Tame functionals on Banach algebras

Abstract

In the present note we introduce tame functionals on Banach algebras. A functional $f\in A^{\ast }$ on a Banach algebra $A$ is tame if the naturally defined linear operator $A\to A^{\ast },a\mapsto f\cdot a$ 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, Ülger 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.

12 pages