A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces

Introduction

The study of contractive projections and contractively complemented subspaces is one of the classical themes of the isometric theory of Banach spaces. Given a Banach space $X$ and a closed subspace $Y \subset X$, the basic problem is to decide whether there exists a projection $P : X \to X$ such that

and, when such a projection exists, to understand both the intrinsic geometry of $Y$ and the structure of $P$. These two questions are closely related. Indeed, on a smooth Banach space, a subspace can be the range of at most one projection of norm one, see [24]. Since noncommutative $L^p$-spaces are smooth for $1 < p < \infty$, the classification of their contractively complemented subspaces is, in this range, essentially equivalent to the classification of their contractive projections.

For classical $L^p$-spaces, this problem admits a rigid answer. By the work of Ando [2], completed in greater generality by Bernau and Lacey [11], a contractively complemented subspace of a classical $L^p$-space, for $1 < p < \infty$ and $p \ne2$, is itself isometric to an $L^p$-space. Moreover, the corresponding projections are described in terms of weighted conditional expectations. Thus, in the commutative setting, norm-one complementation essentially preserves the category of $L^p$-spaces.

The noncommutative situation is substantially richer. The first obstruction already occurs for Schatten spaces. If $\sigma: S^p \to S^p$ denotes the transpose map, then

is a contractive projection whose range is the space of symmetric matrices. This range is not, in general, a Schatten space. More generally, the seminal work of Arazy and Friedman [3] gives a complete description of contractively complemented subspaces of Schatten spaces in terms of the $L^p$-versions of Cartan factors of types I–IV. In particular, rectangular operator spaces, symmetric and antisymmetric operator spaces, and spin factors all occur naturally. The same work explicitly raises the problem of describing the contractively complemented subspaces of general noncommutative $L^p$-spaces.

This problem remains one of the basic structural questions concerning noncommutative $L^p$-spaces. Its difficulty is not merely technical. The examples above indicate that the range of a contractive projection need not carry an associative structure and that Jordan and ternary structures are unavoidable. At the endpoint $p=1$, this phenomenon is reflected by the appearance of preduals of $JBW^*$-triples, see [39]. On the other hand, once the canonical operator space structure of a noncommutative $L^p$-space is taken into account, rectangular structures become especially natural. In the Schatten setting, Le Merdy, Ricard and Roydor [57] showed that completely 1-complemented subspaces are precisely, up to the appropriate direct sums, spaces of the form $S^p(H,K)$. These are the $L^p$-counterparts of ternary rings of operators.

This suggests that the problem of contractive complementation has several levels of rigidity. A scalar norm-one condition may detect a Jordan structure, whereas matricial contractivity should remember the non-symmetrized ternary product. This principle is consistent with a number of results obtained in recent years. The range of a 2-positive contractive projection on an arbitrary noncommutative $L^p$-space is completely order and completely isometrically isomorphic to another noncommutative $L^p$-space [6]; in the $\sigma$-finite case the projection can be viewed as a conditional expectation modulo a symmetric two-sided change of density. Positive contractive projections lead more generally to Jordan conditional expectations and to nonassociative $L^p$-spaces associated with $JW^*$-algebras, see [4, 7]. In a different direction, contractively decomposable projections are naturally related to $W^*$-ternary rings of operators and to rectangular noncommutative $L^p$-spaces [5]. Recent results give a more intrinsic description of their ranges in terms of rectangular $L^p$-spaces [BoI26, Arh26c].

The general contractive case, however, is much less understood. In particular, there is no general mechanism which starts only from the metric inequality

and recovers the hidden Jordan or ternary structure of the range. The main purpose of the present paper is to introduce such a mechanism. Our approach is based on the differential geometry of the norm, and more precisely on the Levi forms of plurisubharmonic functions on complex Banach spaces.

Differential methods have appeared before in the study of norm-one projections. In particular, second derivatives of norms were successfully used in [56] to study contractive complementation in vector-valued spaces. Here we take a different point of view. Instead of considering only the real Hessian of the norm, we exploit the complex structure and the associated Levi form. Levi forms on complex Banach spaces were investigated in [58]. The key observation is that a contractive projection produces, at any point of its range, a canonical Hilbertian geometry.

More precisely, let $\Omega$ be an open subset of a complex Banach space $X$, let $\Phi\colon\Omega\to\mathbb{R}$ be a $C^2$ plurisubharmonic function and suppose that a projection $P\colon X \to X$ satisfies $P(\Omega) \subseteq\Omega$ and

If $a \in\Omega\cap\operatorname{Ran} P$, then the Levi form $\mathcal{L}_{\Phi,a}$ satisfies

After quotienting by the null space of the Levi form and completing, one obtains a Hilbert space on which $P$ becomes an orthogonal projection. Consequently, although the complement $\mathrm{Id}-P$ need not be contractive for the original Banach norm, both $P$ and $\mathrm{Id}-P$ are contractive for every tangent Levi geometry based at a point of $\operatorname{Ran} P$. Equivalently, the involution

becomes a unitary involution on each of these tangent Hilbert spaces.

Thus a single Banach space projection gives rise to a whole family of Hilbertian geometries indexed by the points of its range. At first order, contractivity is governed by duality mappings and norming functionals, in accordance with the classical characterization of Calvert [23]. The first-order condition identifies the directions annihilated by the corresponding norming functional. Differentiating this condition along the range yields the orthogonality relations for the Levi form. The role of the second-order viewpoint is therefore not to provide an independent contractivity condition, but to package these relations into a Hilbertian geometry. When the Levi form is positive, the decomposition

becomes orthogonal for this tangent geometry, making relative positive operators and their spectral calculus available.

Comparing two such Hilbertian geometries leads naturally to relative positive operators. Suppose first that $\operatorname{Ran} P$ is finite-dimensional and that the restrictions to $\operatorname{Ran} P$ of the Levi forms based at two points $a,b \in\Omega\cap\operatorname{Ran} P$ are nondegenerate. If $\langle\cdot,\cdot\rangle_a$ and $\langle\cdot,\cdot\rangle_b$ denote the corresponding inner products on $\operatorname{Ran} P$, there exists a unique positive invertible operator $\Delta_{b|a}\colon\operatorname{Ran} P\to\operatorname{Ran} P$ characterized by

It is selfadjoint for the first Hilbertian structure, and its imaginary powers form a one-parameter group of isometries for both tangent Hilbertian geometries.

A stronger phenomenon appears when the relative operator can be constructed on a common ambient Hilbert completion. More precisely, if the Levi forms at $a$ and $b$ are nondegenerate on $X$ and define equivalent Hilbertian norms, the corresponding ambient relative operator $\widehat{\Delta}_{b|a}$ commutes with the orthogonal extension of $P$. Consequently, the closures of $\operatorname{Ran} P$ and $\operatorname{Ker} P$ are reducing subspaces for $\widehat{\Delta}_{b|a}$, and

Thus the relative one-parameter group dynamically preserves the decomposition determined by the projection. This is the origin of the modular terminology used in this paper. The appearance of positive relative operators and their imaginary powers is reminiscent of modular constructions associated with standard subspaces, see for instance [28]. The important difference is that here the modular object is not given in advance: it is extracted directly from a norm-one projection and from the local differential geometry of the ambient Banach space.

The case of noncommutative $L^4$-spaces is particularly well suited to this approach. There are two reasons for this. First, the fourth power of the norm has an explicit Levi form. If

then

Thus the tangent Hilbert geometry separates naturally into a left part and a right part. This is precisely the two-sided structure which is needed in order to recover rectangular operator spaces.

Second, the duality mapping of $L^4(\mathcal{M})$ is cubic:

Polarization of this cubic mapping produces the Jordan triple product

The special role of the exponent 4 in noncommutative $L^p$-geometry also appears in the theory of 2-isometries, where selfpolar forms can be used to recover operator-algebraic structure; see [48]. Our situation is different: we start with a projection rather than an isometry, and no positivity or matricial assumption is imposed initially. Nevertheless, the cubic duality mapping allows us to extract a ternary structure directly from scalar contractivity.

Indeed, let $P:L^4(\mathcal{M})\to L^4(\mathcal{M})$ be a contractive projection and set

where $P^\dagger$ denotes the adjoint projection under the $L^4$–$L^{4/3}$ trace duality. We prove that

In particular, the nonlinear image $j_4(Y)$ is a linear subspace of $L^{4/3}(\mathcal{M})$. Polarization then yields

Thus scalar contractivity forces Jordan triple relations taking values in the dual range. At this stage, however, the product is not internal to $Y$, so these relations do not yet equip $Y$ with a Jordan triple structure. Recovering an internal ternary product requires the additional deweighting argument developed below.

The first nontrivial matrix level yields a stronger conclusion. One of the main results of this paper is that for projections on $L^4$ the level 2 condition already contains all matrix levels. More precisely, we prove that

Hence, for projections on noncommutative $L^4$-spaces, complete contractivity can be tested on $2\times2$ matrices. This collapse of an infinite family of matricial norm inequalities to a single matrix level is one of the main rigidity phenomena established here.

By trace-adjoint duality, the same equivalence holds for projections on $L^{4/3}(\mathcal{M})$, see Corollary 6.6.

For Schatten spaces, the equivalence between 2-contractive and completely contractive complementation follows from [57 Theorem 1.1]. The contribution of (1.17) is a direct duality-mapping proof beyond the Schatten setting, together with the intrinsic criterion $YY^*Y\subset Y^\sharp$. Theorem 7.3 extends this equivalence to Haagerup $L^4$-spaces associated with arbitrary von Neumann algebras. We refer also to [Arl26], [15] and [76] for related results.

The mechanism behind (17) is transparent. Scalar contractivity gives only the symmetrized expression (13). By applying the scalar result to the amplification $\operatorname{Id}_{S_4^d} \otimes P$ and choosing suitable $2 \times2$ block matrices, one can separate the two outer variables and isolate the non-symmetrized product $xy^*z$. Once

is known, the same property at every matrix level follows entrywise, and the cubic characterization of contractivity implies complete contractivity. In this sense, scalar contractivity detects Jordan triple geometry, whereas 2-contractivity detects TRO geometry.

The same polynomial principle gives a finite-level criterion at every even exponent. If $p=2m$ for some integer $m \ge2$ and

we prove that, for an arbitrary von Neumann algebra $\mathcal{M}$, an $N_m$-contractive projection on its Haagerup space $L^{2m}(\mathcal{M})$ is completely contractive.

The proof extracts an arbitrary alternating product of length $2m-1$ as a torus Fourier coefficient of the power duality mapping at the matrix level $N_m$. Trace-adjoint duality gives the analogous result at the conjugate exponent $\frac{2m}{2m-1}$. In particular, level 2 suffices for $p=6$ and $p=\frac{6}{5}$. We do not claim that the level in (19) is optimal. Indeed, for Schatten spaces, level 2 is already known to suffice for any $1<p<\infty$, $p\ne2$, by [57 Theorem 1.1].

The remaining difficulty is that the ternary products still belong a priori to the dual range $Y^\#$, not to $Y$. In the finite-dimensional setting, the unsymmetrized ternary relation yields natural left and right multiplication maps from $Y$ into $Y^\#$. Choose a basis $x_1,\ldots,x_k$ of $Y$ and define

We prove the exact two-sided relation

This algebraic relation has a natural tangent-Hilbert interpretation. Indeed, if $\mathcal{H}^{\ell}$ and $\mathcal{H}^{r}$ denote the sums of the left and right tangent forms associated with the points $x_1,\ldots,x_k$, then