Similarity degree of Fourier algebras

TL;DR

This paper proves that for certain classes of locally compact groups, their Fourier algebras satisfy a strong similarity property with degree at most 2, improving previous results and characterizing when the algebra is completely isomorphic to an operator algebra.

Contribution

It establishes a completely bounded similarity degree of at most 2 for Fourier algebras of specific groups, extending and strengthening earlier findings.

Findings

Fourier algebras of certain groups satisfy a similarity property with degree ≤ 2.

The similarity degree is 1 if and only if the group is finite.

The results improve previous bounds by Brannan and Samei, Brannan, Daws, and Samei.

Abstract

We show that for a locally compact group $G$, amongst a class which contains amenable and small invariant neighbourhood groups, that its Fourier algebra $A(G)$ satisfies a completely bounded version Pisier's similarity property with similarity degree at most $2$. Specifically, any completely bounded homomorphism $\pi: A(G)\to B(H)$ admits an invertible $S$ in $B(H)$ for which $\|S\|\|S^{-1}\|\leq ||\pi||_{cb}^2$ and $S^{-1}\pi(\cdot)S$ extends to a $*$-representation of the $C^*$-algebra $C_0(G)$. This significantly improves some results due to Brannan and Samei (J. Funct. Anal. 259, 2010) and Brannan, Daws and Samei (M\"{u}nster J. Math 6, 2013). We also note that $A(G)$ has completely bounded similarity degree $1$ if and only if it is completely isomorphic to an operator algebra if and only if $G$ is finite.