Spectra for Infinite Tensor Product Type Actions of Compact Groups

Gootman, Elliot C. ; Lazar, Aldo J.

Canadian journal of mathematics, Tome 48 (1996) no. 2, pp. 330-342

Résumé

We present explicit calculations of the Arveson spectrum, the strong Arveson spectrum, the Connes spectrum, and the strong Connes spectrum, for an infinite tensor product type action of a compact group. Using these calculations and earlier results (of the authors and C. Peligrad) relating the various spectra to the ideal structure of the crossed product algebra, we prove that the topology of G influences the ideal structure of the crossed product algebra, in the following sense: if G contains a nontrivial connected group as a direct summand, then the crossed product algebra may be prime, but it is never simple; while if G is discrete, the crossed product algebra is simple if and only if it is prime. These results extend to compact groups analogous results of Bratteli for abelian groups. In addition, we exhibit a class of examples illustrating that for compact groups, unlike the case for abelian groups, the Connes spectrum and strong Connes spectrum need not be stable.

Détail

DOI : 10.4153/CJM-1996-018-6

Mots-clés : 46L55, 22D25

Bibliographie

[1] 1. Bratteli, , Crossed products ofUHF algebras byproduct type actions, Duke Math. J. 46 (1979), 1—23. Google Scholar

[2] 2. Bratteli, , Kishimoto, A. and Robinson, D.W., Embedding product type actions into C*-dynamical systems, J. Funct. Anal. 75 (1987), 188–210. Google Scholar

[3] 3. Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318—340. Google Scholar

[4] 4. Gootman, E.C. and Lazar, A.J., Compact group actions on C*-algebras: an application of non-commutative duality, J. Funct. Anal. 91 (1990), 237–245. Google Scholar

[5] 5. Gootman, E.C., Lazar, A.J. and Peligrad, C., Spectra for compact group actions, J. Operator Theory 31 (1994), 381–399. Google Scholar

[6] 6. Kishimoto, A., Simple crossed products of C*-algebras by locally compact abelian groups, Yokohama Math. J. 28 (1980), 69–85. Google Scholar

[7] 7. Landstad, M.B., Algebras of spherical functions associated with covariant systems over a compact group, Math. Scand. 47 (1980), 137–149. Google Scholar

[8] 8. Landstad, M.B., Phillips, J., Raeburn, I. and Sutherland, C.E., Representations of crossed products by coactions and principal bundles, Trans. Amer. Math. Soc. 299 (1987), 747–784. Google Scholar

[9] 9. Olesen, D. and Pedersen, G.K., Applications of the Connes spectrum to C* -dynamical systems, J. Funct. Anal. 30 (1978), 179–197. Google Scholar

[10] 10. Pedersen, G.K.,C*-algebras and their automorphism groups, Academic Press, London, New York, 1979. Google Scholar

[11] 11. Peligrad, C., Locally compact group actions on C* -algebras and compact subgroups, J. Funct. Anal. 76 (1988), 126–139. Google Scholar

[12] 12. Vallin, J.-M., C*-algebres de Hopfet C*-algebres de Kac, Proc. London Math. Soc. (3) 50 (1985), 131–174. Google Scholar