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

TL;DR

This paper establishes that tame and weakly almost periodic functionals on the Banach algebra L^{1}(G) correspond exactly to those on the group G, confirming a deep connection between algebraic and functional properties.

Contribution

It proves the equality of tame and almost periodic functionals between L^{1}(G) and G, answering a question by Megrelishvili and extending known results.

Findings

Tame functionals on L^{1}(G) are equivalent to tame functions on G.

Asp and WAP properties are preserved between L^{1}(G) and G.

The results hold for all locally compact groups.

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