A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

Abstract

For a locally compact group G and 1<p<∞ let Ap(G) be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G, A(G) (p=2). It is shown that for any amenable group H, a proper affine map α:Y⊂H→G induces a p-completely contractive algebra homomorphism ϕα:Ap(G)→Ap(H) by setting ϕα(u)=u∘α on Y and ϕα(u)=0 off of Y. Moreover, we show that if both G and H are amenable then any p-completely contractive algebra homomorphism ϕ:Ap(G)→Ap(H) is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.