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Abstract

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