Homomorphisms of Fourier–Stieltjes algebras

Abstract

Every homomorphism ϕ : B(G) → B(H) between Fourier-Stieltjes algebras on locally compact groups G and H is determined by a continuous mapping α : Y → ∆(B(G)), where Y is a set in the open coset ring of H and ∆(B(G)) is the Gelfand spectrum of B(G) (a *-semigroup). We exhibit a large collection of maps α for which ϕ = jα : B(G) → 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 ϕ : B(G) → 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.