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Mackey Borel Structure for the Quasi-Dual of a Separable C*-Algebra

Published online by Cambridge University Press:  20 November 2018

Herbert Halpern*
Affiliation:
University of Cincinnati, Cincinnati, Ohio
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Let A be a separable C*-algebra. Two representations π and π1 of A on the Hilbert spaces H and H1, respectively are said to be quasi-equivalent (denoted by π ~ π1) if projections of HH1 on the invariant subspaces H and H1 of (ππ1)(A) have the same central support in the commutant (ππ1) (A)′ of (ππ1) (A), or equivalently, if there is an isomorphism ϕ of π(A)″ onto π1(A)″ such that ϕ(π(x)) = π(x) for all xA (cf. [5, § 5]). A representation π of A is said to be a factor representation if the center of π(A)″ consists of scalar multiples of the identity.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Auslander, L. and Moore, C. C., Unitary representation of solvable Lie groups, Memoirs Amer. Math. Soc. 62 (1966).Google Scholar
2. Bourbaki, N., Topologie générale Ch. 9., Actualités Scientifiques et industrielles No. 1045 (Hermann, Paris, 1958).Google Scholar
3. Combes, F., Représentations d'une C*-algèbres et formes linéaires positives, C. R. Acad. Sci. Paris Ser. A-B 260 (1965), 59935996.Google Scholar
4. Dixmier, J., Quasi-dual d'une ideal dans une C*-algèbre, Bull. Sci. Math. 87 (1963), 711.Google Scholar
5. Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1964).Google Scholar
6. Efïros, E., Transformation groups and C*-algebras, Ann. of Math. 81 (1965), 3855.Google Scholar
7. Efïros, E., The canonical measures for a separable C*-algebra, Amer. J. Math. 92 (1970), 5660.Google Scholar
8. Gardner, L. T., On the Mackey Borel structure, Can. J. Math. 23 (1971), 674678.Google Scholar
9. Halpern, H., Open projections and Borel structures for C*-algebras (to appear in Pacific J. Math.).Google Scholar
10. Sakai, S., C*-algebras and W*-algebras (Springer-Verlag, New York, 1971).Google Scholar