CONVOLUTION OPERATORS AND HOMOMORPHISMS OF LOCALLY COMPACT GROUPS

Let $1{\TL }p{\TL }\infty $, let G and H be locally compact groups and let ω be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp(G) of the p-convolution operators on G into CVp(H) which extends the usual definition of the image of a bounded measure by ω. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let Gd denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, $\|\HSP | \mu \|\HSP | _{CV_{p}(G)}\leq \|\HSP | \mu \|\HSP | _{CV_{p}(G_{d})}$, for Gd amenable. For arbitrary G, we also obtain $\|\HSP | \mu \|\HSP | _{CV_{p}(G_{d})}\leq \|\HSP | \mu \|\HSP | _{CV_{p}(G)}$. These inequalities were already known for p=2 . The proofs presented in this paper avoid the use of the Hilbertian techniques which are not applicable to $p\neq 2$. Finally, for Gd amenable, we construct a natural map of CVp (G) into CVp (Gd