Completely Bounded Representations Into Von Neumann Algebras And Connes Embedding Problem

TL;DR

For a unital separable $C^*$-algebra $\mathcal{A}$ and a von Neumann algebra $\mathcal{M}$ with the QWEP property, the paper proves that every unital completely bounded representation $\phi:\mathcal{A}\to\mathcal{M}$ is similar to a $*$-representation via a positive invertible $S\in\mathcal{M}$, i.e., $S\phi(\cdot)S^{-1}$ is a $*$-representation. This main theorem is contextualized within Pisier's nuclearity criterion and Kirchberg's equivalences between QWEP and ultrapower embeddings, yielding a route to the Connes embedding problem and its negative resolution in this framework. The work also presents a second approach under separability, revisits the classical similarity problem via Hadwin's decomposition and Haagerup-type results, and includes a detailed appendix with Pop's lifting lemma. Collectively, the results illuminate how CB representations into von Neumann algebras interact with tensor-product properties and ultrapower embeddability, and they establish concrete obstructions to universal embedding phenomena in operator algebras.

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 $φ:\, \mathcal{A}\rightarrow \mathcal{M}$ is a unital completely bounded representation, then there is an invertible operator $S\in \mathcal{M}$ such that $Sφ(\cdot) S^{-1}$ 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 $φ$ of $\mathcal{A}$ into an arbitrary von Neumann algebra $\mathcal{M}$ there is an invertible operator $S\in \mathcal{M}$ such that $Sφ(\cdot) S^{-1}$ 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.

Theorems & Definitions (41)

• Theorem 1.1
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• Theorem 2.1
• Definition 2.2
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• Definition 2.5
• Theorem 2.6