A note on strong similarity and the Connes embedding problem

TL;DR

This paper demonstrates the existence of a completely bounded homomorphism from a specific $C^*$-algebra into a von Neumann algebra that cannot be approximated by *-homomorphisms, highlighting limitations related to the Connes embedding problem.

Contribution

It constructs a specific example of a c.b. homomorphism that is not strongly similar to a *-homomorphism, revealing new insights into the structure of such maps and their relation to the Connes embedding problem.

Findings

Existence of a c.b. homomorphism not strongly similar to a *-homomorphism

The homomorphism cannot be lifted into the WEP $C^*$-algebra

An analogue for strong similarity of Haagerup's formula is established

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 $C=C^*(F_\infty)$ the full $C^*$-algebra of the free group $F_\infty$ with infinitely many generators and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $F_\infty$. Incidentally we observe an analogue for strong similarity of Haagerup's (and Paulsen's) similarity formula for the cb-norm : if $C$ is any unital $C^*$-algebra and $N$ any von Neumann algebra then for any bounded unital homomorphism $u: C \to N$ we have $$\|u\|_{mb}= \inf\{ \|S\|\|S^{-1}\| \}$$ where the inf (which is attained) runs over all invertible $S\in N$ such that $S u(.) S^{-1}$ is a $*$-homomorphism. We end the note by a quick proof of the main point using the mb-norm and the space $R_n\cap C_n$.