Hyperrigidity and convexity in two variables

Introduction

The classical approximation theorem due to P. P. Korovkin [4] states that the asymptotic behaviour of any sequence of positive linear maps on $C[0,1]$ is completely determined by its behaviour on the three functions $1$, $x$ and $x^{2}$. The underlying reason for this phenomenon was revealed by Yu. A. Šaškin [7], who characterized all such test sets in terms of the Choquet boundary. To state his result, let $X$ be a compact metric space and let $G \subseteq C(X)$ be a set whose linear span $S$ contains the constant function $1$. The Choquet boundary $\partial_{S}X$ of $S$ is the set of all points $t \in X$ such that the Dirac measure $\delta_{t}$ is the only Borel probability measure $\mu$ on $X$ satisfying $f(t) = \int_{X} f\,d\mu$ for all $f \in S$. Šaškin’s theorem asserts that $\partial_{S}X = X$ if and only if for any sequence of positive linear maps $\Phi_{k}: C(X) \to C(X)$ ($k \in\mathbb{N}$),

In other words, the asymptotic behaviour of the sequence $\{\Phi_{k}\}_{k=1}^{\infty}$ on the $C^{*}$-algebra $C(X)$ is uniquely determined by the set $G$ precisely when every point of $X$ belongs to the Choquet boundary of $S$.

Šaškin’s theorem thus shows that the Choquet boundary lies at the heart of classical approximation theory. Inspired by this observation, and by the central role that approximation plays in the modern theory of operator algebras, W. Arveson [1] introduced the notion of hyperrigidity, a “noncommutative” counterpart of the Korovkin property that turns out to encompass a wide range of significant operator-algebraic phenomena.

For the reader’s convenience, we recall Arveson’s definition (see [1 Definition 1.1]). Here and below, UCP stands for unital completely positive.

In the spirit of Šaškin’s theorem, Arveson [1 Conjecture 4.3] conjectured that a generating set is hyperrigid whenever every irreducible representation of the generated $C^*$-algebra is a boundary representation. The converse implication is always valid. Bilich and Scherer [3] recently constructed counterexamples to Arveson’s hyperrigidity conjecture in commutative $C^*$-algebras, showing that the boundary condition does not imply the unique extension property for arbitrary representations.

Let $I\subset\mathbb{R}$ be a compact interval and let $f\in C(I)$ be real-valued and strictly convex. The function system $S_f=\operatorname{span}\{1,x,f\}$ has Choquet boundary $I$ (see [1 Propositions 9.1 and 9.2]). Arveson [1 Remark 9.5] asked whether $\{I,T,f(T)\}$ is hyperrigid in the unital $C^*$-algebra generated by $T$ whenever $T$ is selfadjoint with spectrum in $I$. Brown [2] answered this question affirmatively as a consequence of the following weak-to-strong convergence theorem. Below, $\sigma(T)$ denotes the spectrum of an operator $T$.

The present paper extends this rigidity phenomenon to strictly convex functions of two variables, under a regularity assumption.

Preliminaries

Throughout the paper, $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{N}$ denote the sets of real numbers, complex numbers and positive integers, respectively. For a topological space $X$, we write $\mathcal{B}(X)$ for the $\sigma$-algebra of its Borel subsets. The symbol $1_D$ stands for the characteristic function of a set $D$, and $\operatorname{lin}Z$ denotes the complex linear span of a subset $Z$ of a complex vector space.

Let $\mathcal{H}$ be a complex Hilbert space. We denote by $B(\mathcal{H})$ the $C^*$-algebra of all bounded linear operators on $\mathcal{H}$, with identity operator $I=I_{\mathcal{H}}$. An operator $T\in B(\mathcal{H})$ is positive if $\langle Th,h\rangle\geq0$ for every $h\in\mathcal{H}$.

Proof of Theorem 1.3

Choose an open neighbourhood $U\subset\mathbb{R}^2$ of $K$ and a function $\widetilde{f}\in C^2(U,\mathbb{R})$ such that $\widetilde{f}|_K=f$. We denote by $D^2\widetilde{f}(u)=(\partial_i\partial_j\widetilde{f}(u))_{i,j=1}^2$ the Hessian matrix of $\widetilde{f}$ at $u\in U. For each $a \in K$, define $\psi_a: K \to\mathbb{R}$ by $\psi_a(t)=f(t)-f(a)-\langle\nabla\widetilde{f}(a),t-a\rangle$. Strict convexity implies that $\psi_a(t)\geq0$, with equality if and only if $t=a$. Taylor’s formula with integral remainder yields

where $L>0$ satisfies $L\geq\frac{1}{2}\sup_{u\in K}\lVert D^2\widetilde{f}(u)\rVert$. Strict convexity of $f$ implies that $\psi_a(t)\geq0$ for all $t\in K$, with equality if and only if $t=a$.

Proof. Choose a square $Q_0\subset\mathbb{R}^2$ of side length $L>0$ such that $K\subset Q_0$. We shall repeatedly subdivide each square into four congruent subsquares by bisecting its sides. At every stage, boundary points are assigned to exactly one of the adjacent subsquares. We refer to the resulting Borel sets as dyadic cells. Thus, each dyadic cell is the disjoint union of its four children and is contained in a closed square of the corresponding side length.

Fix $h\in\mathcal{H}$ and set $\eta=T^{1/2}h$. We may assume that $\eta\neq0$. Define $\mu:\mathcal{B}(\mathbb{R}^2)\to[0,\infty)$ by

Then $\mu$ is a finite positive Borel measure on $\mathbb{R}^2$, supported on $K$, and

Fix $N\geq1$. Among the four children of $Q_0$, choose a dyadic cell $Q_1$ satisfying $\mu(Q_1)\geq\mu(Q_0)/4$, and continue inductively within the cell so obtained. This yields a nested sequence of dyadic cells

such that, for $0\leq j\leq N$,

Since $\mu(Q_N)>0$ and $\mu$ is supported on $K$, we have $K\cap Q_N\neq\varnothing$. Fix $a\in K\cap Q_N$.

For $0\leq j\leq N$, set

By the definition of $\mu$ and the choice of the cells $Q_j$,

Define $d_N:K\to[0,\infty)$ by

Then

Fix $t\in K$, and let $m\in\{0,\ldots,N\}$ be the largest index such that $t\in Q_m$. Since the cells are nested and $a\in Q_N\subset Q_m$, we have

Consequently,

Multiplying (5) by $d_N(t)$ and applying the spectral calculus, we obtain

Since $\eta=T^{1/2}h$, the Cauchy–Schwarz inequality, the assumption $T\le cG_a$, and (6) give

As $s_N>0$, division by $s_N$ and (4) yield

Although the selected cells and the point $a$ may depend on $h$ and $N$, the constant on the right-hand side is independent of both. The estimate also holds when $T^{1/2}h=0$. Since $h\in\mathcal{H}$ is arbitrary, it follows that $\lVert T\rVert\le8cL^2/[3(N+1)]$ for every $N\ge1$. Letting $N\to\infty$ gives $T=0$. $\square$

Proof of Theorem 1.3. The coordinate functions separate the points of $K$, so $\{1,x_1,x_2,f\}$ generates $C(K)$. By the unique extension characterization of hyperrigidity, it suffices to show that any unital representation $\pi:C(K)\to B(\mathcal{H})$ and any UCP map $\Phi:C(K)\to B(\mathcal{H})$ satisfying

coincide on $C(K)$. Let $E$ be the spectral measure of $\pi$, and let $F$ be the normalized semispectral measure representing $\Phi$.

For $a\in K$, set

Strict convexity and Taylor’s formula give

where $L>0$ is independent of $a$ and $t$. Moreover, since $\psi_a\in\operatorname{span}\{1,x_1,x_2,f\}$,

Let $A,B\subset K$ be nonempty disjoint compact sets. Joint continuity of $(a,t)\mapsto\psi_a(t)$ and (3.8) imply that

For every $a\in A$, the inequality $m1_B\leq\psi_a$ holds on $K$. Integrating with respect to $F$, using (3.9), and compressing to $E(A)\mathcal{H}$, we obtain

If $E(A)\ne0$, the restriction of $E$ to $A$, regarded as a spectral measure on $E(A)\mathcal{H}$, is normalized. Thus Lemma 3.1 applied to (3.10) yields $E(A)F(B)E(A)=0$. This equality is immediate if $E(A)=0$. Positivity of $F(B)$ now gives

Fix a nonempty compact set $B\subset K$ and put

The sets $A_n$ are compact, disjoint from $B$, and increase to $K\setminus B$. By (3.11) and the strong countable additivity of $E$,

Since $0\leq F(B)\leq I$, it follows that

The same inequality holds trivially for $B=\varnothing$.

Let $D\in\mathcal{B}(K)$ and $h\in\mathcal{H}$. Regularity of the finite positive Borel measure $D\mapsto\langle F(D)h,h\rangle$ and (3.12) imply that

Consequently, $F(D) \le E(D)$ for every $D \in\mathcal{B}(K)$. Applying this inequality to $K \setminus D$ and using $F(K) = E(K) = I$, we obtain the reverse inequality. Hence $F = E$, and integration gives $\Phi= \pi$. $\square$

Wydział Matematyki i Informatyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland

Email address: [email protected]