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A weak form of amenability of topological semigroups and its applications in ergodic and fixed point theories. (English) Zbl 1512.43001

Authors’ abstract: In this paper, we introduce a weak form of amenability on topological semigroups that we call \(\varphi\)-amenability, where \(\varphi\) 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 \(\varphi\)-amenable, then \(S\) is amenable. Moreover, \(\varphi\)-ergodicity for a topological semigroup \(S\) is introduced and it is proved that under some conditions on \(S\) and a Banach space \(X\), \(\varphi\)-amenability and \(\varphi\)-ergodicity of any antirepresntation defined by a right action \(S\) on \(X\), are equivalent. A relation between \(\varphi\)-amenability of topological semigroups and the existence of a common fixed point is investigated and by this relation, Hahn-Banach property of topological semigroups in the sense of \(\varphi\)-amenability defined and studied.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
22D15 Group algebras of locally compact groups
22A20 Analysis on topological semigroups
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References:

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