Optimisation in some Banach Algebras related to the Fourier Algebra

Abstract

Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally compact group $G$, thus $A_2(G)$ is the Fourier Algebra of $G$. If $G$ is commutative then $A_2(G)=L^1(\hat{G}){\hat{}}$. Let $A^r_p(G)=A_p\cap L^r(G)$ with norm $||u||_{A_p^r}=||u||_{A_p}+||u||_{L^r}$. We investigate a property which insures not only existence of solutions to optimization problems but moreover, facility in testing that an algorithm converges to such solutions namely the RNP. Theorem(a): If $G$ is weakly amenable then $A_p^r$ is a dual Banach space with RNP if $1\leq r\leq p'$. This does not hold if $G=SL(2,R)$, $p=2$ and $r>2$. Theorem(b): If $G$ is weakly amenable and second countable and $A^t_p$ has the RNP for $t=s$, then it has the RNP for all $1\leq t\leq s$, where $s=\infty$ is allowed. In particular second countable noncompact groups $G$, for which $A_p(G)$ has RNP, namely Fell groups, have to satisfy that $A_p^r(G)$ has the RNP for all $1\leq r<\infty$. The results are new, even if $G=\mathbb{Z}$, the additive integers.