Semigroup representations, positive definite functions and abelian -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 | References | Similar Articles | Additional Information

Abstract: It is shown that every -representation of a commutative semigroup 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 . 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 -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 -algebras.

**1.**Christian Berg, Jens Peter Reus Christensen, and Paul Ressel,*Harmonic analysis on semigroups*, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. MR**747302****2.**Christian Berg and P. H. Maserick,*Exponentially bounded positive definite functions*, Illinois J. Math.**28**(1984), no. 1, 162–179. MR**730718****3.**P. H. Maserick,*Spectral theory of operator-valued transformations*, J. Math. Anal. Appl.**41**(1973), 497–507. MR**0343084****4.**Paul Ressel,*Integral representations on convex semigroups*, Math. Scand.**61**(1987), no. 1, 93–111. MR**929398****5.**Walter Rudin,*Functional analysis*, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR**0365062**

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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

DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04814-X

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