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Proceedings of the American Mathematical Society

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Semigroup representations, positive definite functions and abelian $C^*$-algebras

Authors: P. Ressel and W. J. Ricker
Journal: Proc. Amer. Math. Soc. 126 (1998), 2949-2955
MSC (1991): Primary 47A67, 47B15, 47D03, 47D25
MathSciNet review: 1486749
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Abstract: It is shown that every $*$-representation of a commutative semigroup $S$ with involution via operators on a Hilbert space has an integral representation with respect to a unique, compactly supported, selfadjoint Radon spectral measure defined on the Borel sets of the character space of $S$. The main feature is that the proof, which is based on the theory of positive definite functions, makes no use what-so-ever (directly or indirectly) of the theory of $C^*$-algebras or more general Banach algebra arguments. Accordingly, this integral representation theorem is used to give a new proof of the Gelfand-Naimark theorem for abelian $C^*$-algebras.

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Additional Information

P. Ressel
Affiliation: Math.-Geogr. Fakultät, Katholische Universität Eichstätt, D-85071 Eichstätt, Germany

W. J. Ricker
Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales, 2052 Australia

Received by editor(s): March 3, 1997
Additional Notes: The support of the Deutscher Akademischer Austauschdienst (DAAD) is gratefully acknowledged by the second author
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society