On $${{\varvec{p}}}$$-completely contractive homomorphisms of Figà-Talamanca–Herz algebras

Abstract

For a locally compact group G and $$1<p<\infty $$, let $$A_p(G)$$ be the Figa-Talamanca–Herz algebras, which include in particular the Fourier algebra of G, A(G) ($$p=2$$). In this note, we show that it is possible to describe an algebra homomorphism $$\Phi : A_p(G)\rightarrow A_p(H)$$ in terms of proper affine maps between the groups, under conditions that involve norm bounds on the second amplification of the homomorphism. This improves results by the first and second author (J Math Anal Appl 419(1):273–284, 2014; J Oper Theory 78:227–243, 2017). In particular, our result applies to p-completely bounded idempotent multipliers. As a corollary, we obtain that Figa-Talamanca–Herz algebras determine the underlying group up to p-completely contractive algebra isomorphisms.