Homomorphisms of Fourier-Stieltjes algebras

2020/10/13 by Stokke, Ross
#43A22 #43A30 #43A70 #47L25 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2010.06650

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 → Δ(B(G)), where Y is a set in the open coset ring of H and Δ(B(G)) is the Gelfand spectrum of B(G) (a *-semigroup). We exhibit a large collection of maps α for which φ=j_α: 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.

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