Decomposition of $B(G)$

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 $b\in B(G)$ that satisfy the property: for any $\epsilon > 0$ and any compact subset $K\subset G$, there is an $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$ such that $\vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon .$ $A(G)$ is characterized by the following: an element $b\in B(G)$ is in $A(G)$ if and only if, for any $\epsilon > 0,$ there is a compact subset $K\subset G$ such that $\vert \langle f, b \rangle \vert < \epsilon$ for all $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $\Vert 1 + a\Vert = 1 + \Vert a\Vert$ for all $a\in A(G)$ 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 $b\in B(G)$ with the property that for any compact subset $K\subseteq G$, $\Vert b\Vert = \sup \, \{ \, \Vert a b\Vert : \, a\in A(G), \; supp(a) \subseteq K^{c} \; \text{ and } \; \Vert a\Vert \le 1 \, \}$.