Homomorphisms of L1 algebras and Fourier algebras

We investigate conditions for the extendibility of continuous algebra homomorphisms ct. 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 infinity algebras which are weak* continuous. When ct. is completely bounded and F is amenable, it is induced by a piecewise affine map alpha : Y-+ F where Y subset of G. We show that extendibility of ct. is equivalent to alpha being an open map. We also study the dual problem for contractive homomorphisms ct. : L1(F)-+ M(G). We show that ct. induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map theta (Greenleaf [1965], Stokke [2011]) is a proper map. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article investigates the conditions for extending a continuous algebra homomorphism from the Fourier algebra of one locally compact group to the Fourier-Stieltjes algebra of another locally compact group. When the mapping is completely bounded and the original group is amenable, it can be induced by a piecewise affine map. The dual problem is also studied.