Homomorphisms of Fourier-Stieltjes algebras

Abstract

Every homomorphism $\phi :B(G)\to B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $\alpha :Y\to \Delta (B(G))$, where $Y$ is a set in the open coset ring of $H$ and $\Delta (B(G))$ is the Gelfand spectrum of $B(G)$ (a $\ast$-semigroup). We exhibit a large collection of maps $\alpha$ for which $\phi =j_{\alpha }:B(G)\to B(H)$ 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 $\phi :B(G)\to B(H)$ 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.

40 pages