Conditional expectations on CVp(G). Applications

We obtain for H, a closed amenable subgroup of a locally compact group G, the existence of an invariant conditional expectation of CVp(G) onto {T∈CVp(G)|suppT⊂H}. As a consequence we prove that H is locally p-Ditkin in G. We also establish relations between cvp(H), {T∈cvp(G)|suppT⊂H} and cvp(G) where cvp is the norm closure, in the set of all bounded operators of Lp, of the set all convolution operators with compact support.