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Representations of involutive semigroups. (English) Zbl 0814.20047

An involutive semigroup \(S\) is a semigroup together with an involutive antiautomorphism \(* : S \rightarrow S,\;s \mapsto s^*\). For discrete semigroups the correspondence between cyclic representations of \(S\) and exponentially bounded positive definite functions is the result of Theorem II.13. For topological involutive semigroups the continuous exponentially bounded positive definite functions correspond to locally bounded strongly continuous representations (Theorem III.8). These results are applied to prove an Extension Theorem which guarantees that, under certain natural conditions, a continuous locally bounded representation of a semigroup ideal extends continuously to the whole semigroup.
Reviewer: A.K.Guts (Omsk)

MSC:

20M30 Representation of semigroups; actions of semigroups on sets
22A25 Representations of general topological groups and semigroups
43A35 Positive definite functions on groups, semigroups, etc.
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