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Some groups with T1 primitive ideal spaces

Part of: Locally compact groups and their algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

A. L. Carey  and
W. Moran
A. L. Carey
Affiliation:
Department of Pure Mathematics The University of AdelaideAdelaide, SA5001, Australia
W. Moran
Affiliation:
Department of Pure Mathematics The University of AdelaideAdelaide, SA5001, Australia
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Abstract

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Let G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

MSC classification

Secondary: 22D10: Unitary representations of locally compact groups 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 46L55: Noncommutative dynamical systems

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 38 , Issue 1 , February 1985 , pp. 55 - 64
Copyright
Copyright © Australian Mathematical Society 1985

References

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