On C*-Diagonals

Kumjian, Alexander

Canadian journal of mathematics, Tome 38 (1986) no. 4, pp. 969-1008

Résumé

Preface. The impetus for this study arose from the belief that the structure of a C*-algebra is illuminated by an understanding of the manner in which abelian subalgebras embed in it. Posed in its full generality, the question concerning abelian subalgebras would seem impossible to answer. A notion of diagonal subalgebra is, however, proposed which has the virtue that one can associate a “topological”; invariant to the pair consisting of the diagonal and the ambient algebra, from which these algebras may be retrieved.In the setting of von Neumann algebras, the analogous question was addressed in the seminal work of Feldman and Moore [13]. Their definition of Cartan subalgebra permits the abstraction of a complete invariant consisting of a Borel equivalence relation together with a certain cohomology class on the relation from which the Cartan pair may be recovered. Our development parallels theirs in spirit; differences in substance derive from the topological flavor of C*-theory.

Détail

DOI : 10.4153/CJM-1986-048-0

Bibliographie

[1] 1. Idrissi, A. Alami, Sur le théorème de Riesz dans les algebres stellaires, Thèse de 3 ème cycle, Paris VI (1979). Google Scholar

[2] 2. Anderson, J., Extensions, restrictions, and representations of states on C*-algebras, Trans. Amer. Math. Soc. 227 (1977), 63–107. Google Scholar

[3] 3. Archbold, R. J., Extensions of states of C*-algebras, J. London Math. Soc. 21 (1980), 351–354 Google Scholar

[4] 4. Archbold, R. J., Bunce, J. W. and Gregson, K. D., Extensions of states of C*-algebras II, Proc. Royal Soc. Edinburgh 92A (1982), 113–122. Google Scholar

[5] 5. Blackadar, B. and Kumjian, A., Skew products of relations and the structure of simple C*-algebras, Math. Zeitschrift 189 (1985), 55–63. Google Scholar

[6] 6. Brown, L. G., Green, P. and Rieffel, M. A., Stable isomorphism and strong Morita equivalence of C*-algebras, Pacific J. Math. 71 (1977), 349–363. Google Scholar

[7] 7. Disney, S. A. R. and Raeburn, I., Homogeneous C*-algebras whose spectra are tori, J. Aust. Math. Soc. 55 (1985), 9–39. Google Scholar

[8] 8. Disney, S. A. R., Elliott, G. A., Kumjian, A. and Raeburn, I., On the classification of noncommutative tori, C.R. Math. Rep. Acad. Sci. Canada 7 (1985), 137–141. Google Scholar

[9] 9. Dixmier, J., C*-algebras (North-Holland, Amsterdam, 1977). Google Scholar

[10] 10. Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. of Algebra 38 (1976), 29–44. Google Scholar

[11] 11. Elliott, G. A., On totally ordered groups, and K , Ring theory Waterloo (1978), Lecture notes in Math. 734 (Springer-Verlag, Berlin, 1979). Google Scholar

[12] 12. Elliott, G. A., On the K-theory of the C*-algebra generated by a projective representation of a torsion-free discrete abelian group, in Operator algebras and group representations, Vol. I (Pitman, London, 1984). Google Scholar

[13] 13. Feldman, J. and Moore, C. C., Ergodic equivalence relations, cohomology and von Neumann algebras I, II, Trans. Amer. Math. Soc. 234 (1977), 289–359. Google Scholar

[14] 14. Greenberg, M., Lectures on algebraic topology (Benjamin, New York, 1967). Google Scholar

[15] 15. Husemoller, D., Fibre bundles (McGraw-Hill, New York, 1966). Google Scholar | DOI

[16] 16. Jensen, C. U., Les fondeurs dérivés de lim et leurs applications en théorie des modules, Lecture Notes in Math. 254 (Springer-Verlag, Berlin, 1972). Google Scholar

[17] 17. Kaminker, J. and Schochet, C., K-theory and Steenrod homology. Applications to the Brown-Douglas-Fillmore theory of operator algebras, Trans. Amer. Math. Soc. 227 (1977), 63–107. Google Scholar

[18] 18. Kasparov, G. G., Hilbert C*-modules: Theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), 133–150. Google Scholar

[19] 19. Kelley, J. L., General topology (Van Nostrand, Princeton, 1955). Google Scholar

[20] 20. Kumjian, A., On localizations and simple C*-algebras, Pacific J. Math. 772 (1984), 141–192. Google Scholar

[21] 21. Kumjian, A., Preliminary algebras arising from local homeomorphisms, Math. Scand. 52 (1983), 269–278. Google Scholar

[22] 22. Kumjian, A., On C*-diagonals and twisted relations, Semesterbericht Funktionalanalysis, Universität Tübingen, Wintersemester (1982/83). Google Scholar

[23] 23. Kumjian, A., Diagonals in algebras of continuous trace, OATE proceedings, Busteni (1983), Lecture Notes in Math. 1132 (Springer-Verlag, Berlin, 1985), 297–311. Google Scholar

[24] 24. MacLane, S., Homology (Springer-Verlag, Berlin, 1967). Google Scholar

[25] 25. Masuda, T., Groupoid dynamical systems and crossed products, I, II, the case of C*-systems, Publ. R.I.M.S. Kyoto Univ. 20 (1984), 959–970. Google Scholar

[26] 26. Olesen, D., Pedersen, G. K. and Takesaki, M., Ergodic actions of compact abelian groups, J. Operator Theory 3 (1980), 237–269. Google Scholar

[27] 27. Paschke, W., Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. Google Scholar

[28] 28. Pedersen, G. K. and Petersen, N. H., Ideals in a C*-algebra, Math. Scand. 27 (1970), 193–204. Google Scholar

[29] 29. Pedersen, G. K., C*-algebras and their automorphism groups (Academic Press, San Francisco, 1979). Google Scholar

[30] 30. Phillips, J. and Raeburn, I., Automorphisms of C*-algebras and second Cech cohomology, Indiana Univ. Math. J. 29 (1980), 799–822. Google Scholar

[31] 31. Raeburn, I., On the Picard group of a continuous trace C*-algebra, Trans. Amer. Math. Soc. 263 (1981), 183–205. Google Scholar

[32] 32. Raeburn, I. and Taylor, J., Continuous trace C*-algebras with given Dixmier-Douady class, J. Aust. Math. Soc. (Series A) 38 (1985), 394–407. Google Scholar

[33] 33. Raeburn, I. and Williams, D., Pull-backs of C*-algebras and crossed products by certain diagonal actions, Trans. Amer. Math. Soc. 287 (1985), 755–777. Google Scholar

[34] 34. Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Match. 793 (Springer-Verlag, Berlin, 1980). Google Scholar | DOI

[35] 35. Renault, J., C*-algebras of groupoids and foliations, Proc. of Symposia in Pure Math. 38 (1982), 339–350. Google Scholar

[36] 36. Renault, J., Two applications of the dual groupoid of a C*-algebra, OATE proceedings, Busteni (1983), Lecture Notes in Math. 1132 (Springer-Verlag, Berlin, 1985), 434–445. Google Scholar

[37] 37. Rieffel, M. A., Induced representations of C*-algebras, Advances Math. 13 (1974), 176–257. Google Scholar

[38] 38. Rieffel, M. A., Strong Morita equivalence of certain transformation group C*-algebras, Math. Ann. 222 (1976), 7–22. Google Scholar

[39] 39. Rieffel, M. A., C*-algebras associated with irrational rotations, Pacific J. Math. 95 (1981), 415–429. Google Scholar

[40] 40. Shultz, F., Pure states as a dual object for C*'-algebras, Comm. Math. Phys. 82 (1982), 497–507. Google Scholar

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

[42] 42. Strǎtilǎ, Ş and Voiculescu, D., Representations of AF algebras and of the group U(∞), Lecture notes in Math. 486 (Springer-Verlag, Berlin, 1975). Google Scholar | DOI

[43] 43. Tomiyama, J., On the projection of norm one in W*-algebras, Proc. Japan Acad. 33 (1957), 608–612. Google Scholar