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Decomposition of
Authors:Tianxuan Miao
Affiliation:Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario P7E 5E1 Canada
Abstract:For any locally compact group $G$, let $A(G)$ and $B(G)$ be the Fourier and the Fourier-Stieltjes algebras of $G$, respectively. $B(G)$ is decomposed as a direct sum of $A(G)$ and $ B^{s}(G)$, where $ B^{s}(G)$ is a subspace of $B(G)$ consisting of all elements $ bin B(G)$ that satisfy the property: for any $epsilon > 0$ and any compact subset $ Ksubset G$, there is an $ fin L^{1}(G) $ with $Vert fVert _{C^{*}(G)} le 1$ and $ supp(f) subset K^{c}$ such that $vert langle f, b rangle vert > Vert bVert - epsilon .$ $A(G)$ is characterized by the following: an element $bin B(G)$ is in $A(G)$ if and only if, for any $epsilon > 0,$ there is a compact subset $ Ksubset G$ such that $ vert langle f, b rangle vert < epsilon $ for all $ fin L^{1}(G) $ with $Vert fVert _{C^{*}(G)} le 1 $ and $ supp(f) subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $Vert 1 + aVert = 1 + Vert aVert $ for all $ain A(G)$ if $G$ is noncompact. We will apply this characterization of $B^{s}(G)$ to investigate the general properties of $B^{s}(G)$ and we will see that $B^{s}(G)$ is not a subalgebra of $B(G)$ even for abelian locally compact groups. If $G$ is an amenable locally compact group, then $B^{s}(G)$ is the subspace of $B(G)$ consisting of all elements $bin B(G)$ with the property that for any compact subset $Ksubseteq G$, $Vert bVert = sup , { , Vert a bVert : , ain A(G), ; supp(a) subseteq K^{c} ; text{ and } ; Vert aVert le 1 , }$.

Keywords:Locally compact groups   amenable groups   the Fourier algebra of a locally compact group   the Fourier-Stieltjes algebra of a locally compact group   the Lebesgue-type decomposition of the Fourier-Stieltjes algebra
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