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A note on strong similarity and the Connes embedding problem

Gilles Pisier

Abstract

We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^*$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $*$-homomorphism, i.e. the similarities that ``orthogonalize" $u$ (which exist since $u$ is c.b.) cannot belong to the von Neumann algebra $N$. Moreover, the map $u$ does not admit any c.b. lifting up into the WEP $C^*$-algebra of which $N$ is a quotient. We can take $C=C^*(G)$ (full $C^*$-algebra) where $G$ is any nonabelian free group and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $G$.

A note on strong similarity and the Connes embedding problem

Abstract

We show that there exists a completely bounded (c.b. in short) homomorphism from a -algebra with the lifting property (in short LP) into a QWEP von Neumann algebra that is not strongly similar to a -homomorphism, i.e. the similarities that ``orthogonalize" (which exist since is c.b.) cannot belong to the von Neumann algebra . Moreover, the map does not admit any c.b. lifting up into the WEP -algebra of which is a quotient. We can take (full -algebra) where is any nonabelian free group and where is the von Neumann algebra generated by the reduced -algebra of .
Paper Structure (3 theorems, 43 equations)

This paper contains 3 theorems, 43 equations.

Key Result

Proposition 2

Consider $u: C \to N$ a unital cb homomorphism with values in N assumed to be a QWEP von Neumann algebra in $B(H)$. Let $v: N' \to B(H)$ be the inclusion map. Then $u$ is strongly similar to a $*$-homomorphism iff the map $u . v : C \otimes N' \to B(H)$ (defined by $u . v (a\otimes b)= u(a)v(b)$ ext where the (attained) inf runs over all invertible $T\in N$ such that $T u(.) T^{-1}$ is a $*$-homom

Theorems & Definitions (8)

  • Remark 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Remark 5: Free group variant