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Transactions of the American Mathematical Society

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

 

 

Primitive ideals of $ C\sp{\ast} $-algebras associated with transformation groups


Author: Elliot C. Gootman
Journal: Trans. Amer. Math. Soc. 170 (1972), 97-108
MSC: Primary 22D25; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9947-1972-0302818-X
MathSciNet review: 0302818
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Abstract: We extend results of E. G. Effros and F. Hahn concerning their conjecture that if $ (G,Z)$ is a second countable locally compact transformation group, with $ G$ amenable, then every primitive ideal of the associated $ {C^ \ast }$-algebra arises as the kernel of an irreducible representation induced from an isotropy subgroup. The conjecture is verified if all isotropy subgroups lie in the center of $ G$ and either (a) the restriction of each unitary representation of $ G$ to some open subgroup contains a one-dimensional subrepresentation, or (b) $ G$ has an open abelian subgroup and orbit closures in $ Z$ are compact and minimal.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0302818-X
Keywords: Locally compact transformation group, amenable group, primitive ideal, $ {C^ \ast }$-algebra, kernel of a representation, isotropy subgroup, induced representation, positive-definite measure, weak containment, quasi-orbit, orbit closure
Article copyright: © Copyright 1972 American Mathematical Society