arXiv subjects
Publications and source records attributed to David P. Blecher.
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).
We prove a noncommutative version of Bishop's peak interpolation-set theorem.
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 interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.
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.
We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In closing we introduce the notion of `noncommutative Jensen measures', and show that as in the classical case representing measures of logmodular algebras are Jensen measures. The proofs of the two main cycles of results rely on the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction theorem, etc.
We present some foundations for a theory of real operator algebras and real Jordan operator algebras, and the various morphisms between these. A common theme is the ingredient of real positivity from papers of the first author with Read, Neal, Wang, and other coauthors, which we import to the real scalar setting here.
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 the theory of real positivity of operator algebras initiated by the author and Charles Read. Then we present several new general results (mostly joint work with Matthew Neal) about real positive maps. The key point is that real positivity is often the right replacement in a general algebra $A$ for positivity in C*-algebras. We then apply this to studying contractive projections (`conditional expectations') and isometries of operator algebras. For example we generalize and find variants of certain classical results on positive projections on C*-algebras and JB algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In previous work with Neal we had done the `completely contractive' case; we focus here on describing the real positive contractive case from recent work with Neal. We also prove here several new and complementary results on this topic due to the author, indeed this new work constitutes most of Section 5. Finally, in the last section we describe a related part of some recent joint work with Labuschagne on what we consider to be a good noncommutative generalization of the `characters' (i.e. homomorphisms into the scalars) on an algebra. Such characters are a special case of the projections mentioned above, and are shown to be intimately related to conditional expectations. The idea is to try to use these to generalize certain classical function algebra results involving characters.
Let A be a closed subalgebra of a C*-algebra, that is a closed algebra of Hilbert space operators. We generalize to such operator algebras $A$ several key theorems and concepts from the theory of classical function algebras. In particular we consider several problems that arise when generalizing classical function algebra results involving characters ((contractive) homomorphisms into the scalars) on the algebra. For example, the Jensen inequality, the related Bishop-Ito-Schreiber theorem, and the theory of Gleason parts. We will usually replace characters (classical function algebra case) by D-characters, certain completely contractive homomorphisms $Φ: A \to D$, where D is a C*-subalgebra of A. We also consider some D-valued variants of the classical Gleason-Whitney theorem.
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 JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which are foundational to the study of such maps, and of interest in their own right. We also prove a new Banach-Stone type theorem for isometries between operator algebras or Jordan operator algebras. An application of this is given to the characterization of symmetric real positive projections.
We study possible noncommutative (operator algebra) variants of the classical Hoffman-Rossi theorem from the theory of function algebras. In particular we give a condition on the range of a contractive weak* continuous homomorphism defined on an operator algebra A, which is necessary and sufficient (in the setting we explain) for a positive weak* continuous extension to any von Neumann algebra containing A.
We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on L^p spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of L^2-operator algebras and Banach algebras, to algebras of bounded operators on Lp spaces.
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 algebras, (complexifications) of real operator algebras, and an operator algebraic version of the {\em complex symmetric} operators studied by Garcia, Putinar, Wogen and others. We investigate the general theory of involutive operator algebras, and give many applications.
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 of associative operator algebras, in particularly surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.
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 space, or with separable predual.) To achieve this extension completely new strategies had to be invented at certain key points, ultimately resulting in a more operator algebraic proof of the result. Ueda showed in the case of finite von Neumann algebras that his peak set theorem is the fountainhead of many other very elegant results, like the uniqueness of the predual of such subalgebras, a highly refined F and M Riesz type theorem, and a Gleason-Whitney theorem. The same is true in our more general setting, and indeed we obtain a quite strong variant of the last mentioned theorem. We also show that set theoretic issues dash hopes for extending the theorem to some other large general classes of von Neumann algebras, for example finite or semi-finite ones. Indeed certain cases of Ueda's peak set theorem, for a von Neumann algebra M, may be seen as `set theoretic statements' about M that require the sets to not be `too large'.
Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with $a^2 \in A$ for all $a \in A$. We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a `full' noncommutative topology in the C*-algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of C*-algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about $L+L^*$ for a left ideal $L$ in a C*-algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in $C^*$-algebras and various subspaces of $C^*$-algebras, related to open and closed projections, and technical $C^*$-algebraic results of Brown.
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 Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.
In a previous paper we generalized the theory of W*-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel "Rigged modules II", giving many new results about weak*-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty}(D)$, and prove that for a weak* closed function algebra A, the weak Picard group of A is a semidirect product of the automorphism group of A, and the subgroup consisting of symmetric equivalence bimodules.
In a previous paper with Kashyap we generalized the theory of $W^*$-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. The present paper and its contemporaneous predecessor comprise the sequel which we promised at that time would be forthcoming. We give many new results about rigged and weak*-rigged modules and their tensor products, such as an Eilenberg-Watts type theorem.