Tame functionals on Banach algebras

Abstract

In the present note we introduce tame functionals on Banach algebras. A functional f ∈ A * on a Banach algebra A is tame if the naturally defined linear operator A → A * , a → f • a factors through Rosenthal Banach spaces (i.e., not containing a copy of l 1 ). Replacing Rosenthal by reflexive we get a well known concept of weakly almost periodic functionals. So, always WAP(A) ⊆ Tame(A). We show that tame functionals on l 1 (G) are induced exactly by tame functions (in the sense of topological dynamics) on G for every discrete group G. That is, Tame(l 1 (G)) = Tame(G). Many interesting tame functions on groups come from dynamical systems theory. Recall that WAP(L 1 (G)) = WAP(G) (Lau [19], Ülger [28]) for every locally compact group G. It is an open question if Tame(L 1 (G)) = Tame(G) holds for (nondiscrete) locally compact groups.