## Abstract

To every Banach space V we associate a compact right topological affine semigroup ℰ(V). We show that a separable Banach space V is Asplund if and only if ℰ(V) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l _{1}) if and only if ℰ(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in ℰ(V). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily nonsensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

Original language | English |
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Title of host publication | Asymptotic Geometric Analysis |

Subtitle of host publication | Proceedings of the Fall 2010 Fields Institute Thematic Program |

Editors | Monika Ludwig, Vladimir Pestov, Vitali Milman, Nicole Tomczak-Jaegermann |

Pages | 75-144 |

Number of pages | 70 |

DOIs | |

State | Published - 2013 |

### Publication series

Name | Fields Institute Communications |
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Volume | 68 |

ISSN (Print) | 1069-5265 |

### Bibliographical note

Cited By :7Export Date: 6 March 2022

Correspondence Address: Megrelishvili, M.; Department of Mathematics, , 52900 Ramat-Gan, Israel; email: [email protected]

## Keywords

- Affine compactification
- Affine flow
- Asplund space
- Enveloping semigroup
- Nonsensitivity
- Right topological semigroup
- Semigroup compactification
- Tame system
- Weakly almost periodic