# Homomorphisms with small bound between Fourier algebras

## TL;DR

This paper characterizes algebra homomorphisms between Fourier algebras with small bounds and establishes conditions under which group isomorphisms correspond to such homomorphisms, extending previous structural theorems.

## Contribution

It provides new bounds for completely bounded norms that characterize group isomorphisms via Fourier algebra homomorphisms, generalizing prior results.

## Key findings

- Algebra homomorphisms with bounded cb-norm imply group isomorphism.
- Characterization of isomorphic groups via Fourier algebra homomorphisms with small norm distortion.
- Extension of Walter's and Lau's theorems on Fourier algebras.

## Abstract

Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism $T$ between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm $\| T \|_{cb} < \sqrt {3/2}$ (resp. $ \| T \|_{cb} < \sqrt {5}/2$). We show similar results involving the norm distortion $\| T \| \| T ^{-1} \|$ with universal but non-explicit bound. Our results subsume Walter's well-known structural theorems and also Lau's theorem on second conjugate of Fourier algebras.