  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.
