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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Integral decomposition of functionals on $ C\sp{\ast} $-algebras


Author: Herbert Halpern
Journal: Trans. Amer. Math. Soc. 168 (1972), 371-385
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-1972-0296710-7
MathSciNet review: 0296710
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Abstract: The spectrum of the center of the weak closure of a $ {C^ \ast }$-algebra with identity on a Hilbert space is mapped into a set of quasi-equivalence classes of representations of the $ {C^ \ast }$-algebra so that every positive $ \sigma $-weakly continuous functional on the algebra can be written in a central decomposition as an integral over the spectrum of a field of states whose canonical representations are members of the respective quasi-equivalence classes except for a nowhere dense set. Various questions relating to disjointness of classes, factor classes, and uniformly continuous functionals are studied.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0296710-7
Keywords: $ {C^ \ast }$-algebras, von Neumann algebras, positive functionals, representations, quasi-equivalence classes of representations, decomposition theory
Article copyright: © Copyright 1972 American Mathematical Society