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Dixmier-Douady Classes of Dynamical Systems and Crossed Products

Published online by Cambridge University Press:  20 November 2018

Iain Raeburn  and
Dana P. Williams
Iain Raeburn
Affiliation:
Department of Mathematics University of Newcastle Newcastle, New South Wales 2308 Australia
Dana P. Williams
Affiliation:
Department of Mathematics Dartmouth College Hanover, New Hampshire 03755-3551 USA
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Abstract

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Continuous-trace C*-algebras A with spectrum T can be characterized as those algebras which are locally Monta equivalent to C0(T). The Dixmier-Douady class δ(A) is an element of the Čech cohomology group Ȟ3(T, ℤ) and is the obstruction to building a global equivalence from the local equivalences. Here we shall be concerned with systems (A, G, α) which are locally Monta equivalent to their spectral system (C0(T),G, τ), in which G acts on the spectrum T of A via the action induced by α. Such systems include locally unitary actions as well as N-principal systems. Our new Dixmier-Douady class δ (A, G, α) will be the obstruction to piecing the local equivalences together to form a Monta equivalence of (A, G, α) with its spectral system. Our first main theorem is that two systems (A, G, α) and (B, G, β) are Monta equivalent if and only if δ (A, G, α) = δ (B, G, β). In our second main theorem, we give a detailed formula for δ (A ⋊α G) when (A, G, α) is N-principal.

Keywords

46L5546L05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 1032 - 1066
Copyright
Copyright © Canadian Mathematical Society 1993

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