Optimisation in some Banach Algebras related to the Fourier Algebra

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 $A_2(G)=L^1(\widehat{G})\widehat{}$. Let $A_p^r(G)=A_p\cap L^r(G)$ with norm $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{A_p^r}=\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{A_p}+\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{L^r}$. 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 $1\le r\le p^{\mathrm{\prime }}$. 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 $A_p^t$ has the RNP for $t=s$, then it has the RNP for all $1\le t\le s$, where $s=\mathrm{\infty }$ 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 $1\le r<\mathrm{\infty }$. The results are new, even if $G=\mathbb{Z}$, the additive integers.