Abstract. Let PMp(G) be the space of all p-pseudomeasures on a locally compact group G. We show the existence of a conditional expectation from PMp(G) onto PMp(H) where H is a closed normal subgroup of G. As an application we give a new proof of the fact that H is a set of p-synthesis in G; we also get an inequality involving the operator norm of bounded measures on G. Moreover, in analogy with a theorem of Reiter, we obtain a result concerning the closed ideals of the Fig~-Talamanca Herz algebra of G.
Let G be a locally compact group and let pAð1; NÞ: Let A be any of the Banach spaces C d;p ðGÞ; PF p ðGÞ; M p ðGÞ; AP p ðGÞ; WAP p ðGÞ; UC p ðGÞ; PM p ðGÞ; of convolution operators on L p ðGÞ: It is shown that PF p ðGÞ 0 can be isometrically embedded into UC p ðGÞ 0 : The structure of maximal regular ideals of A 0 (and of MA p ðGÞ 00 ; B p ðGÞ 00 ; W p ðGÞ 00 ) is studied. Among other things it is shown that every maximal regular left (right, two sided) ideal in A 0 is either weak à dense or is the annihilator of a unique element in the spectrum of A p ðGÞ: Minimal ideals of A 0 is also studied. It is shown that a left ideal M in A 0 is minimal if and only if M ¼ CC; where C is either a right annihilator of A 0 or is a topologically x-invariant element (for some xAG). Some results on minimal right ideals are also given. r 2004 Elsevier Inc. All rights reserved.
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