) , where zero padding of h p a d h^{\mathrm{pad}} h pad
Identify the trace on its prescribed finite ideal. Fix a regularized trace ρ \rho ρ Q ( V ) = Fin ( σ ) Q(V)=\operatorname{Fin}(\sigma) Q ( V ) = Fin ( σ ) Fin ( F σ ) = P ( V ) \operatorname{Fin}(F\sigma)=P(V) Fin ( F σ ) = P ( V )
ω = ρ ∘ p , ν = m ⊗ F σ . \omega=\rho\circ p,\qquad\nu=m\otimes F\sigma. ω = ρ ∘ p , ν = m ⊗ F σ . The trace ρ \rho ρ J ( V ) J(V) J ( V ) s = f ⊗ a s=f\otimes a s = f ⊗ a V V V
ω ( s 2 ) = ρ ( h p a d ( s ) 2 ) = ∫ f 2 d m F σ ( a 2 ) = ν ( s 2 ) < ∞ . (88) \omega(s^2)=\rho(h^{\mathrm{pad}}(s)^2)=\int f^2\,\mathrm{d}m\,F\sigma(a^2)=\nu(s^2)<\infty. \tag*{(88)} ω ( s 2 ) = ρ ( h pad ( s ) 2 ) = ∫ f 2 d m F σ ( a 2 ) = ν ( s 2 ) < ∞. (88) Apply the same formula to the positive powers in the supply. The cuts of these localized elementary tests generate S ( V ) S(V) S ( V ) V V V
S ( V ) ⊂ Fin ( ω ) , ω ∣ S ( V ) = ν ∣ S ( V ) . (89) S(V)\subset\operatorname{Fin}(\omega),\qquad\omega|_{S(V)}=\nu|_{S(V)}. \tag*{(89)} S ( V ) ⊂ Fin ( ω ) , ω ∣ S ( V ) = ν ∣ S ( V ) . (89) Exclude an additional finite region. We show that the first inclusion is equality. Suppose the open set corresponding to Fin ( ω ) \operatorname{Fin}(\omega) Fin ( ω ) L × U L\times U L × U ( 0 , 1 ] × V (0,1]\times V ( 0 , 1 ] × V L L L a a a P ( U ) P(U) P ( U ) Fin ( ω ) \operatorname{Fin}(\omega) Fin ( ω ) ω \omega ω s = f ⊗ a s=f\otimes a s = f ⊗ a c ∈ Q ( U a ) + c\in Q(U_a)_+ c ∈ Q ( U a ) + ε > 0 \varepsilon>0 ε > 0 M M M η > 0 \eta>0 η > 0
[ ( c − ε ) + ] ≤ M [ ( p ( s ) − η ) + ] in Cu ( D ) . (90) [(c-\varepsilon)_+]\le M[(p(s)-\eta)_+]\quad\text{in }\operatorname{Cu}(D). \tag*{(90)} [( c − ε ) + ] ≤ M [( p ( s ) − η ) + ] in Cu ( D ) . (90) It follows that
d σ ( ( c − ε ) + ) = d ρ ( ( c − ε ) + ) ≤ M d ω ( s ) < ∞ . d_{\sigma}((c-\varepsilon)_+)=d_{\rho}((c-\varepsilon)_+)\le M d_{\omega}(s)<\infty. d σ (( c − ε ) + ) = d ρ (( c − ε ) + ) ≤ M d ω ( s ) < ∞. Hence σ \sigma σ Q ( U a ) Q(U_a) Q ( U a ) U a ⊂ V U_a \subset V U a ⊂ V U a U_a U a U U U U ⊂ V U \subset V U ⊂ V
Fin ( ω ) = S ( V ) = Fin ( ν ) . (91) \operatorname{Fin}(\omega)=S(V)=\operatorname{Fin}(\nu). \tag*{(91)} Fin ( ω ) = S ( V ) = Fin ( ν ) . (91) Both traces take value infinity on positive elements outside this ideal. Together with (89) , this proves (7.25) . The argument also applies when V V V
Point models and approximate intertwining
We now construct homomorphisms P → Q P \to Q P → Q Q → P Q \to P Q → P
The suspension, crossed-product projection, and subsequence extraction also occur in [15 ] . The suspension, crossed-product, and full-corner strategy adapts the proof of [16 ] . The latter theorem has an O ∞ \mathcal{O}_{\infty} O ∞ Ad ( w ) ( a ) = w a w ∗ \operatorname{Ad}(w)(a)=waw^* Ad ( w ) ( a ) = w a w ∗
The line and its translation
Proposition 8.1 (A covariant line model). There is a model
λ : C 0 ( R ) ⊗ P ⟶ D \lambda:C_0(\mathbb{R})\otimes P\longrightarrow D λ : C 0 ( R ) ⊗ P ⟶ D for Lebesgue measure and F F F α ( f ) ( t ) = f ( t − 1 ) \alpha(f)(t)=f(t-1) α ( f ) ( t ) = f ( t − 1 ) P P P λ n \lambda_n λ n u n ∈ M ( Q ) u_n\in M(Q) u n ∈ M ( Q )
∥ Ad ( u n ) ( λ n ( s ) ) − λ n ( α ( s ) ) ∥ ⟶ 0 ( s ∈ C 0 ( R ) ⊗ P ) . (92) \left\|\operatorname{Ad}(u_n)(\lambda_n(s))-\lambda_n(\alpha(s))\right\|\longrightarrow0 \qquad(s\in C_0(\mathbb{R})\otimes P). \tag*{(92)} ∥ Ad ( u n ) ( λ n ( s )) − λ n ( α ( s )) ∥ ⟶ 0 ( s ∈ C 0 ( R ) ⊗ P ) . (92) Proof. For j ≥ 2 j\ge2 j ≥ 2 I j = ( − j , j ] I_j=(-j,j] I j = ( − j , j ] (75) , transported along a homeomorphism ( 0 , 1 ] ≅ I j (0,1]\cong I_j ( 0 , 1 ] ≅ I j λ ( j ) : C 0 ( I j ) ⊗ P → D \lambda^{(j)}:C_0(I_j)\otimes P\to D λ ( j ) : C 0 ( I j ) ⊗ P → D 0 ≤ η j ≤ 1 0\le\eta_j\le1 0 ≤ η j ≤ 1 C c ( ( − j , j ) ) C_c((-j,j)) C c (( − j , j )) [ − j + 1 , j − 1 ] [-j+1,j-1] [ − j + 1 , j − 1 ]
R j : C 0 ( R ) ⊗ P ⟶ C 0 ( I j ) ⊗ P , R j ( s ) = η j s ∣ I j , R_j:C_0(\mathbb{R})\otimes P\longrightarrow C_0(I_j)\otimes P,\qquad R_j(s)=\eta_j s|_{I_j}, R j : C 0 ( R ) ⊗ P ⟶ C 0 ( I j ) ⊗ P , R j ( s ) = η j s ∣ I j , are c.p.c.: restriction followed by multiplication by the scalar positive contraction η j \eta_j η j C 0 ( I j ) ⊗ P C_0(I_j)\otimes P C 0 ( I j ) ⊗ P λ ( j ) \lambda^{(j)} λ ( j ) R j R_j R j j j j η j = 1 \eta_j=1 η j = 1
Take c.p.c. coordinate lifts of each λ ( j ) \lambda^{(j)} λ ( j ) [4 ] , and compose with R j R_j R j (19) to an increasing dense list of compactly supported algebra tests and the elementary rank tests of (15) . Here the tests are fixed on the line before increasing j j j ( − j + 1 , j − 1 ) (-j+1,j-1) ( − j + 1 , j − 1 ) R j R_j R j Q Q Q D D D j j j j − 1 j^{-1} j − 1 λ n \lambda_n λ n λ \lambda λ
Lebesgue measure is translation invariant, including the extended product convention. Hence λ \lambda λ λ ∘ α \lambda\circ\alpha λ ∘ α (92) .
The covariance unitaries belong to M ( Q ) M(Q) M ( Q ) Q Q Q
C = ∏ n = 1 ∞ M ( Q ) / ⨁ n = 1 ∞ M ( Q ) . \mathcal{C} = \prod_{n=1}^{\infty} M(Q) \mathbin{/} \bigoplus_{n=1}^{\infty} M(Q). C = n = 1 ∏ ∞ M ( Q ) / n = 1 ⨁ ∞ M ( Q ) . The product here consists of bounded sequences and the direct sum of norm-null sequences. There is a natural faithful inclusion D ⊆ C D \subseteq C D ⊆ C D D D C C C u ∈ C u \in C u ∈ C Ad ( u ) λ ( s ) = λ ( α ( s ) ) \operatorname{Ad}(u)\lambda(s) = \lambda(\alpha(s)) Ad ( u ) λ ( s ) = λ ( α ( s ))
Ψ : ( C 0 ( R ) ⊗ P ) ⋊ α Z ⟶ D . (93) \Psi\colon(C_0(\mathbb{R}) \otimes P) \rtimes_{\alpha} \mathbb{Z} \longrightarrow D. \tag*{(93)} Ψ : ( C 0 ( R ) ⊗ P ) ⋊ α Z ⟶ D . (93) This also applies when λ \lambda λ C C C H \mathcal{H} H λ ( C 0 ( R ) ⊗ P ) H \lambda(C_0(\mathbb{R}) \otimes P)\mathcal{H} λ ( C 0 ( R ) ⊗ P ) H ∑ n λ ( s n ) u n \sum_n \lambda(s_n)u^n ∑ n λ ( s n ) u n D D D P P P Z \mathbb{Z} Z
E ⊗ P , E = C 0 ( R ) ⋊ α Z . E \otimes P, \qquad E = C_0(\mathbb{R}) \rtimes_{\alpha} \mathbb{Z}. E ⊗ P , E = C 0 ( R ) ⋊ α Z . The identification follows also by matching the commuting representations of P P P C 0 ( R ) C_0(\mathbb{R}) C 0 ( R ) E E E v v v v f v ∗ = α ( f ) vfv^* = \alpha(f) v f v ∗ = α ( f )
An explicit trace-normalized corner
For any projection p ∈ E p \in E p ∈ E a ↦ p ⊗ a a \mapsto p \otimes a a ↦ p ⊗ a P P P E ⊗ P E \otimes P E ⊗ P Ψ \Psi Ψ E E E
Lemma 8.2 (The corner projection). There are ζ ∈ C c ( R ) + \zeta\in C_c(\mathbb{R})_+ ζ ∈ C c ( R ) + and a projection p ∈ E p \in E p ∈ E such that
∑ n ∈ Z ζ ( t − n ) 2 = 1 , ∫ R ζ ( t ) 2 d t = 1 , \sum_{n \in\mathbb{Z}} \zeta(t-n)^2 = 1, \qquad\int_{\mathbb{R}} \zeta(t)^2\,\mathrm{d}t = 1, n ∈ Z ∑ ζ ( t − n ) 2 = 1 , ∫ R ζ ( t ) 2 d t = 1 , and, for any C ∗ C^* C ∗ algebra R 0 R_0 R 0 any extended trace ω \omega ω on E ⊗ R 0 E \otimes R_0 E ⊗ R 0 and every a ∈ ( R 0 ) + a \in(R_0)_+ a ∈ ( R 0 ) +
ω ( p ⊗ a ) = ω ( ζ 2 ⊗ a ) . \omega(p \otimes a) = \omega(\zeta^2 \otimes a). ω ( p ⊗ a ) = ω ( ζ 2 ⊗ a ) . In particular, if the restriction of ω \omega ω to C 0 ( R ) ⊗ R 0 C_0(\mathbb{R}) \otimes R_0 C 0 ( R ) ⊗ R 0 is Lebesgue measure tensored with ν \nu ν then ω ( p ⊗ a ) = ν ( a ) \omega(p \otimes a) = \nu(a) ω ( p ⊗ a ) = ν ( a ) including the value infinity. Moreover, p p p is full and p E p ≅ C ( R / Z ) pEp \cong C(\mathbb{R}/\mathbb{Z}) pE p ≅ C ( R / Z ) Proof. Choose η ∈ C c ( R ) + \eta\in C_c(\mathbb{R})_+ η ∈ C c ( R ) + [ 0 , 1 ] [0,1] [ 0 , 1 ]
H ( t ) = ∑ n ∈ Z η ( t − n ) 2 , ζ ( t ) = H ( t ) − 1 / 2 η ( t ) . H(t)=\sum_{n\in\mathbb{Z}}\eta(t-n)^2,\qquad\zeta(t)=H(t)^{-1/2}\eta(t). H ( t ) = n ∈ Z ∑ η ( t − n ) 2 , ζ ( t ) = H ( t ) − 1/2 η ( t ) . The sum defining H H H ζ \zeta ζ [ 0 , 1 ] [0,1] [ 0 , 1 ]
p = ∑ n ∈ Z p n v n , p n ( t ) = ζ ( t ) ζ ( t − n ) . (94) p=\sum_{n\in\mathbb{Z}}p_nv^n,\qquad p_n(t)=\zeta(t)\zeta(t-n). \tag*{(94)} p = n ∈ Z ∑ p n v n , p n ( t ) = ζ ( t ) ζ ( t − n ) . (94) Only finitely many coefficients are nonzero. The involution formula gives α n ( p − n ) = p n \alpha_n(p_{-n})=p_n α n ( p − n ) = p n p ∗ = p p^*=p p ∗ = p k ∈ Z k\in\mathbb{Z} k ∈ Z
( p 2 ) k ( t ) = ζ ( t ) ζ ( t − k ) ∑ n ∈ Z ζ ( t − n ) 2 = p k ( t ) . (p^2)_k(t)=\zeta(t)\zeta(t-k)\sum_{n\in\mathbb{Z}}\zeta(t-n)^2=p_k(t). ( p 2 ) k ( t ) = ζ ( t ) ζ ( t − k ) n ∈ Z ∑ ζ ( t − n ) 2 = p k ( t ) . Thus p p p
Choose finitely many real functions q i ∈ C c ( R ) q_i\in C_c(\mathbb{R}) q i ∈ C c ( R )
q : = ∑ i q i 2 = 1 on supp ( ζ ) . q:=\sum_i q_i^2=1\quad\text{on }\operatorname{supp}(\zeta). q := i ∑ q i 2 = 1 on supp ( ζ ) . For completeness, choose bump functions r i r_i r i supp ( ζ ) \operatorname{supp}(\zeta) supp ( ζ ) κ \kappa κ q i = κ r i / ( ∑ j r j 2 ) 1 / 2 q_i=\kappa r_i/(\sum_j r_j^2)^{1/2} q i = κ r i / ( ∑ j r j 2 ) 1/2 p p p
q p = p = p q , ∑ i q i p q i = ζ 2 . qp=p=pq,\qquad\sum_i q_i p q_i=\zeta^2. q p = p = pq , i ∑ q i p q i = ζ 2 . To verify the second identity, its n n n
ζ ( t ) ζ ( t − n ) ∑ i q i ( t ) q i ( t − n ) . \zeta(t)\zeta(t-n)\sum_i q_i(t)q_i(t-n). ζ ( t ) ζ ( t − n ) i ∑ q i ( t ) q i ( t − n ) . This vanishes when n ≠ 0 n\ne0 n = 0 ζ ( t ) 2 \zeta(t)^2 ζ ( t ) 2 n = 0 n=0 n = 0
For a ∈ ( R 0 ) + a\in(R_0)_+ a ∈ ( R 0 ) + y i = q i p ⊗ a 1 / 2 y_i=q_i p\otimes a^{1/2} y i = q i p ⊗ a 1/2 E ⊗ R 0 E\otimes R_0 E ⊗ R 0 (8.5) gives the exact positive identities
∑ i y i ∗ y i = p ⊗ a , ∑ i y i y i ∗ = ζ 2 ⊗ a . \sum_i y_i^*y_i=p\otimes a,\qquad\sum_i y_i y_i^*=\zeta^2\otimes a. i ∑ y i ∗ y i = p ⊗ a , i ∑ y i y i ∗ = ζ 2 ⊗ a . Additivity and ω ( y i ∗ y i ) = ω ( y i y i ∗ ) \omega(y_i^*y_i)=\omega(y_i y_i^*) ω ( y i ∗ y i ) = ω ( y i y i ∗ ) (8.3) . There are only finitely many positive summands. In particular, this calculation uses no subtraction of infinite values and requires no finiteness assumption on ω \omega ω ( ∫ ζ 2 ) ν ( a ) = ν ( a ) (\int\zeta^2)\nu(a)=\nu(a) ( ∫ ζ 2 ) ν ( a ) = ν ( a )
We also verify fullness and the corner description. Equation (8.5) puts ζ 2 \zeta^2 ζ 2 Ideal E ( p ) \operatorname{Ideal}_E(p) Ideal E ( p ) v n v^n v n α n ( ζ 2 ) \alpha_n(\zeta^2) α n ( ζ 2 ) f ∈ C c ( R ) f\in C_c(\mathbb{R}) f ∈ C c ( R )
f = ∑ n ∈ Z f α n ( ζ 2 ) f=\sum_{n\in\mathbb{Z}}f\alpha_n(\zeta^2) f = n ∈ Z ∑ f α n ( ζ 2 ) as a finite sum of elements of that ideal. It therefore contains C 0 ( R ) C_0(\mathbb{R}) C 0 ( R ) E E E p p p
For f ∈ C c ( R ) f \in C_{\mathrm{c}}(\mathbb{R}) f ∈ C c ( R ) n ∈ Z n \in\mathbb{Z} n ∈ Z
p ( f v n ) p = g f , n p , g f , n ( t ) = ∑ k ∈ Z ζ ( t − k ) f ( t − k ) ζ ( t − k − n ) . (95) p(fv^{n})p = g_{f,n}p,\qquad g_{f,n}(t) = \sum_{k\in\mathbb{Z}} \zeta(t-k)f(t-k)\zeta(t-k-n). \tag*{(95)} p ( f v n ) p = g f , n p , g f , n ( t ) = k ∈ Z ∑ ζ ( t − k ) f ( t − k ) ζ ( t − k − n ) . (95) The function g f , n g_{f,n} g f , n g g g g p gp g p E E E g g g E E E g p ∈ p E p gp \in pEp g p ∈ pE p g ↦ g p g \mapsto gp g ↦ g p C ( R / Z ) → p E p C(\mathbb{R}/\mathbb{Z}) \to pEp C ( R / Z ) → pE p g ζ 2 g\zeta^{2} g ζ 2 ζ ≠ 0 \zeta\ne0 ζ = 0 (95) and density of compactly supported Fourier sums prove surjectivity.
Corollary 8.3 (Existence of point models). There is a homomorphism φ : P → Q ∞ \varphi: P \to Q_{\infty} φ : P → Q ∞ such that
ρ ∘ φ = F ( ρ ∣ Q ) \rho\circ\varphi= F(\rho|_{Q}) ρ ∘ φ = F ( ρ ∣ Q ) for every regularized limit ρ \rho ρ
Proof. Use (93) and Lemma 8.2 to define φ ( a ) = Ψ ( p ⊗ a ) \varphi(a) = \Psi(p \otimes a) φ ( a ) = Ψ ( p ⊗ a ) ρ \rho ρ ω = ρ ∘ Ψ \omega= \rho\circ\Psi ω = ρ ∘ Ψ C 0 ( R ) ⊗ P C_{0}(\mathbb{R}) \otimes P C 0 ( R ) ⊗ P F ( σ ) F(\sigma) F ( σ ) σ = ρ ∣ Q \sigma= \rho|_{Q} σ = ρ ∣ Q a ∈ P + a \in P_{+} a ∈ P +
ρ ( φ ( a ) ) = ω ( p ⊗ a ) = ω ( ζ 2 ⊗ a ) = F ( σ ) ( a ) . \rho(\varphi(a)) = \omega(p \otimes a) = \omega(\zeta^{2} \otimes a) = F(\sigma)(a). ρ ( φ ( a )) = ω ( p ⊗ a ) = ω ( ζ 2 ⊗ a ) = F ( σ ) ( a ) . The positive partition calculation applies equally when this common value is infinite.
From a point model to an actual map
The coordinate lifts of a point model need not converge. Uniqueness will make their unitary orbits Cauchy on finite sets. We can then conjugate a subsequence so that its successive errors are summable and its limit lies in Q Q Q
Proposition 8.4 (An actual homomorphism). There is a homomorphism ϕ : P → Q \phi: P \to Q ϕ : P → Q satisfying
τ ∘ ϕ = F ( τ ) ( τ ∈ T ( Q ) ) , Cu ( ϕ ) = G . \tau\circ\phi= F(\tau) \qquad(\tau\in T(Q)), \qquad\operatorname{Cu}(\phi) = G. τ ∘ ϕ = F ( τ ) ( τ ∈ T ( Q )) , Cu ( ϕ ) = G . It is injective, and its constant sequence is a point model.
Proof. Take c.p.c. lifts φ n : P → Q \varphi_n : P \to Q φ n : P → Q {cite:p}choi_1976_completely_positive_lifting`` Theorem 3.10. For every finite E ⊆ P E \subseteq P E ⊆ P ε > 0 \varepsilon> 0 ε > 0 N N N
inf w ∈ U ( M ( Q ) ) max a ∈ E ∥ Ad ( w ) ( φ m ( a ) ) − φ n ( a ) ∥ < ε ( m , n ≥ N ) . (96) \inf_{w\in\mathcal{U}(M(Q))}\max_{a\in E}\left\|\operatorname{Ad}(w)(\varphi_m(a))-\varphi_n(a)\right\| < \varepsilon\qquad(m,n \geq N). \tag*{(96)} w ∈ U ( M ( Q )) inf a ∈ E max ∥ Ad ( w ) ( φ m ( a )) − φ n ( a ) ∥ < ε ( m , n ≥ N ) . (96) If this failed, choose pairs m k , n k → ∞ m_k,n_k \to\infty m k , n k → ∞ ε \varepsilon ε τ k \tau_k τ k n k n_k n k ω \omega ω k k k τ ~ n k = τ k \widetilde{\tau}_{n_k} = \tau_k τ n k = τ k A ∈ ω ~ A \in\widetilde{\omega} A ∈ ω { k : n k ∈ A } ∈ ω \{k : n_k \in A\} \in\omega { k : n k ∈ A } ∈ ω ω ~ \widetilde{\omega} ω ( φ m k ) (\varphi_{m_k}) ( φ m k ) ( φ n k ) (\varphi_{n_k}) ( φ n k )
Let E k E_k E k P P P (97) to choose an increasing subsequence n k n_k n k w k ∈ U ( M ( Q ) ) w_k \in\mathcal{U}(M(Q)) w k ∈ U ( M ( Q ))
max a ∈ E k ∥ Ad ( w k ) ( φ n k + 1 ( a ) ) − φ n k ( a ) ∥ < 2 − k . (97) \max_{a\in E_k}\left\|\operatorname{Ad}(w_k)(\varphi_{n_{k+1}}(a))-\varphi_{n_k}(a)\right\| < 2^{-k}. \tag*{(97)} a ∈ E k max ∥ Ad ( w k ) ( φ n k + 1 ( a )) − φ n k ( a ) ∥ < 2 − k . (97) Choose n k n_k n k k k k W 1 = 1 W_1=1 W 1 = 1 W k + 1 = W k w k W_{k+1}=W_kw_k W k + 1 = W k w k ψ k = Ad ( W k ) ∘ φ n k \psi_k=\operatorname{Ad}(W_k)\circ\varphi_{n_k} ψ k = Ad ( W k ) ∘ φ n k ψ k ( a ) \psi_k(a) ψ k ( a ) a a a a ∈ P a\in P a ∈ P ϕ : P → Q \phi:P\to Q ϕ : P → Q
We check the invariant without assuming continuity of unbounded trace evaluation under norm convergence. Fix τ ∈ T ( Q ) \tau\in T(Q) τ ∈ T ( Q ) ρ τ \rho_\tau ρ τ ψ k \psi_k ψ k ϕ \phi ϕ Q ∞ Q_\infty Q ∞ a ∈ P + a\in P_+ a ∈ P +
τ ( ( ψ k ( a ) − ε ) + ) = τ ( ( φ n k ( a ) − ε ) + ) . \tau((\psi_k(a)-\varepsilon)_+)=\tau((\varphi_{n_k}(a)-\varepsilon)_+). τ (( ψ k ( a ) − ε ) + ) = τ (( φ n k ( a ) − ε ) + ) . Regularization and the subsequence model identity therefore imply
τ ( ϕ ( a ) ) = ρ τ ( ( ψ k ( a ) ) k ) = F ( τ ) ( a ) . \tau(\phi(a))=\rho_\tau((\psi_k(a))_k)=F(\tau)(a). τ ( ϕ ( a )) = ρ τ (( ψ k ( a ) ) k ) = F ( τ ) ( a ) . Here ρ τ ∣ Q = τ \rho_\tau|_Q=\tau ρ τ ∣ Q = τ
d τ ( ϕ ( a ) ) = d F ( τ ) ( a ) = d τ ( G [ a ] ) . d_\tau(\phi(a))=d_{F(\tau)}(a)=d_\tau(G[a]). d τ ( ϕ ( a )) = d F ( τ ) ( a ) = d τ ( G [ a ]) . This calculation also applies to finite matrices. If x = ( x i j ) ∈ M k ( P ) + x=(x_{ij})\in M_k(P)_+ x = ( x ij ) ∈ M k ( P ) +
( τ ⊗ Tr k ) ( ϕ ( k ) ( x ) ) = ∑ i τ ( ϕ ( x i i ) ) = ∑ i F ( τ ) ( x i i ) = ( F ( τ ) ⊗ Tr k ) ( x ) . \begin{aligned}
(\tau\otimes\operatorname{Tr}_k)(\phi^{(k)}(x))&=\sum_i\tau(\phi(x_{ii}))=\sum_iF(\tau)(x_{ii})\\
&=(F(\tau)\otimes\operatorname{Tr}_k)(x).
\end{aligned} ( τ ⊗ Tr k ) ( ϕ ( k ) ( x )) = i ∑ τ ( ϕ ( x ii )) = i ∑ F ( τ ) ( x ii ) = ( F ( τ ) ⊗ Tr k ) ( x ) . including infinite values. Apply this identity to positive powers and then use positive cuts of finite-matrix approximations in P ⊗ K P\otimes\mathcal{K} P ⊗ K Cu ( Q ) \operatorname{Cu}(Q) Cu ( Q ) Cu ( ϕ ) = G \operatorname{Cu}(\phi)=G Cu ( ϕ ) = G G G G ϕ ( a ) = 0 \phi(a)=0 ϕ ( a ) = 0 a a a [ a ] = 0 [a]=0 [ a ] = 0 a = 0 a=0 a = 0 ϕ \phi ϕ ρ \rho ρ σ = ρ ∣ Q \sigma=\rho|_Q σ = ρ ∣ Q
ρ ∘ ϕ = σ ∘ ϕ = F ( σ ) . \rho\circ\phi=\sigma\circ\phi=F(\sigma). ρ ∘ ϕ = σ ∘ ϕ = F ( σ ) . So the constant map is itself a point model.
The final intertwining
Theorem 8.5 (Trace-cone classification). There is an isomorphism Φ : P → Q \Phi:P\to Q Φ : P → Q such that
τ ∘ Φ = F ( τ ) ( τ ∈ T ( Q ) ) , Cu ( Φ ) = G . \tau\circ\Phi=F(\tau)\quad(\tau\in T(Q)),\qquad\operatorname{Cu}(\Phi)=G. τ ∘ Φ = F ( τ ) ( τ ∈ T ( Q )) , Cu ( Φ ) = G . Consequently Theorem 1.1 holds.
Proof. Apply Proposition 8.4 to F F F F − 1 F^{-1} F − 1 ϕ : P → Q \phi:P\to Q ϕ : P → Q ψ : Q → P \psi:Q\to P ψ : Q → P ψ ϕ \psi\phi ψ ϕ id P \operatorname{id}_P id P ϕ ψ \phi\psi ϕ ψ id Q \operatorname{id}_Q id Q P , Q P,Q P , Q
Fix increasing finite sets P n , Q n \mathcal{P}_n,\mathcal{Q}_n P n , Q n ε n = 2 − n \varepsilon_n=2^{-n} ε n = 2 − n f 1 = ϕ f_1=\phi f 1 = ϕ g 0 = ψ g_0=\psi g 0 = ψ E 0 , F 0 \mathcal{E}_0,\mathcal{F}_0 E 0 , F 0 f n f_n f n g n − 1 g_{n-1} g n − 1
E n ⊇ E n − 1 ∪ P n ∪ g n − 1 ( F n − 1 ) . \mathcal{E}_n\supseteq\mathcal{E}_{n-1}\cup\mathcal{P}_n\cup g_{n-1}(\mathcal{F}_{n-1}). E n ⊇ E n − 1 ∪ P n ∪ g n − 1 ( F n − 1 ) . The composition g n − 1 f n g_{n-1}f_n g n − 1 f n v n ∈ U ( M ( P ) ) v_n \in\mathcal{U}(M(P)) v n ∈ U ( M ( P )) g n = Ad ( v n ) ∘ g n − 1 g_n = \operatorname{Ad}(v_n) \circ g_{n-1} g n = Ad ( v n ) ∘ g n − 1
∥ g n f n ( a ) − a ∥ < ε n ( a ∈ E n ) . (98) \lVert g_nf_n(a)-a\rVert< \varepsilon_n \qquad(a \in\mathcal{E}_n). \tag*{(98)} ∥ g n f n ( a ) − a ∥ < ε n ( a ∈ E n ) . (98) Next choose a finite set
F n ⊇ F n − 1 ∪ Q n ∪ f n ( E n ) . \mathcal{F}_n \supseteq\mathcal{F}_{n-1} \cup\mathcal{Q}_n \cup f_n(\mathcal{E}_n). F n ⊇ F n − 1 ∪ Q n ∪ f n ( E n ) . Choose z n ∈ U ( M ( Q ) ) z_n \in\mathcal{U}(M(Q)) z n ∈ U ( M ( Q )) f n + 1 = Ad ( z n ) ∘ f n f_{n+1} = \operatorname{Ad}(z_n) \circ f_n f n + 1 = Ad ( z n ) ∘ f n
∥ f n + 1 g n ( b ) − b ∥ < ε n ( b ∈ F n ) . (99) \lVert f_{n+1}g_n(b)-b\rVert< \varepsilon_n \qquad(b \in\mathcal{F}_n). \tag*{(99)} ∥ f n + 1 g n ( b ) − b ∥ < ε n ( b ∈ F n ) . (99) This second adjustment is possible by the same identity-data argument for f n g n f_ng_n f n g n
The inclusions of the image tests make these sequences Cauchy. If a ∈ E n a \in\mathcal{E}_n a ∈ E n (98) –(99) give
∥ f n + 1 ( a ) − f n ( a ) ∥ ≤ ∥ f n + 1 ( a − g n f n ( a ) ) ∥ + ∥ f n + 1 g n ( f n ( a ) ) − f n ( a ) ∥ < 2 ε n . \begin{aligned}
\lVert f_{n+1}(a)-f_n(a)\rVert
&\leq\lVert f_{n+1}(a-g_nf_n(a))\rVert\\
&\quad+\lVert f_{n+1}g_n(f_n(a))-f_n(a)\rVert< 2\varepsilon_n.
\end{aligned} ∥ f n + 1 ( a ) − f n ( a )∥ ≤ ∥ f n + 1 ( a − g n f n ( a ))∥ + ∥ f n + 1 g n ( f n ( a )) − f n ( a )∥ < 2 ε n . If n ≥ 2 n \geq2 n ≥ 2 b ∈ F n − 1 b \in\mathcal{F}_{n-1} b ∈ F n − 1 g n − 1 ( b ) ∈ E n g_{n-1}(b) \in\mathcal{E}_n g n − 1 ( b ) ∈ E n
∥ g n ( b ) − g n − 1 ( b ) ∥ ≤ ∥ g n ( b − f n g n − 1 ( b ) ) ∥ + ∥ g n f n ( g n − 1 ( b ) ) − g n − 1 ( b ) ∥ < ε n − 1 + ε n . \begin{aligned}
\lVert g_n(b)-g_{n-1}(b)\rVert
&\leq\lVert g_n(b-f_ng_{n-1}(b))\rVert\\
&\quad+\lVert g_nf_n(g_{n-1}(b))-g_{n-1}(b)\rVert
<\varepsilon_{n-1}+\varepsilon_n.
\end{aligned} ∥ g n ( b ) − g n − 1 ( b )∥ ≤ ∥ g n ( b − f n g n − 1 ( b ))∥ + ∥ g n f n ( g n − 1 ( b )) − g n − 1 ( b )∥ < ε n − 1 + ε n . These summable bounds and density give point-norm limits f ∞ : P → Q f_\infty:P \to Q f ∞ : P → Q g ∞ : Q → P g_\infty:Q \to P g ∞ : Q → P (98) and (99) imply
g ∞ f ∞ = id P , f ∞ g ∞ = id Q . g_\infty f_\infty=\operatorname{id}_P,\qquad f_\infty g_\infty=\operatorname{id}_Q. g ∞ f ∞ = id P , f ∞ g ∞ = id Q . For example, contractivity controls the moving input f n ( a ) f_n(a) f n ( a ) f ∞ ( a ) f_\infty(a) f ∞ ( a ) g n g_n g n f ∞ f_\infty f ∞
The isomorphism realizes the prescribed trace data as well. Each f n f_n f n G G G ϕ \phi ϕ a ∈ P + a \in P_+ a ∈ P + ε > 0 \varepsilon>0 ε > 0 n n n ∥ f n ( a ) − f ∞ ( a ) ∥ < ε \lVert f_n(a)-f_\infty(a)\rVert<\varepsilon ∥ f n ( a ) − f ∞ ( a )∥ < ε τ ∈ T ( Q ) \tau\in T(Q) τ ∈ T ( Q )
d F τ ( ( a − ε ) + ) ≤ d τ ( f ∞ ( a ) ) , d τ ( ( f ∞ ( a ) − ε ) + ) ≤ d F τ ( a ) . d_{F\tau}((a-\varepsilon)_+) \leq d_\tau(f_\infty(a)),\qquad d_\tau((f_\infty(a)-\varepsilon)_+) \leq d_{F\tau}(a). d F τ (( a − ε ) + ) ≤ d τ ( f ∞ ( a )) , d τ (( f ∞ ( a ) − ε ) + ) ≤ d F τ ( a ) . Taking the supremum over cuts proves equality of ranks; spectral integration gives τ ∘ f ∞ = F ( τ ) \tau\circ f_\infty=F(\tau) τ ∘ f ∞ = F ( τ ) Cu ( f ∞ ) = G \operatorname{Cu}(f_\infty)=G Cu ( f ∞ ) = G Φ = f ∞ \Phi=f_\infty Φ = f ∞ P = A ⊗ W ⊗ K P=A\otimes\mathcal{W}\otimes\mathcal{K} P = A ⊗ W ⊗ K Q = B ⊗ W ⊗ K Q=B\otimes\mathcal{W}\otimes\mathcal{K} Q = B ⊗ W ⊗ K
