Completely bounded homomorphisms of the Fourier algebra revisited

TL;DR

This paper revisits the characterization of completely bounded homomorphisms between Fourier algebras of locally compact groups, providing a new combinatorial and measure-theoretic approach to address gaps in previous proofs.

Contribution

It introduces an alternative proof strategy for the characterization of completely bounded homomorphisms, moving away from topological arguments to a combinatorial and measure-theoretic framework.

Findings

New proof strategy for the main theorem

Addresses gaps in previous arguments

Extends understanding of Fourier algebra homomorphisms

Abstract

Let $A(G)$ and $B(H)$ be the Fourier and Fourier-Stieltjes algebras of locally compact groups $G$ and $H$, respectively. Ilie and Spronk have shown that continuous piecewise affine maps $\alpha: Y \subseteq H\rightarrow G$ induce completely bounded homomorphisms $\Phi:A(G)\rightarrow B(H)$, and that when $G$ is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure theoretic ideas, following more closely the original ideas of Cohen.