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On the “Third Definition” of the Topology on the Spectrum of a C*-Algebra

Published online by Cambridge University Press:  20 November 2018

L. Terrell Gardner
L. Terrell Gardner*
Affiliation:
University of Toronto, Toronto, Ontario
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0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the form

where the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].

A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 445 - 450
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bichteler, K., A generalization to the non-separable case of Takesaki's duality theorem for C*-algebras, Invent. Math. 9 (1969), 89–98.Google Scholar
2. Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1969).Google Scholar
3. Fell, J. M. G., C*-algebras with smooth dual, Illinois J. Math. 4 (1960), 221–230.Google Scholar
4. Fell, J. M. G., Weak containment and induced representations of groups, Can. J. Math. 14 (1962), 237–268.Google Scholar
5. Takesaki, M., A duality in the representation theory of C*-algebras, Ann. of Math. (2) 85 (1967), 370–382.Google Scholar