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On the primitive ideal space of a finitely generated nilpotent group

On the primitive ideal space of a finitely generated nilpotent group Abstract. Let G be a finitely generated nilpotent discrete group. It is shown that either Prim(C*(G)) is Hausdorff or the set of separated points in Prim(C*(Gi)) has empty interior. The first possibility occurs precisely when the centre of G has finite index in G. We also study the strong closure of Prim(C*(G)) in the space of all ideals in C*(G), with both the weak and the strong topologies, and relate it to a space of positive definite class functions on G. This leads to a group-theoretic criterion for the strong closure of Prim(C*(G)) to consist entirely of primitive ideals and minimal primal ideals. 1991 Mathematics Subject Classification: 22D25, 22D10. Introduction In this paper we consider two aspects of the primitive ideal space Prim(C*(<3)) of the C*-algebra of a nilpotent discrete group G. The main emphasis will be on the case where G is finitely generated. Firstly, we investigate in Section 2 the set of separated points in Prim(C* (<?))> We show in Theorem 2.2 that for a large class of nilpotent discrete groups G (including all those that are finitely generated) there is a dichotomy: either Prim(C*(G)) is Hausdorff or eise the set of separated points has empty http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

On the primitive ideal space of a finitely generated nilpotent group

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References (12)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1997.9.193
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let G be a finitely generated nilpotent discrete group. It is shown that either Prim(C*(G)) is Hausdorff or the set of separated points in Prim(C*(Gi)) has empty interior. The first possibility occurs precisely when the centre of G has finite index in G. We also study the strong closure of Prim(C*(G)) in the space of all ideals in C*(G), with both the weak and the strong topologies, and relate it to a space of positive definite class functions on G. This leads to a group-theoretic criterion for the strong closure of Prim(C*(G)) to consist entirely of primitive ideals and minimal primal ideals. 1991 Mathematics Subject Classification: 22D25, 22D10. Introduction In this paper we consider two aspects of the primitive ideal space Prim(C*(<3)) of the C*-algebra of a nilpotent discrete group G. The main emphasis will be on the case where G is finitely generated. Firstly, we investigate in Section 2 the set of separated points in Prim(C* (<?))> We show in Theorem 2.2 that for a large class of nilpotent discrete groups G (including all those that are finitely generated) there is a dichotomy: either Prim(C*(G)) is Hausdorff or eise the set of separated points has empty

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1997

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