Homomorphisms of Fourier-Stieltjes algebras

TL;DR

This paper characterizes homomorphisms between Fourier-Stieltjes algebras on locally compact groups, identifying when they are completely positive, contractive, or bounded, especially for Euclidean and p-adic motion groups.

Contribution

It provides a new characterization of such homomorphisms via continuous maps into the Gelfand spectrum and introduces the concept of fusion maps for describing these homomorphisms in specific groups.

Findings

Characterization of homomorphisms via continuous maps into the Gelfand spectrum

Complete description of positive/contractive/bounded homomorphisms for Euclidean and p-adic motion groups

Introduction of fusion maps to describe homomorphisms in these cases

Abstract

Every homomorphism $\varphi: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $\alpha: Y \rightarrow \Delta(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $\Delta(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $\alpha$ for which $\varphi=j_\alpha: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractive/completely bounded homomorphisms $\varphi: B(G) \rightarrow B(H)$ when $G$ is a Euclidean- or $p$-adic-motion group. In these cases, our description of the completely positive/completely contractive homomorphisms employs the notion of a "fusion map of a compatible system of homomorphisms/affine maps" and is quite different from the Fourier algebra situation.