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David P. Blecher

Publications and source records attributed to David P. Blecher.

At least 19 recordsLinked to original sources
arXiv2026

Compact convex sets and bases--classical and noncommutative

Matrix and noncommutative convexity constitute an important area of modern noncommutative analysis and have found significant applications in mathematical physics. In the first part of our paper we give an abstract characterization of matrix convex sets, and compact matrix convex sets. Our approach is in some part via a universal Banach space (resp.\ operator space) $X_K$ of an abstract compact convex set (resp.\ matrix convex set) K. This turns out to be a concrete construction of the base norm space (resp.\ nc base norm space) with base K, together with a natural TVS topology. Noncommutative (nc for short) base norm spaces, recently developed by the first author and Hay, are an important class of operator spaces which include duals and preduals of unital C*-algebras and von Neumann algebras, and operator systems, where the `base' is exactly the noncommutative convex set of (matrix) states on these. In the later parts of the paper we give many applications, mostly to base norm spaces (classical and noncommutative). We also refine some of our recent results concerning regularity of convex sets (classical and noncommutative). We give several interesting characterizations of base norm spaces (classical and noncommutative). Any such characterization will correspond by duality to a new characterization of operator systems, or in the classical case, of function systems. For example, (complex) nc dual base norm spaces are the matrix ordered LCTVS's V such that V (at level 1) has a linear base which is compact.

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arXiv2026

Real decomposable maps on operator systems

We initiate and study the theory of ``real decomposable maps" between real operator systems. Formally, this is new even in the complex case, which hitherto has restricted itself to the case where the systems are complex C*-algebras. We investigate how our definition interacts with the existing theory (which it generalizes) and with the complexification. In particular, a surprising term appears in the `Jordan decomposition' of real decomposable maps. This term constitutes a new class of completely bounded maps, a class that also showed up in disguised form in our recent study of real noncommutative (nc) convexity, and whose theory is likely to have applications in that subject. We also check the real case of many important known results related to decomposability, for example results about the weak expectation property or injectivity of von Neumann algebras.

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arXiv2026

Real Non-Commutative Convexity I

We initiate the theory of real noncommutative (nc) convex sets, the real case of the recent and profound complex theory developed by Davidson and Kennedy. The present paper focuses on the real case of the topics from the first several sections of their Memoir. Later results will be discussed in future papers. We develop here some of the infrastructure of real nc convexity, giving many foundational structural results for real operator systems and their associated nc convex sets, and elucidate how the complexification interacts with the basic convexity theory constructions. Several new features appear in the real case, including the novel notion of the complexification of a nc convex set.

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arXiv2026

Real Noncommutative Convexity II: Extremality and nc convex functions

We continue the development of real noncommutative (nc) convexity, building on the recent and profound complex theory of Davidson and Kennedy. The present paper focuses on the theory of nc extreme points (and pure and maximal points) and the nc Choquet boundary in the real setting, as well as on the theory of real nc convex and semicontinuous functions and real nc convex envelopes. Our main emphasis is on how these notions interact with complexification. In particular, parts of the paper analyze in detail how various notions of `extreme' or `maximal' relate to our earlier concept of the complexification of a convex set. Several new features emerge in the real case, especially in the later sections, including the novel notions of the complexification of a nc convex function and of the complexification of the convex envelope of a nc function. With an appendix by T. Russell.

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arXiv2026

Base norm spaces--classical, complex, and noncommutative

We generalize the theory of base norm spaces to the complex case, and further to the noncommutative setting relevant to `quantum convexity'. In particular, we establish the duality between complex Archimedean order unit spaces and complex base norm spaces, as well as the corresponding duality between their noncommutative counterparts. Additional topics include an exploration of natural connections with various notions of quantum convexity and regularity of noncommutative convex sets, and an analysis of how these concepts interact with complexification. We also define, as in the classical case, a class that contains and generates the noncommutative base norm spaces, but is defined by fewer axioms. We show how this may be applied to provide new and interesting examples of noncommutative base norm spaces.

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arXiv2026

Regularity of compact convex sets--classical and noncommutative

The classical theory of regularity of embeddings of compact convex sets was developed in the 1970s, exclusively in the real case, and even there it does not appear to have been stated in its simplest form. We begin by revisiting this setting, showing that under a reasonable condition, every locally convex topological vector space E that contains and is spanned by a compact convex set lying in a hyperplane not passing through the origin, is a (specific) dual Banach space equipped with the weak* topology. Second, we establish the corresponding regularity theory for convex sets in complex LCTVS's. Third, we develop a theory of regular embeddings for complex noncommutative convex sets, in the sense of Davidson and Kennedy. Finally, we use the complex theory to derive a theory of regular embeddings for real noncommutative convex sets. Interestingly, at present there appears to be no direct route to the latter.

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arXiv2025

Commutativity of operator algebras

We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator algebras}: the subalgebras of B(H) on which the transpose map is a complete isometry. In previous work we studied the unital case, where reversibility is equivalent to commutativity. We give many sufficient conditions under which a nonunital reversible or symmetric operator algebra is commutative. We also give many complementary results of independent interest, and solve a few open questions from previous papers. Not every reversible or symmetric operator algebra is commutative, however we show that they all are 3-commutative. That is, order does not matter in the product of three or more elements from A. The proof of this relies on a technical analysis involving the injective envelope. Indeed nonunital algebras are often enormously more complicated than unital ones in regard to the topics we consider. On the positive side, our considerations raise very many questions even for low dimensional matrix algebras, some of which are of a computational nature and might be suitable for undergraduate research. The canonical anticommutation relations from mathematical physics play a significant role.

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arXiv2025

Real operator systems

Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in modern analysis and also in modern quantum physics (such as quantum information theory). They have an extensive theory, and have very important applications in all of these subjects. We present here the real case of the theory of (complex) operator systems, and also the real case of their remarkable tensor product theory, due in the complex case to Paulsen and his coauthors and students (such as Kavruk), building on pioneering earlier work of Kirchberg and others. We uncover several notable differences between the real and complex theory, including the absence of minimal and maximal functors in the category of real operator systems. We also develop very many foundational structural results for real operator systems, and elucidate how the complexification interacts with the basic constructions in the subject. In the final two sections of our paper we study real analogues of the Kirchberg conjectures (and of several important related problems that have attracted much interest recently), and study the deep relationships between them.

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arXiv2025

Null projections and noncommutative function theory in operator algebras

We study projections in the bidual of a $C^*$-algebra $B$ that are null with respect to a subalgebra $A$, that is projections $p\in B^{**}$ satisfying $|ϕ|(p)=0$ for every $ϕ\in B^*$ annihilating $A$. In the separable case, $A$-null projections are precisely the peak projections in the bidual of $A$ at which the subalgebra $A$ interpolates the entire $C^*$-algebra $B$. These are analogues of null sets in classical function theory, on which several profound results rely. This motivates the development of a noncommutative variant, which we use to find appropriate `quantized' versions of some of these classical facts. Through a delicate generalization of a theorem of Varopoulos, we show that, roughly speaking, sufficiently regular interpolation projections are null precisely when their atomic parts are. As an application, we give alternative proofs and sharpenings of some recent peak-interpolation results of Davidson and Hartz for algebras on Hilbert function spaces, also illuminating thereby how earlier noncommutative peak-interpolation theory may be applied. In another direction, given a convex subset of the state space of $B$, we characterize when the associated Riesz projection is null. This is then applied to various important topics in noncommutative function theory, such as the F.& M. Riesz property, the existence of Lebesgue decompositions, the description of Henkin functionals, and Arveson's noncommutative Hardy spaces (maximal subdiagonal algebras).

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arXiv2025

$\mathrm{M}$-ideals: from Banach spaces to rings

We introduce and investigate a class of ring ideals, termed ring $\mathrm{M}$-ideals, inspired by the Alfsen--Effros theory of $\mathrm{M}$-ideals in Banach spaces. We show that $\mathrm{M}$-ideals extend the classical notion of essential ideals and subsume them as a subclass. The central theorem provides a full characterization: an ideal is an $\mathrm{M}$-ideal if and only if it is either essential or relatively irreducible. This dichotomy reveals the abundant and diverse nature of $\mathrm{M}$-ideals, encompassing both essential and minimal ideals, and admits natural generalizations in rings beyond the commutative and unital settings. We systematically study the algebraic stability of $\mathrm{M}$-ideals under standard constructions such as intersection, quotient, direct product, and Morita equivalence and establish their behavior in topological rings and operator algebras. In certain rings such as $\mathbb{Z}_n$ and C*-algebras, we completely classify $\mathrm{M}$-ideals and relate them to algebraically minimal projections and central idempotents. The ring $\mathrm{M}$-ideals in $C(K)$ are shown to be precisely the essential ideals or those minimal ideals corresponding to isolated points. Structurally, we show that the absence of proper $\mathrm{M}$-ideals characterizes simplicity, while rings in which every proper $\mathrm{M}$-ideal is a direct summand must decompose as finite direct sums of simple rings. In closing, we introduce the notion of $\mathrm{M}$-complements, drawing an analogy with essential extensions in module theory, and demonstrate their existence.

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arXiv2025

M-ideals in real operator algebras

In a recent paper we showed that a subspace of a real JBW*-triple is an M-summand if and only if it is a weak*-closed triple ideal. As a consequence, M-ideals of real JB*-triples, including real C*-algebras, real JB*-algebras and real TROs, correspond to norm-closed triple ideals. In the present paper we extend this result to (possibly non-selfadjoint) real operator algebras and Jordan operator algebras, where the argument is necessarily different. We also give simple characterizations of one-sided M-ideals in real operator algebras, and give some applications to that theory.

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arXiv2024

Geometric influences on quantum Boolean cubes

In this work, we study three problems related to the $L_1$-influence on quantum Boolean cubes. In the first place, we obtain a dimension free bound for $L_1$-influence, which implies the quantum $L^1$-KKL Theorem result obtained by Rouze, Wirth and Zhang. Beyond that, we also obtain a high order quantum Talagrand inequality and quantum $L^1$-KKL theorem. Lastly, we prove a quantitative relation between the noise stability and $L^1$-influence. To this end, our technique involves the random restrictions method as well as semigroup theory.

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arXiv2024

A missing theorem on dual spaces

We answer in the affirmative the surprisingly difficult questions: If a complex Banach space possesses a real predual X, then is X a complex Banach space? If a complex Banach space possesses a real predual, then does it have a complex predual? We also answer the analogous questions for operator spaces, that is spaces of operators on a Hilbert space, up to complete isometry. Indeed we use operator space methods to solve the Banach space question above.

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arXiv2024

Real operator spaces and operator algebras

We verify that a large portion of the theory of complex operator spaces and operator algebras (as represented by the 2004 book by the author and Le Merdy for specificity) transfers to the real case. We point out some of the results that do not work in the real case. We also discuss how the theory and standard constructions interact with the complexification. For example, we develop the real case of the theory of operator space multipliers and the operator space centralizer algebra, and discuss how these topics connect with the complexification. This turns out to differ in some important details from the complex case. We also characterize real structure in complex operator spaces; and give `real' characterizations of some of the most important objects in the subject.

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arXiv2024

$M$-ideals, yet again: the case of real JB$^*$-triples

We prove that a subspace of a real JBW$^*$-triple is an $M$-summand if and only if it is a weak$^*$-closed triple ideal. As a consequence, $M$-ideals of real JB$^*$-triples correspond to norm-closed triple ideals. As in the setting of complex JB$^*$-triples, a geometric property is characterized in purely algebraic terms. This is a newfangled treatment of the classical notion of $M$-ideal in the real setting by a fully new approach due to the unfeasibility of the known arguments in the setting of complex C$^*$-algebras and JB$^*$-triples. The results in this note also provide a full characterization of all $M$-ideals in real C$^*$-algebras, real JB$^*$-algebras and real TROs.

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arXiv2023

On a class of subdiagonal algebras

We investigate some new classes of operator algebras which we call semi-$σ$-finite subdiagonal and Riesz approximable. These constitute the most general setting to date for a noncommutative Hardy space theory based on Arveson's subdiagonal algebras. We develop this theory and study the properties of these new classes.

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arXiv2023

Operator space complexification transfigured

Given a finite group G, a central subgroup H of G, and an operator space X equipped with an action of H by complete isometries, we construct an operator space $X_G$ equipped with an action of G which is unique under a `reasonable' condition. This generalizes the operator space complexification $X_c$ of $X$. As a linear space $X_G$ is the space obtained from inducing the representation of H to G (in the sense of Frobenius).

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