Year: 2017
Type: preprint
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, 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.
Source: arXiv (Cornell University)
Author: Megrelishvili, Michael
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Israel Science Foundation
Awards: G7033326859, G865570889
OpenAlex ID: W4298417334