A note on strong similarity and the Connes embedding problem

Abstract

We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^{\ast }$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $\ast$-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 $C^{\ast }$-algebra of which $N$ is a quotient. We can take $C=C^{\ast }(F_{\infty })$ the full $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-algebra and $N$ any von Neumann algebra then for any bounded unital homomorphism $u:C\to N$ we have $$\Vert u{\Vert }_{mb}=\inf \{\Vert S\Vert \Vert S^{-1}\Vert \}$$ where the inf (which is attained) runs over all invertible $S\in N$ such that $Su(.)S^{-1}$ is a $\ast$-homomorphism. We end the note by a quick proof of the main point using the mb-norm and the space $R_n\cap C_n$.

v4 Further expanded exposition, addition of a new proof of main point