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David P. Blecher

Publications and source records attributed to David P. Blecher.

At least 55 records · Page 3Linked to original sources
arXiv2016

Quantum measurable cardinals

We investigate states on von Neumann algebras which are not normal but enjoy various forms of infinite additivity, and show that these exist on $B(H)$ if and only if the cardinality of an orthonormal basis of $H$ satisfies various large cardinal conditions. For instance, there is a singular countably additive pure state on $B(l^2(κ))$ if and only if $κ$ is Ulam measurable, and there is a singular ${<}\,κ$-additive pure state on $B(l^2(κ))$ if and only if $κ$ is measurable. The proofs make use of Farah and Weaver's theory of quantum filters. Applications to Ueda's peak set theorem for von Neumann algebras are discussed in the final section.

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arXiv2016

Completely contractive projections on operator algebras

The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.

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arXiv2015

Roots in operator and Banach algebras

We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the `sign' and `geometric mean' of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.

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arXiv2015

Generalization of C*-algebra methods via real positivity for operator and Banach algebras

With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. As motivation note that the `completely' real positive maps on C*-algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C*-algebra it generates. We have continued this work together with Read, and also with Matthew Neal. Recently with Narutaka Ozawa we have investigated the parts of the theory that generalize further to Banach algebras. In the present paper we describe some of this work which is connected with generalizing various C*-algebraic techniques initiated by Richard V. Kadison. In particular Section 2 is in part a tribute to him in keeping with the occasion of this volume, and also discusses some of the origins of the theory of positivity (in our sense) in the setting of algebras, which the later parts of our paper develops further. The most recent work will be emphasized.

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arXiv2015

Real positivity and approximate identities in Banach algebras

Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.

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arXiv2014

Operator algebras with contractive approximate identities: A large operator algebra in c0

We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity, such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0.

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arXiv2014

Order theory and interpolation in operator algebras

We continue our study of operator algebras with and contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algebra and the C*-algebra it generates. In much of this it is not necessary that the algebra have an approximate identity. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.

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arXiv2014

Operator algebras with contractive approximate identities: Weak compactness and the spectrum

We continue our study of operator algebras with contractive approximate identities (cais) by presenting a couple of interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series, for example about whether, roughly speaking, `weak compactness' of an operator algebra, or the lack of it, can be seen in the spectra of its elements.

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arXiv2014

On positivity and roots in operator algebras

In earlier papers the second author and Charles Read have introduced and studied a new notion of positivity for operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. The present paper consists of complements to some facts in the just mentioned papers, concerning this notion of positivity. For example we prove a result on the numerical range of products of the roots of commuting operators with numerical range in a sector.

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arXiv2013

Operator algebras with contractive approximate identities, III

We continue our study of operator algebras with contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the first part of this paper we do a more systematic development of this positivity and its associated ordering, proving many foundational facts. In much of this it is not necessary that the algebra have an approximate identity. In the second part we study interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series. Indeed the present work solves almost all of the open questions raised in these papers. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.

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arXiv2013

Noncommutative peak interpolation revisited

Peak interpolation is concerned with a foundational kind of mathematical task: building functions in a fixed algebra $A$ which have prescribed values or behaviour on a fixed closed subset (or on several disjoint subsets). In this paper we do the same but now $A$ is an algebra of operators on a Hilbert space. We briefly survey this {\em noncommutative peak interpolation}, which we have studied with coauthors in a long series of papers, and whose basic theory now appears to be approaching its culmination. This program developed from, and is based partly on, theorems of Hay and Read whose proofs were spectacular, but therefore inaccessible to an uncommitted reader. We give short proofs of these results, using recent progress in noncommutative peak interpolation, and conversely give examples of the use of these theorems in peak interpolation. For example, we prove a useful new noncommutative peak interpolation theorem.

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arXiv2013

Outers for noncommutative H^p revisited

We continue our study of outer elements of the noncommutative H^p spaces associated with Arveson's subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in H^p actually satisfy the stronger condition that there exist a_n in A with h a_n in Ball(A) and h a_n \to 1 in p-norm.

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arXiv2012

Ideals and hereditary subalgebras in operator algebras

This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (HSA's), which are in some sense generalization of ideals. Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which are `weakly compact'. We also give several examples answering natural questions that arise in such an investigation.

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arXiv2012

Open projections in operator algebras II: Compact projections

We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of `peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is weak* closed.

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arXiv2012

Metric characterizations II

The present paper is a sequel to our paper "Metric characterization of isometries and of unital operator spaces and systems". We characterize certain common objects in the theory of operator spaces (unitaries, unital operator spaces, operator systems, operator algebras, and so on), in terms which are purely linear-metric, by which we mean that they only use the vector space structure of the space and its matrix norms. In the last part we give some characterizations of operator algebras (which are not linear-metric in our strict sense described in the paper).

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arXiv2012

Open projections in operator algebras I: Comparison theory

We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the authors and Hay), and the open partial isometries (tripotents) introduced by the authors.

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arXiv2011

Operator algebras with contractive approximate identities

We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for $C^*$-algebras, and we give many other applications.

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