Decomposition of 𝐵(𝐺)

Abstract

For any locally compact group G G , let A ( G ) A(G) and B ( G ) B(G) be the Fourier and the Fourier-Stieltjes algebras of G G , respectively. B ( G ) B(G) is decomposed as a direct sum of A ( G ) A(G) and B s ( G ) B^{s}(G) , where B s ( G ) B^{s}(G) is a subspace of B ( G ) B(G) consisting of all elements b ∈ B ( G ) b\in B(G) that satisfy the property: for any ϵ > 0 \epsilon > 0 and any compact subset K ⊂ G K\subset G , there is an f ∈ L 1 ( G ) f\in L^{1}(G) with ‖ f ‖ C ∗ ( G ) ≤ 1 \Vert f\Vert _{C^{*}(G)} \le 1 and s u p p ( f ) ⊂ K c supp(f) \subset K^{c} such that | ⟨ f , b ⟩ | > ‖ b ‖ − ϵ . \vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon . A ( G ) A(G) is characterized by the following: an element b ∈ B ( G ) b\in B(G) is in A ( G ) A(G) if and only if, for any ϵ > 0 , \epsilon > 0, there is a compact subset K ⊂ G K\subset G such that | ⟨ f , b ⟩ | > ϵ \vert \langle f, b \rangle \vert > \epsilon for all f ∈ L 1 ( G ) f\in L^{1}(G) with ‖ f ‖ C ∗ ( G ) ≤ 1 \Vert f\Vert _{C^{*}(G)} \le 1 and s u p p ( f ) ⊂ K c supp(f) \subset K^{c} . Note that we do not assume the amenability of G G . Consequently, we have   ‖ 1 + a ‖ = 1 + ‖ a ‖ \Vert 1 + a\Vert = 1 + \Vert a\Vert for all a ∈ A ( G ) a\in A(G) if G G is noncompact. We will apply this characterization of B s ( G ) B^{s}(G) to investigate the general properties of B s ( G ) B^{s}(G) and we will see that B s ( G ) B^{s}(G) is not a subalgebra of B ( G ) B(G) even for abelian locally compact groups. If G G is an amenable locally compact group, then B s ( G ) B^{s}(G) is the subspace of B ( G ) B(G) consisting of all elements b ∈ B ( G ) b\in B(G) with the property that for any compact subset K ⊆ G K\subseteq G , ‖ b ‖ = sup { ‖ a b ‖ : a ∈ A ( G ) , s u p p ( a ) ⊆ K c  and  ‖ a ‖ ≤ 1 } \Vert b\Vert = \sup \, \{ \, \Vert a b\Vert : \, a\in A(G), \; supp(a) \subseteq K^{c} \; \text { and } \; \Vert a\Vert \le 1 \, \} .