Representations of dynamical systems on Banach spaces

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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 7→ 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 l1(G) are induced exactly by tame functions
(in the sense of topological dynamics) on G for 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 [19], Ulger ¨
[28]) for every locally compact group G. It is an open question if Tame(L1(G)) =
Tame(G) holds for (nondiscrete) locally compact groups.
Original languageEnglish
Title of host publicationRecent progress in general topology. III
EditorsP. Simon, J. van Mill, K.P. Hart
Pages399-470
Number of pages72
Volume3
ISBN (Electronic)978-94-6239-024-9
DOIs
StatePublished - 2014

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