Semigroup representations, positive definite functions and abelian $C^*$-algebras
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- by P. Ressel and W. J. Ricker PDF
- Proc. Amer. Math. Soc. 126 (1998), 2949-2955 Request permission
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.References
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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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2949-2955
- MSC (1991): Primary 47A67, 47B15, 47D03, 47D25
- DOI: https://doi.org/10.1090/S0002-9939-98-04814-X
- MathSciNet review: 1486749