Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-06T17:55:33.617Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Absract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Busby, R. C., ‘Double centralizers and extensions of C*-algebras,’ Trans. Amer. Math. Soc. 132 (1968), 79–99.Google Scholar
[2]Busby, R. C. and Smith, H. A., ‘Representations of twisted group algebras,’ Trans. Amer. Math. Soc. 147 (1970), 503–537.CrossRefGoogle Scholar
[3]Dixmier, J., C*–algebras, (North-Holland, New York, 1977).Google Scholar
[4]Doplicher, S., Kastler, D., and Robinson, D. W., ‘Covariance algebras in field theory and statistical mechanics,’ Comm. Math. Phys. 3 (1966), 1–28.CrossRefGoogle Scholar
[5]Dupré, M., ‘The classification and structure of C*-algebra bundles,’ Mem. Amer. Math. Soc. 21 (1979), No. 222.Google Scholar
[6]Fell, J. M. G., ‘The structure of algebras of operator fields,’ Acta Math. 106 (1961), 233–280.CrossRefGoogle Scholar
[7]Fell, J. M. G., ‘An extension of Mackey's method to Banach *-algebraic bundles,’ Mem. Amer. Math. Soc. 90 (1969).Google Scholar
[8]Green, P., ‘The local structure of twisted covariance algebras,’ Acta Math. 140 (1978), 191–250.CrossRefGoogle Scholar
[9]Green, P., ‘The structure of imprimitivity algebras,’ J. Funct. Anal. 36 (1980), 88–104.CrossRefGoogle Scholar
[10]Glimm, J., ‘Locally compact transformation groups,’ Trans. Amer. Math. Soc. 101 (1961), 124–128.CrossRefGoogle Scholar
[11]Gootman, E. C., ‘The type of some C* - and W*-algebras associated with transformation groups,’ Pacific J. Math. 48 (1973), 93–106.CrossRefGoogle Scholar
[12]Gootman, E. C. and Rosenburg, J., ‘The structure of crossed product C*-algebras: A proof of the Generalized Effros-Hahn conjecture,’ Invent. Math. 52 (1979), 283–298.CrossRefGoogle Scholar
[13]Lee, R. -Y., ‘On the C*-algebras of operator fields,’ Indiana Univ. Math. J. 25 (1976), 303–314.CrossRefGoogle Scholar
[14]Pedersen, G. K., C*-algebras and their automorphism groups, (Academic Press, London, 1979).Google Scholar
[15]Rieffel, M. A., ‘Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner,’ Studies in Analysis, Adv. Math. Suppl. Studies 4 (1979), 43–81.Google Scholar
[16]Takesaki, M., ‘Covariant representations of C*-algebras and their locally compact automorphism groups’, Acta Math. 119 (1967), 273–303.CrossRefGoogle Scholar
[17]Williams, D. P., ‘The topology on the primitive ideal space of transformation group C*-algebras and C.C.R. transformation group C*-algebras,’ Trans. Amer. Math. Soc. 266 (1981), 335–359.Google Scholar