A Weak Form of Amenability of Topological Semigroups and its Applications in Ergodic and Fixed Point Theories

Ali Jabbari; Ali Ebadian; Madjid Eshaghi Gordji

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arXiv · 2005.08089

A Weak Form of Amenability of Topological Semigroups and its Applications in Ergodic and Fixed Point Theories

Ali Jabbari·Ali Ebadian·Madjid Eshaghi Gordji

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Abstract

In this paper, we introduce a weak form of amenability on topological semigroups that we call $φ$-amenability, where $φ$ is a character on a topological semigroup. Some basic properties of this new notion are obtained and by giving some examples, we show that this definition is weaker than the amenability of semigroups. As a noticeable result, for a topological semigroup $S$, it is shown that if $S$ is $φ$-amenable, then $S$ is amenable. Moreover, $φ$-ergodicity for a topological semigroup $S$ is introduced and it is proved that under some conditions on $S$ and a Banach space $X$, $φ$-amenability and $φ$-ergodicity of any antirepresntation defined by a right action $S$ on $X$, are equivalent. A relation between $φ$-amenability of topological semigroups and existance of a common fixed point is investigated and by this relation, Hahn-Banach property of topological semigroups in the sense of $φ$-amenability defined and studied.

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BibTeX RIS

Ali Jabbari, Ali Ebadian, Madjid Eshaghi Gordji. 2020-05-16. A Weak Form of Amenability of Topological Semigroups and its Applications in Ergodic and Fixed Point Theories. https://arxiv.org/abs/2005.08089

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