Completely Bounded Representations Into Von Neumann Algebras And Connes Embedding Problem

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.

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