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Transactions of the American Mathematical Society

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



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

Author: Elliot C. Gootman
Journal: Trans. Amer. Math. Soc. 170 (1972), 97-108
MSC: Primary 22D25; Secondary 46L05
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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Keywords: Locally compact transformation group, amenable group, primitive ideal, <!– MATH ${C^ \ast }$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img6.gif" ALT="${C^ \ast }$">-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