Abstract
Let (G,A,α) be a separable C*-dynamical system, with G abelian and A type I. We prove that all points in a quasi-orbit in  have the same isotropy subgroup and determine cocycles of this subgroup of the same class. These results are then used to prove that if  has a non-transitive quasi-orbit and either the (common) dimension of all representations in this quasi-orbit is finite or the (common) isotropy group is discrete, then the crossed product algebra is non-type I. While this latter result has long been known, we present a new proof using Takai duality. An example is also given of a non-smooth action of ℝ2 on A for which the crossed product algebra is nevertheless type I. Finally, we characterize the Connes spectrum in terms of the separated primitive ideals of the crossed product algebra. When A=C0 (X) is commutative, we determine which primitive ideals of the crossed product algebra are separated in terms of the behaviour of isotropy groups and orbit closures.
Partially supported by a grant from the National Science Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Auslander and C. C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62(1996).
L. Baggett and A. Kleppner, Multiplier representations of abelian groups, J. Functional Analysis 14(1973), 299–324.
J. Dixmier, C*-algebras, North-Holland Mathematical Library, volume 15, Amsterdam, 1977.
S. Doplicher, D. Kastler and D. W. Robinson, Covariance algebras in field theory and statistical mechanics, Comm. Math. Phys. 3(1966), 1–28.
E. G. Effros and F. Hahn, Locally compact transformation groups and C*-algebras, Mem. Amer. Math. Soc. 75(1967).
E. C. Gootman, The type of some C* and W⋆-algebras associated with transformation groups, Pacific J. Math. 48(1973), 93–106.
E. C. Gootman and D. Olesen, Spectra of actions on type I C*-algebras, Math. Scand. 47(1980), 329–349.
____, Minimal abelian group actions on type I C*-algebras, in Operator Algebras and Applications, R. V. Kadison, ed., Proc. Symp. Pure Math., vol. 38, part I, pp. 323–325, Amer. Math. Soc., Providence, R.I., 1982.
E. C. Gootman and J. Rosenberg, The structure of crossed product C*-algebras: A proof of the generalized Effros-Hahn conjecture, Invent. Math. 52(1979), 283–298.
P. Green, The local structure of twisted covariance algebras, Acta Math. 140(1978), 191–250.
R. R. Kallman, Certain quotient spaces are countably separated, Illinois J. Math. 19(1975), 378–388.
D. Olesen and G. K. Pedersen, Applications of the Connes spectrum to C*-dynamical systems, J. Functional Anal. 30(1978), 179–197.
_____, Applications of the Connes spectrum to C*-dynamical systems, III, J. Functional Analysis 45(1982), 357–390.
G. K. Pedersen, C*-algebras and their automorphism groups, London Math. Soc. Monographs 14, Academic Press, London/New York, 1979.
H. Takai, On a duality for crossed products of C*-algebras, J. Functional Analysis 19(1975), 25–39.
M. Takesaki, Covariant representations of C*-algebras and their locally compact automorphism groups, Acta Math. 119(1967), 273–303.
D. Williams, The topology on the primitive ideal space of transformation group C*-algebras and C.C.R. transformation group C*-algebras, Trans. Amer. Math. Soc. 226(1981), 335–359.
____, Transformation group C*-algebras with Hausdorff spectrum, Illinois J. Math. 26(1982), 317–321.
R.R. Kallman, Certain quotient spaces are countably separated, II, J. Functional Analysis 21(1976), 52–62.
R.R. Kallman, Certain quotient spaces are countably separated, III, J. Functional Analysis 22(1976), 225–241.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1985 Springer-Verleg
About this paper
Cite this paper
Gootman, E.C. (1985). Abelian group actions on type I C*-algebras. In: Araki, H., Moore, C.C., Stratila, ŞV., Voiculescu, DV. (eds) Operator Algebras and their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics, vol 1132. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074884
Download citation
DOI: https://doi.org/10.1007/BFb0074884
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15643-7
Online ISBN: 978-3-540-39514-0
eBook Packages: Springer Book Archive