Ergodic Actions of Compact Groups on Operator Algebras II: Classification of Full Multiplicity Ergodic Actions

Wassermann, Antony

Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1482-1527

Résumé

In the first paper of this series [17], we set up some general machinery for studying ergodic actions of compact groups on von Neumann algebras, namely, those actions for which . In particular we obtained a characterisation of the full multiplicity ergodic actions:THEOREM A. If α is an ergodic action of G on , then the following conditions are equivalent: (1) Each spectral subspace has multiplicity dim π for π in . (2) Each π in admits a unitary eigenmatrix in . (3) The W* crossed product is a (Type I) factor. (4) The C* crossed product of the C* algebra of norm continuity is isomorphic to the algebra of compact operators on a Hilbert space.

Détail

DOI : 10.4153/CJM-1988-068-4

Bibliographie

[1] 1. Albeverio, S. and Høegh-Krøhn, R., Ergodic actions by compact groups on operator algebras, Math. Z. 174 (1980), 1–17. Google Scholar

[2] 2. Burnside, W., The theory of groups of finite order, 2nd ed. (Dover, New York, 1955). Google Scholar

[3] 3. Dixmier, J., Les algèbres d'opérateurs dans Vespace hilbertien (Gauthier-Villars, Paris, 1957 and 1969). Google Scholar

[4] 4. Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1964 and 1968). Google Scholar

[5] 5. Drinfeld, D. G., Quantum groups, Report to the I.CM., Berkeley, (1986). Google Scholar

[6] 6. Green, P., The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250. Google Scholar

[7] 7. Hegh-Krøhn, R., Landstad, M. and Størmer, E., Compact ergodic groups of automorphisms, Ann. of Math. 114 (1981), 75–86. Google Scholar

[8] 8. Howlett, R. B. and Isaacs, I. M., On groups of central type, Math. Z. 179 (1982), 555–569. Google Scholar

[9] 9. Kleppner, A., Multipliers on abelian groups, Math. Ann. 158 (1965), 11–34. Google Scholar

[10] 10. Kleppner, A., Non-type I multiplier representations of abelian groups, unpublished manuscript (1974). Google Scholar

[11] 11. Landstad, M., unpublished manuscript. Google Scholar

[12] 12. Nakagami, Y. and Takesaki, M., Duality for crossed products of von Neumann algebras, Lect. Notes in Math. 731 (Springer-Verlag, 1979). Google Scholar | DOI

[13] 13. Olesen, D., Pedersen, G. and Takesaki, M., Ergodic actions of compact abelian groups, J. Operator Theory 3 (1980), 237–270. Google Scholar

[14] 14. Sah, C.-H., Automorphisms of finite groups, J. Algebra 10 (1968), 47–68; Addenda, J. Algebra 44(1977), 573–575. Google Scholar

[15] 15. Slawny, J., On factor representations and the C*-algebra of canonical commutation relations, Comm. Math. Phys. 24 (1972), 151–170. Google Scholar

[16] 16. Wassermann, A.J., Automorphic actions of compact groups on operator algebras, Ph.D. Dissertation, University of Pennsylvania (1981). Google Scholar

[17] 17. Wassermann, A.J., Ergodic actions of compact groups on operator algebras I: General theory, Ann. of Math., to appear. Google Scholar | DOI

[18] 18. Wassermann, A.J., Ergodic actions of compact groups on operator algebras III: Classification for SU(2), Inv. Math. 93 (1988), 309–355. Google Scholar

[19] 19. Wassermann, A.J., Coactions and Yang-Baxter equations for ergodic actions and subfactors, in Operator algebras and applications, L.M.S. lecture notes, Cambridge Univ. Press, to appear. Google Scholar

[20] 20. Wenzl, H. G., On sequences of projections, C.R. Math. Rep. Acad. Sci. Canada 9 (1987), 5–9. Google Scholar

[21] 21. Wolf, J. A., Spaces of constant curvature (McGraw-Hill, New York, 1967). Google Scholar