Optimisation in some Banach Algebras related to the Fourier Algebra

TL;DR

This paper studies the Radon-Nikodym property in certain Banach algebras associated with locally compact groups, revealing conditions under which solutions to optimization problems exist and algorithms converge.

Contribution

It establishes new conditions for the Radon-Nikodym property in Banach algebras related to Fourier algebras, especially for weakly amenable groups.

Findings

A_p^r is a dual Banach space with RNP if G is weakly amenable and 1≤r≤p'.

The RNP property in A_p^r depends on group properties, failing for SL(2,R) when r>2.

For second countable groups with RNP in A_p(G), A_p^r(G) also has RNP for all 1≤r<∞.

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.