Homomorphisms of L1 algebras and Fourier algebras

Abstract

We investigate conditions for the extendibility of continuous algebra homomorphisms ϕ from the Fourier algebra A(F) of a locally compact group F to the Fourier-Stieltjes algebra B(G) of a locally compact group G to maps between the corresponding L∞ algebras which are weak* continuous. When ϕ is completely bounded and F is amenable, it is induced by a piecewise affine map α:Y→F where Y⊆G. We show that extendibility of ϕ is equivalent to α being an open map. We also study the dual problem for contractive homomorphisms ϕ:L1(F)→M(G). We show that ϕ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map θ (Greenleaf [1965], Stokke [2011]) is a proper map.