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