J Austral Math Soc Ser A 47 pp226--235, 1989.

The Structure of Crossed Products by Smooth Actions

Dana P. Williams

Abstract

Let x br a C*-bundle over T with fibres {A_t}_{t Î A}. Suppose that A is the C*-algebra of sections of x which vanish at infinity, and that (A, G, a) is a C*-dynamical system such that, for each t Î T, the ideal I_t = {f Î A | f(t) = 0} is G-invariant. If an addition, the stabiliser group of each P Î Prim(A) is amenable, then A ×_a G is the section algebra of a C*-bundle with fibres {A_t ×_a G}_{t Î T}. The above theorem may be used to prove a structure theorem for crossed products built from C*-dynamical systems (A, G, a) where the action of G on A is smooth. Assuming that the stabiliser groups are amenable, then A ×_a G has a composition series such that each quotient is a section algebra of a C*-bundle where the fubres are of the form A_d ×_a G; moreoverm the A_d correspond to locally closed subsets of Prim(A), and G acts transitively on Prim(A_d). In many cases, in particular when (G, A) is separable, the A_d ×_a G have been computed explicitly by other authors. These results are actually proved for twisted C*-dynamical systems.

1980 AMS Subject Classification (1985 Revision): 46L05

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Authors

Dana P. Williams

Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, U.S.A.