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Keywords:
weakly almost periodicity; functional on Banach algebra; bornology; Rosenthal space; tame family; Asplund space; group algebra
weakly almost periodicity; functional on Banach algebra; bornology; Rosenthal space; tame family; Asplund space; group algebra
Summary:
We give an affirmative answer to a question due to M. Megrelishvili, and show that for every locally compact group $G$ we have Tame$(L^{1}(G)) = $ 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 RUC$_b(G)$, being small as a function and as a functional is the same. This proves that Asp$(L^{1}(G)) = $ Asp$(G)$ and reaffirms a well-known, similar result which states that WAP$(G) = $ WAP$(L^{1}(G))$.
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