Approximation properties and approximate identities of $A_{p}(G)$

Abstract:

For a locally compact group $G$ and $1 < p < \infty$, let $A_{p}(G)$ be the Figà-Talamanca-Herz algebra. Then the multiplier algebra $MA_{p}(G)$ of $A_{p}(G)$ is a dual space. We say that $A_{p}(G)$ has the approximation property (or simply, AP) in $MA_{p}(G)$ if there is a net $\{ u_{\alpha } \}$ in $A_{p}(G)$ such that $u_{\alpha }\rightarrow 1$ in the associated $weak^{*}$ topology. We prove that $A_{p}(G)$ has the AP in $MA_{p}(G)$ if and only if there exists a net $\{ a_{\alpha } \}$ in $A_{p}(G)$ such that $\Vert a_{\alpha } a - a\Vert _{A_{p}(G)}\rightarrow 0$ uniformly for $a$ in any compact subset of $A_{p}(G)$. Consequently, we have that if $A_{p}(G)$ has the AP in $MA_{p}(G)$, then $A_{p}(G)$ has the approximation property as a Banach space in the sense of Grothendieck for a discrete group $G$. We also study the relationship between the AP of $A_{p}(G)$ in $MA_{p}(G)$ and the weak amenability of $G$.