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

For a locally compact group G and 1 < p < ∞ let A p ( 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 ϕ α : A p ( G ) → A p ( H ) by setting ϕ α ( u ) = u ∘ α on Y and ϕ α ( u ) = 0 off of Y. Moreover, we show that if both G and H are amenable then any p-completely contractive algebra homomorphism ϕ : A p ( G ) → A p ( 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.