Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T17:54:39.588Z Has data issue: false hasContentIssue false
  • English
  • Français

On C*-Diagonals

Published online by Cambridge University Press:  20 November 2018

Alexander Kumjian
Alexander Kumjian*
Affiliation:
University of New South Wales, Kensington, Australia
Rights & Permissions [Opens in a new window]

Extract

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.

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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 4 , 01 August 1986 , pp. 969 - 1008
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Idrissi, A. Alami, Sur le théorème de Riesz dans les algebres stellaires, Thèse de 3 ème cycle, Paris VI (1979).Google Scholar
2. Anderson, J., Extensions, restrictions, and representations of states on C*-algebras, Trans. Amer. Math. Soc. 227 (1977), 63–107.Google Scholar
3. Archbold, R. J., Extensions of states of C*-algebras, J. London Math. Soc. 21 (1980), 351–354 Google Scholar
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. 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. 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. 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. 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. Dixmier, J., C*-algebras (North-Holland, Amsterdam, 1977).Google Scholar
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. Elliott, G. A., On totally ordered groups, and K0 , Ring theory Waterloo (1978), Lecture notes in Math. 734 (Springer-Verlag, Berlin, 1979).Google Scholar
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. 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. Greenberg, M., Lectures on algebraic topology (Benjamin, New York, 1967).Google Scholar
15. Husemoller, D., Fibre bundles (McGraw-Hill, New York, 1966).CrossRefGoogle Scholar
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. 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. Kasparov, G. G., Hilbert C*-modules: Theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), 133–150.Google Scholar
19. Kelley, J. L., General topology (Van Nostrand, Princeton, 1955).Google Scholar
20. Kumjian, A., On localizations and simple C*-algebras, Pacific J. Math. 772 (1984), 141–192.Google Scholar
21. Kumjian, A., Preliminary algebras arising from local homeomorphisms, Math. Scand. 52 (1983), 269–278.Google Scholar
22. Kumjian, A., On C*-diagonals and twisted relations, Semesterbericht Funktionalanalysis, Universität Tübingen, Wintersemester (1982/83).Google Scholar
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. MacLane, S., Homology (Springer-Verlag, Berlin, 1967).Google Scholar
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. Olesen, D., Pedersen, G. K. and Takesaki, M., Ergodic actions of compact abelian groups, J. Operator Theory 3 (1980), 237–269.Google Scholar
27. Paschke, W., Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468.Google Scholar
28. Pedersen, G. K. and Petersen, N. H., Ideals in a C*-algebra, Math. Scand. 27 (1970), 193–204.Google Scholar
29. Pedersen, G. K., C*-algebras and their automorphism groups (Academic Press, San Francisco, 1979).Google Scholar
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. Raeburn, I., On the Picard group of a continuous trace C*-algebra, Trans. Amer. Math. Soc. 263 (1981), 183–205.Google Scholar
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. 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. Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Match. 793 (Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar
35. Renault, J., C*-algebras of groupoids and foliations, Proc. of Symposia in Pure Math. 38 (1982), 339–350.Google Scholar
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. Rieffel, M. A., Induced representations of C*-algebras, Advances Math. 13 (1974), 176–257.Google Scholar
38. Rieffel, M. A., Strong Morita equivalence of certain transformation group C*-algebras, Math. Ann. 222 (1976), 7–22.Google Scholar
39. Rieffel, M. A., C*-algebras associated with irrational rotations, Pacific J. Math. 95 (1981), 415–429.Google Scholar
40. Shultz, F., Pure states as a dual object for C*'-algebras, Comm. Math. Phys. 82 (1982), 497–507.Google Scholar
41. Slawny, J., On factor representations and the C*-algebra of canonical commutation relations, Comm. Math. Phys. 24 (1972), 151–170.Google Scholar
42. Strǎtilǎ, Ş and Voiculescu, D., Representations of AF algebras and of the group U(∞), Lecture notes in Math. 486 (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar
43. Tomiyama, J., On the projection of norm one in W*-algebras, Proc. Japan Acad. 33 (1957), 608–612.Google Scholar