Homomorphisms of Fourier algebras and transference results

TL;DR

This paper investigates the conditions under which homomorphisms between Fourier algebras extend to larger algebras, linking these extensions to properties of the inducing maps and exploring their implications for operator set preservation.

Contribution

It establishes a characterization of when such homomorphisms extend to $L^ Infty$ algebras based on the openness of the inducing map and explores transference results for various operator sets.

Findings

Extension of homomorphisms linked to open maps.

Preservation of operator synthesis sets under certain maps.

Results extend existing theories of operator sets and Fourier algebra homomorphisms.

Abstract

We prove that if $\rho: A(H) \to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $\alpha : G \to H$, then $\rho$ extends to a w*-w* continuous map between the corresponding $L^\infty$ algebras if and only if $\alpha$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $\alpha$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure of $H$, then $(\alpha\times\alpha)^{-1}$ preserves sets of compact operator synthesis, and conversely when $\alpha$ is onto. We also prove similar preservation properties for operator Ditkin sets and operator M-sets, obtaining preservation properties for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.