On the ideal structure of some Banach algebras related to convolution operators on Lp(G)

Abstract

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.

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

“…The equivalence of (i) and (iii) in the next proposition is [5,Theorem 3.7] in the special case when p = 2.…”

“…Duals of introverted subspaces of VN(G) are also studied in [5,15,[21][22][23]25], as well as many other papers.…”

“…For any locally compact group, the map π was defined by Lau and Losert in [21], and a version of π was defined for algebras related to the generalized Fourier algebras A p (G) by Derighetti, Filali, and Sangani Monfared [5].…”

“…Most parts of the following lemma are either implicit in [21, p. 10], or are stated explicitly as in [21,Proposition 4.2]. The fact that π is a homomorphism was established for amenable groups in [19], and in greater generality in [5]. Regarding part (ii) of the lemma, we remark that π is the analogue of the natural embedding of M(G) into LUC(G) * as defined by Wong [32], and in this case Lau proved that the image of M(G) is precisely the topological centre of LUC(G) * [20].…”

Finite-dimensional left ideals in the duals of introverted spaces

“…and A p (G) * * , Baker-Filali [3,4], Filali [13], Filali-Pym [16], and Filali-Salmi [17] for (among other things) X * , L 1 (G) * * , and LU C(G) * , where X is an introverted subspace of C(G), Derighetti et al [10] for duals of introverted subspaces of ppseudo measures P M p (G), and the most recent work by the present authors and Neufang [14] on A(G) * * and UC 2 (G) * . Let G be a locally compact group, 1 < p < ∞, and L (L p (G)) be the space of continuous linear operators on L p (G).…”