## 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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### References:

 [1] [B84] Berg, C., Christensen, J. P. R., and P. Ressel, ”Harmonic analysis on semigroups,” Graduate Texts in Math., Springer, 1984. [2] [Bou67] Bourbaki, N., ”Espaces vectoriels topologiques, Chap. III–V,” Hermann, Paris, 1967. [3] [Dix64] Dixmier, J., ”LesC *-algèbres et leurs représentations”, Gauthier-Villars, Paris, 1964. [4] [Dz84] Dzinotyiweyi, H. A. M., ”The analogue of the group algebra for topological semigroups,” Pitman, Boston, London, Melbourne, 1984. · Zbl 0581.43004 [5] [La84] Lawson, J. D.,Points of Continuity for semigroups actions, Transactions of the AMS284 (1984), 183–202. [6] [HN93] Hilgert, J. and K.-H Neeb, ”Lie semigroups and their applications,” Springer Verlag, Lecture Notes in Math.1552, 1993. [7] [Ne93a] Neeb, K.-H., ”Holomorphic Representation Theory and Coadjoint Orbits of Convexity Type,” Habilitation Thesis Technische, Hochschule Darmstadt, 1993. [8] [Ne93b] Neeb, K.-H.,Holomorphic representation theory I–III submitted. [9] [Ols82a] Ol’shanskiî, G. I.,Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. and Appl.15 (1982), 275–285. · Zbl 0503.22011 [10] [Ols82b] Ol’shanskiî, G. I.,Complex Lie semigroups, Hardy spaces and the Gelfand Gindikin program, in Russian, Conference Report, 1982. [11] [Wal92] Wallach, N. R., ”Real reductive groups II”, Academic Press Inc., Boston, New York, Tokyo, 1992. · Zbl 0785.22001
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