Weak*- continuous homomorphisms of Fourier–Stieltjes algebras
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Abstract
Abstract For a locally compact group G , let B ( G ) denote its Fourier–Stieltjes algebra. Any continuous, piecewise affine map α: Y ⊂ H → G induces a completely bounded algebra homomorphism j α : B ( G ) → B ( H ) [ 14, 15 ] and we prove that j α is w * – w * continuous if and only if α is an open map. This extends one of the main results in [ 3 ], due to M.B. Bekka, E. Kaniuth, A.T. Lau and G. Schlichting. Several classical theorems regarding isomorphisms and extensions of homomorphisms on group algebras of abelian groups are extended to the setting of Fourier–Stieltjes algebras of amenable groups. As applications, when G is amenable we provide complete characterizations of those maps between Fourier–Stieltjes algebras that are either associated to a piecewise affine mapping, or are completely bounded and w * – w * continuous.
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