Completely Bounded Representations Into Von Neumann Algebras And Connes Embedding Problem

TL;DR

This paper explores conditions under which completely bounded representations of $C^*$-algebras into von Neumann algebras can be conjugated into *-representations, relating to the QWEP property and the Connes Embedding Problem.

Contribution

It extends previous results by establishing new links between QWEP, completely bounded representations, and the Connes Embedding Problem.

Findings

If $ ext{QWEP}$ holds, certain representations are conjugate to *-representations.

Existence of non-QWEP von Neumann algebras is demonstrated.

Connections between QWEP, embeddability, and the Connes Embedding Problem are clarified.

Abstract

In this paper, we prove that if $\mathcal{A}$ is a unital separable $C^*$-algebra, $\mathcal{M}$ is a von Neumann algebra which has the Kirchberg's quotient weak expectation property (QWEP), and $\phi:\, \mathcal{A}\rightarrow \mathcal{M}$ is a unital completely bounded representation, then there is an invertible operator $S\in \mathcal{M}$ such that $S\phi(\cdot) S^{-1}$ is a $\ast$-representation. On the other hand, Gilles Pisier proved the following result: a unital $C^*$-algebra $\mathcal{A}$ is nuclear if and only if for every unital completely bounded representation $\phi$ of $\mathcal{A}$ into an arbitrary von Neumann algebra $\mathcal{M}$ there is an invertible operator $S\in \mathcal{M}$ such that $S\phi(\cdot) S^{-1}$ is a $\ast$-representation. This implies that there exist von Neumann algebras which are not QWEP. Eberhard Kirchberg showed that every von Neumann algebra has QWEP if and only if every tracial von Neumann algebra embeds into the ultrapower $\mathcal{R}^w$ of the hyperfinite type ${\rm II}_1$ factor $\mathcal{R}$. This provides a negative answer to the Connes Embedding Problem. This paper relies on previous work of Gilles Pisier and Florin Pop.