## 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.

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 language | American English |
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Title of host publication | Recent progress in general topology. III |

Editors | P. Simon, J. van Mill, K.P. Hart |

Publisher | Atlantis Press, Paris |

Pages | 399-470 |

Number of pages | 72 |

Volume | 3 |

ISBN (Electronic) | 978-94-6239-024-9 |

ISBN (Print) | 978-94-6239-023-2 |

DOIs | |

State | Published - 2014 |