Homomorphisms of $L^1$ algebras and Fourier algebras

TL;DR

This paper characterizes when algebra homomorphisms between Fourier and Fourier-Stieltjes algebras extend to $L^ Infty$ algebras, linking extendibility to the openness of certain maps and exploring dual problems for group measure algebras.

Contribution

It provides new criteria for the extendibility of homomorphisms between Fourier algebras and their duals, connecting algebraic properties with topological map characteristics.

Findings

Extendibility of homomorphisms is equivalent to the associated map being open.

Homomorphisms induce weak* continuous maps between von Neumann algebras under properness conditions.

Characterization of when algebra homomorphisms extend to $L^ Infty$ algebras in terms of topological properties.

Abstract

We investigate conditions for the extendibility of continuous algebra homomorphisms $\phi$ from the Fourier algebra $A(F)$ of a locally compact group $F$ to the Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$ to maps between the corresponding $L^\infty$ algebras which are weak* continuous. When $\phi$ is completely bounded and $F$ is amenable, it is induced by a piecewise affine map $\alpha: Y\to F$ where $Y\subseteq G$. We show that extendibility of $\phi$ is equivalent to $\alpha$ being an open map. We also study the dual problem for contractive homomorphisms $\phi: L^1(F)\to M(G)$. We show that $\phi$ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map $\theta$ (Greenleaf [1965], Stokke [2011]) is a proper map.