The Banach Algebra $L^{1}(G)$ and Tame Functionals

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 $\operatorname{WAP}(G) = \operatorname{WAP}(L^{1}(G))$.