The Banach Algebra $L^{1}(G)$ and Tame Functionals
Matan Komisarchik
TL;DR
The paper resolves Megrelishvili's question by proving that for any locally compact group $G$, the tameness of a functional on $L^{1}(G)$ coincides with tameness as a function on $G$, i.e. $\operatorname{Tame}(L^{1}(G))=\operatorname{Tame}(G)$. This equivalence is extended in a general framework: for any norm-saturated, convex vector bornology $\mathcal{B}$ on $\mathrm{RUC}_{b}(G)$, being small as a function and as a functional are equivalent, yielding $\operatorname{Asp}(L^{1}(G))=\operatorname{Asp}(G)$ and recovering the known equality $\operatorname{WAP}(G)=\operatorname{WAP}(L^{1}(G))$. The approach develops a robust theory of small functionals via vector bornologies, the $\mathrm{UEB}$ topology, and dual Banach-algebra actions, along with a duality result between tameness and cotameness in the dual space. The results unify function- and functional-based notions of smallness across several classical classes (WAP, Asp, Tam e) and illuminate the interplay between group dynamics and Banach algebra actions, with open problems guiding future extensions beyond right uniformly continuous functions. The work thus consolidates a coherent bridge between harmonic analysis on groups and the dynamical systems perspective in functional analysis.
Abstract
We give an affirmative answer to a question due to M. Megrelishvili, and show that for every locally compact group $G$ we have $\operatorname{Tame}(L^{1}(G)) = \operatorname{Tame}(G)$, which means that a functional is tame over $L^{1}(G)$ if and only if it is tame as a function over $G$. In fact, it is proven that for every norm-saturated, convex vector bornology on $\operatorname{RUC}_{b}(G)$, being small as a function and as a functional is the same. This proves that $\operatorname{Asp}(L^{1}(G)) = \operatorname{Asp}(G)$ and reaffirms a well-known, similar result which states that $\operatorname{WAP}(G) = \operatorname{WAP}(L^{1}(G))$.