Tame functionals on Banach algebras

TL;DR

This paper introduces tame functionals on Banach algebras, linking them to tame functions in topological dynamics, and explores their properties on group algebras, highlighting connections to dynamical systems.

Contribution

It defines tame functionals on Banach algebras, relates them to known concepts like WAP functionals, and characterizes tame functionals on group algebras via dynamical systems.

Findings

Tame functionals on $l_1(G)$ correspond exactly to tame functions on $G$.

WAP functionals are a subset of tame functionals.

Open question on whether $Tame(L_1(G))=Tame(G)$ for locally compact groups.

Abstract

In the present note we introduce tame functionals on Banach algebras. A functional $f \in A^*$ on a Banach algebra $A$ is tame if the naturally defined linear operator $A \to A^*, a \mapsto f \cdot a$ factors through Rosenthal Banach spaces (i.e., not containing a copy of $l_1$). Replacing Rosenthal by reflexive we get a well known concept of weakly almost periodic functionals. So, always $WAP(A) \subseteq Tame(A)$. We show that tame functionals on the group algebra $l_1(G)$ are induced exactly by tame functions (in the sense of topological dynamics) on $G$ for every discrete group $G$. That is, $Tame(l_1(G))=Tame(G)$. Many interesting tame functions on groups come from dynamical systems theory. Recall that $WAP(L_1(G))=WAP(G)$ (Lau 1977, \"{U}lger 1986) for every locally compact group $G$. It is an open question if $Tame(L_1(G))=Tame(G)$ holds for (nondiscrete) locally compact groups.