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The similarity problem and hyperreflexivity of von Neumann algebras

G. K. Eleftherakis, E. Papapetros

Abstract

The similarity problem is one of the most famous open problems in the theory of $C^*$-algebras. We say that a $C^*$-algebra $\cl A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $u\colon \cl A\to \cl B(H)$ is similar to a $*$-homomorphism and that a von Neumann algebra $\cl A$ satisfies the weak similarity property ((WSP) for short) if every $\mathrm{w}^*$-conitnuous unital and bounded homomorphism $u\colon \cl A\to \cl B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $\cl A$ satisfies (WSP) if and only if the algebras $\cl A^{\prime}\bar \otimes \cl B(\ell^2(I))$ are hyperreflexive for all cardinals $I.$ In the case in which $\cl A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $\cl A^\prime \bar \otimes \cl B(\ell^2(\bb{N}))$ is hyperreflexive. We also introduce the hypothesis {\bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {\bf (CHH)}, all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $\cl A\bar \otimes \cl B,$ where $\cl A$ is an injective von Neumann algebra and $\cl B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the $\text{w}^*$-similarity degree $d_{*}(\cl A\bar \otimes \cl B).$

The similarity problem and hyperreflexivity of von Neumann algebras

Abstract

The similarity problem is one of the most famous open problems in the theory of -algebras. We say that a -algebra satisfies the similarity property ((SP) for short) if every bounded homomorphism is similar to a -homomorphism and that a von Neumann algebra satisfies the weak similarity property ((WSP) for short) if every -conitnuous unital and bounded homomorphism where is a Hilbert space, is similar to a -homomorphism. We prove that a von Neumann algebra satisfies (WSP) if and only if the algebras are hyperreflexive for all cardinals In the case in which is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra is hyperreflexive. We also introduce the hypothesis {\bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {\bf (CHH)}, all -algebras satisfy (SP). Finally, we prove that the spatial tensor product where is an injective von Neumann algebra and is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the -similarity degree
Paper Structure (4 sections, 32 theorems, 163 equations)

This paper contains 4 sections, 32 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Weak Similarity property and hyperreflexivity
  3. (CHH) implies that all C*-algebras satisfy (SP)
  4. Injective von Neumann algebras and w*-similarity property

Key Result

Lemma 1.1

Let $\mathcal{A}$ be a unital $C^*$-algebra. The algebra $\mathcal{A}$ satisfies (SP) if and only if the second dual algebra $\mathcal{A}^{**}$ satisfies (WSP).

Theorems & Definitions (64)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • ...and 54 more