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 $\mathrm{Tame}(L^1(G))=\mathrm{Tame}(G)$, 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 ${\mathrm{RUC}}_b(G)$, being small as a function and as a functional is the same. This proves that $\mathrm{Asp}(L^1(G))=\mathrm{Asp}(G)$ and reaffirms a well-known, similar result which states that $\mathrm{WAP}(G)=\mathrm{WAP}(L^1(G))$.