Abstract: 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)).
Keywords: weakly almost periodicity; functional on Banach algebra; bornology; Rosenthal space; tame family; Asplund space; group algebra
DOI: DOI 10.14712/1213-7243.2025.013
AMS Subject Classification: 43A60 43A20 46H05 46A17