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The structure of crossed products by smooth actions

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Dana P. Williams
Dana P. Williams
Affiliation:
Department of MathematicsDartmouth College Hanover, New Hampshire 03755, U.S.A.
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Let ξ be a C*;-bundle over T with fibres {At}t∈A. Suppose that A is the C*-algebra of sections of ξ which vanish at infinity, and that (A, G, α) is a C*-dymanical system that, for each t ∈ T, the ideal It = {f ∈ A|f(t) =; 0} is G-invariant. If in addition, the stabiliser group of each P ∈ Prim(A) is amenable, then A ⋊αG is the section algebra of a C*-bundle with fibres {At ⋊αG}t∈T.

The above theorem may be used to prove a structure theorem for crossed products built from C*-dynamical systems (A, G, α) where the action of G on A is smooth. Assuming that the stabiliser groups are amenable, then A ⋊αG has a composition series such that each quotient is a section algebra of a C*-bundle where the fibres are of the form Aσ ⋊αG; moreover, the Aσ correspond to locally closed subsets of Prim(A), and G acts transitively on Prim(Aσ). In many cases, in particular when (G, A) is separable, the Aσ ⋊αG have been computed explicitly by other authors.

These results are actually proved for twisted C*dynamical systems.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 2 , October 1989 , pp. 226 - 235
Copyright
Copyright © Australian Mathematical Society 1989

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