Characterizations of elements with compact support in the dual spaces of 𝐴_{𝑝}(𝐺)-modules of 𝑃𝑀_{𝑝}(𝐺)

Source abstract

For a locally compact group G G and 1 > p > ∞ 1 > p > \infty , let A p ( G ) A_{p}(G) be the Figà-Talamanca-Herz algebra and let P M p ( G ) PM_{p}(G) be its dual Banach space. For a Banach A p ( G ) A_{p}(G) -module X X of P M p ( G ) PM_{p}(G) , we denote the norm closure of the subspace of the elements in X ∗ X^{*} with compact support by A p , X ( G ) A_{p,X}(G) . We prove that an element u u of X ∗ X^{*} is in A p , X ( G ) A_{p,X}(G) if and only if for any ϵ > 0 \epsilon > 0 , there exists a compact subset K K of G G such that | ⟨ u , f ⟩ | > ϵ \vert \langle u, f \rangle \vert > \epsilon for all f ∈ X f\in X with ‖ f ‖ ≤ 1 \Vert f\Vert \le 1 and s u p p ( f ) ⊆ G ∼ K supp \, (f)\subseteq G\sim K . In particular, we have that an element b b of W p ( G ) W_{p}(G) is in A p ( G ) A_{p}(G) if and only if for any ϵ > 0 \epsilon > 0 , there exists a compact subset K K of G G such that | ⟨ u , f ⟩ | > ϵ \vert \langle u, f \rangle \vert > \epsilon for all f ∈ L 1 ( G ∼ K ) f\in L^{1}(G\sim K) with ‖ f ‖ ≤ 1 \Vert f\Vert \le 1 . If A p , X ( G ) A_{p, X}(G) has an orthogonal complement A p , X s ( G ) A_{p, X}^{s}(G) in X ∗ X^{*} , we characterize A p , X s ( G ) A_{p, X}^{s}(G) by the following condition: u ∈ X ∗ u\in X^{*} is in A p , X s ( G ) A_{p, X}^{s}(G) if and only if for any ϵ > 0 \epsilon > 0 and any compact subset K K of G G , there exists some f ∈ X f\in X with ‖ f ‖ ≤ 1 \Vert f\Vert \le 1 and s u p p ( f ) ⊆ G ∼ K supp\, (f)\subseteq G\sim K such that | ⟨ u , f ⟩ | > ‖ u ‖ − ϵ \vert \langle u, f \rangle \vert > \Vert u\Vert - \epsilon . Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking p = 2 p=2 and X X as some C ∗ C^{*} -subalgebras of P M p ( G ) PM_{p}(G) .