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

Let G be a locally compact group and let p∈(1,∞). Let A be any of the Banach spaces Cδ,p(G), PFp(G), Mp(G), APp(G), WAPp(G), UCp(G), PMp(G), of convolution operators on Lp(G). It is shown that PFp(G)′ can be isometrically embedded into UCp(G)′. The structure of maximal regular ideals of A′ (and of MAp(G)″, Bp(G)″, Wp(G)″) is studied. Among other things it is shown that every maximal regular left (right, two sided) ideal in A′ is either weak∗ dense or is the annihilator of a unique element in the spectrum of Ap(G). Minimal ideals of A′ is also studied. It is shown that a left ideal M in A′ is minimal if and only if M=CΨ, where Ψ is either a right annihilator of A′ or is a topologically x-invariant element (for some x∈G). Some results on minimal right ideals are also given.