Tame functionals on Banach algebras

In the present note, we introduce tame functionals on Banach algebras. A func- tional f∈A^*on a Banach algebra Ais tame if the naturally defined linear operator A→A^*, a→f a factors through Rosenthal Banach spaces (i. e., not containing a copy of l1). 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 the group algebra l1(G)are induced exactly by tame functions (in the sense of topolog- ical dynamics) on Gfor every discrete group G. That is, Tame(l1(G))=Tame(G). Many interesting tame functions on groups come from dynamical systems theory. Recall that WAP(L₁(G))=WAP(G)(Lau [18], Ülger [27]) for every locally compact group G. It is an open question if Tame(L₁(G))=Tame(G)holds for (nondiscrete) locally compact groups.