Allan M. Sinclair

A Characterization of Operator Algebras

Abstract An operator algebra is a uniformly closed algebra of bounded operators on a Hilbert space. In this paper we give a characterization of unital operator algebras in terms of their matricial norm structure. More precisely if A is an L ∞ -matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1, then there is a completely isometric isomorphism of A onto a unital operator algebra. Indeed the multiplication on A need not be assumed to be associative for this conclusion to follow. Examples are given to show that the condition on the identity is necessary. It follows from the above that the quotient of an operator algebra by a closed two-sided ideal (with the natural matricial structure) is again an operator algebra up to complete isometric isomorphism.

Representations of completely bounded multilinear operators

A deli&ion of a completely bounded multilinear operator from one C*-algebra into another is introduced. Each completely bounded multilinear operator from a (‘*-algebra into the algebra of bounded linear operators on a Hilbert space is shown to be representable in terms of *-representations of the C*-algebra and interlacing operators. This result extends Wittstock’s Theorem that decomposes a completely bounded linear operator from a C*-algebra into an injective C*-algebra into completely positive linear operators. 0 1987 Academic Press, Inc Stinespring’s Theorem gives a useful representation for a completely positive linear operator from a C*-algebra into the algebra BL(H) for continuous linear operators on a Hilbert space H [12, 11. Using this representation with Wittstock’s Theorem that decomposes a completely bounded linear operator as a finite linear combination of completely positive linear operators [14, lo], one obtains a representation of a completely bounded linear operator from a C*-algebra into BL(H). Our main result is a representation theorem for completely bounded multilinear operators from a C*-algebra into BL(H), which generalizes this representation of completely bounded linear operators. Corollaries give multilinear generalizations of several results known for completely bounded linear operators. We shall briefly describe the type of representations to be 151

Hochschild cohomology of von Neumann algebras

Preface 1. Introduction 2. Completely bounded operators 3. Derivations 4. Averaging in continuous and normal cohomology 5. Completely bounded cohomology 6. Hyperfinite subalgebras 7. Continuous cohomology 8. Stability of products 9Appendix Bibliography Notation Index.

STRONG SINGULARITY FOR SUBALGEBRAS OF FINITE FACTORS

In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type

Strongly singular masas in type II1 factors

\tto

Perturbations of nuclear C*-algebras

factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity.

Equivalence of norms on operator space tensor products of *-algebras

Abstract. In this paper we introduce and study strongly singular maximal abelian self-adjoint subalgebras of type II1 factors. We show that certain elements of free groups and of non-elementary hyperbolic groups generate such masas, and these also give new examples of masas for which Popas invariant

Perturbations of C*-Algebraic Invariants

\delta(\cdot)

The Laplacian MASA in a free group factor

is 1. We also explore the connection between Popas invariant and strong singularity.

Bounded approximate identities, factorization, and a convolution algebra

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.This paper provides the details of the results announced in Christensen et al. Proc. Natl. Acad. Sci. USA 107 (2010), 587–591.