David P. Blecher
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View article: Commutativity of operator algebras
Commutativity of operator algebras Open
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 operato…
View article: Real decomposable maps on operator systems
Real decomposable maps on operator systems Open
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.) W…
View article: Real Non-Commutative Convexity I
Real Non-Commutative Convexity I Open
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 sect…
View article: M$M$‐Ideals in real operator algebras
M$M$‐Ideals in real operator algebras Open
In a recent paper, we showed that a subspace of a real ‐triple is an ‐summand if and only if it is a ‐closed triple ideal. As a consequence, ‐ideals of real ‐triples, including real ‐algebras, real ‐algebras and real TROs, correspond to no…
View article: Real operator systems
Real operator systems Open
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 i…
View article: Geometric influences on quantum Boolean cubes
Geometric influences on quantum Boolean cubes Open
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, …
View article: M-ideals in real operator algebras
M-ideals in real operator algebras Open
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…
View article: A missing theorem on dual spaces
A missing theorem on dual spaces Open
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 pred…
View article: Null projections and noncommutative function theory in operator algebras
Null projections and noncommutative function theory in operator algebras Open
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 projec…
View article: $M$-ideals, yet again: the case of real JB$^*$-triples
$M$-ideals, yet again: the case of real JB$^*$-triples Open
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 comp…
View article: Real operator spaces and operator algebras
Real operator spaces and operator algebras Open
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…
View article: On a class of subdiagonal algebras
On a class of subdiagonal algebras Open
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…
View article: Operator space complexification transfigured
Operator space complexification transfigured Open
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' conditio…
View article: Real structure in operator spaces, injective envelopes and $G$-spaces
Real structure in operator spaces, injective envelopes and $G$-spaces Open
We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and $C^*$-envelope. We consider the interacti…
View article: Real operator spaces and operator algebras
Real operator spaces and operator algebras Open
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…
View article: A noncommutative Bishop peak interpolation-set theorem
A noncommutative Bishop peak interpolation-set theorem Open
We prove a noncommutative version of Bishop's peak interpolation-set theorem.
View article: Abelian von Neumann algebras, measure algebras and L^\infty-spaces
Abelian von Neumann algebras, measure algebras and L^\infty-spaces Open
We give a fresh account of the astonishing interplay between abelian von Neumann algebras, L^\infty-spaces and measure algebras, including an exposition of Maharam's theorem from the von Neumann algebra perspective.
View article: Real positive maps and conditional expectations on operator algebras
Real positive maps and conditional expectations on operator algebras Open
Most of this article is an expanded version of our conference talk. It is essentially a survey, but some part, like most of the lengthy Section 5, is comprised of new results whose proofs are unpublished elsewhere. We begin by reviewing th…
View article: Contractive projections and real positive maps on operator algebras
Contractive projections and real positive maps on operator algebras Open
We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and J…
View article: Jordan operator algebras revisited
Jordan operator algebras revisited Open
Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with a^2 in A for all a in A. In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory…
View article: Noncommutative topology and Jordan operator algebras
Noncommutative topology and Jordan operator algebras Open
Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We sh…
View article: Involutive operator algebras
Involutive operator algebras Open
Examples of operator algebras with involution include the operator $*$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix …
View article: Operator ∗‐correspondences in analysis and geometry
Operator ∗‐correspondences in analysis and geometry Open
An operator -algebra is a non-self-adjoint operator algebra with completely isometric involution. We show that any operator -algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the o…
View article: Ueda’s peak set theorem for general von Neumann algebras
Ueda’s peak set theorem for general von Neumann algebras Open
We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras, to sigma-finite von Neumann algebras (that is, von Neumann algebras with a faithful state; which includes those on a separable Hilbert sp…
View article: Jordan operator algebras: Basic theory
Jordan operator algebras: Basic theory Open
Jordan operator algebras are norm-closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jorda…