Roger R. Smith

Schur Products and Matrix Completions

We prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain Schur product map defined on a certain subspace of matrices is a positive map. By analyzing the positive elements of this subspace, we obtain new proofs of the results of H. Dym and I. Gohberg and Grone, Johnson, Sa, and Wolkowitz (Linear Algebra Appl.58 (1984), 109–124). (Linear Algebra Appl.36 (1981), 1–24). We also obtain a new proof of a result of U. Haagerup (Decomposition of completely bounded maps on operation algebras, preprint), characterizing the norm of Schur product maps, and a new Hahn-Banach type extension theorem for these maps. Finally, we obtain generalizations of many of these results to matrices of operators, which we apply to the study of representations of certain subalgebras of the n × n matrices.

Multilinear maps and tensor norms on operator systems

Abstract We extend work of Christensen and Sinclair on completely bounded multilinear forms to the case of subspaces of C ∗ algebras, and obtain a representation theorem and a Hahn-Banach extension theorem for such maps. In the second part of the paper the Haagerup norms on tensor products are investigated, and we obtain new characterizations of these quantities.

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.

M-ideal structure in Banach algebras

Abstract It is shown that M -ideals in a Banach algebra with identity are subalgebras, and that they are ideals if the algebra is commutative. Counterexamples demonstrate that these are the strongest results available. The theory is then applied to familiar classes of Banach spaces.

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

Perturbations of nuclear C*-algebras

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Perturbations of C*-Algebraic Invariants

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.

The Laplacian MASA in a free group factor

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.

Finite dimensional injective operator spaces

Kadison and Kastler introduced a metric on the set of all C*-algebras on a fixed Hilbert space. In this paper structural properties of C*-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison’s similarity problem transfers to close C*-algebras. In establishing this result we answer questions about closeness of commutants and tensor products when one algebra satisfies the similarity property. We also examine K-theory and traces of close C*-algebras, showing that sufficiently close algebras have isomorphic Elliott invariants when one algebra has the similarity property.

Completely contractive factorizations of C∗-algebras

The Laplacian (or radial) masa in a free group factor is generated by the sum of the generators and their inverses. We show that such a masa B is strongly singular and has Popa invariant δ(B) = 1. This is achieved by proving that the conditional expectation E B onto B is an asymptotic homomorphism. We also obtain similar results for the free product of discrete groups, each of which contains an element of infinite order.