Matan Komisarchik

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

Let G be a locally compact group, and L^{1}(G) be the convolution Banach algebra of G. Also, let \mathrm{WAP}(G) be the Banach algebra of weakly almost periodic (w.a.p.) functions, and \mathrm{WAP}(L^{1}(G)) be the Banach algebra of w.a.p. functionals on L^{1}(G).
It is known for some time that {\mathrm{WAP}(G) = \mathrm{WAP}(L^{1}(G))} [3]. This fact, and many other interesting results ([1], for example) make the relationship between functions on groups and functionals on the group algebra a central and important topic in topological algebra.
In this lecture we give a partial answer to a question due to M. Megrelishvili [2], and show that \mathrm{Tame}(L^{1}(G))\subseteq \mathrm{Tame}(G), which means that every tame functional over L^{1}(G) is also tame as a function over G. Similarly, for the case of Asplund functionals: \mathrm{Asp}(L^{1}(G))\subseteq \mathrm{Asp}(G). Next we show that for some large class containing weak-star saturated bornologies, being “small” as a function and as a functional are the same. Thus, reaffirming Ülger’s classical result [3].

[1] M. Filali, M. Neufang, and M. Sangani Monfared. Representations of Banach algebras subordinate to topologically introverted spaces. Trans. Amer. Math. Soc., 367 : 8033—8050.

[2] M. Megrelishvili. Tame functionals on Banach algebras. In M. Filali, editor, Banach Algebras and Applications, pages 213—226. De Gruyter, Berlin, Boston, 2020.

[3] A. Ülger. Continuity of weakly almost periodic functionals on L_{1}(G). Q. J. Math., 37(4) : 495—497, 1986.