C*-Algebras With Real Rank Zero and The Internal Structure of Their Corona and Multiplier Algebras Part III

Zhang, Shuang

Canadian journal of mathematics, Tome 42 (1990) no. 1, pp. 159-190

Résumé

In this part, we shall be concerned with the structure of projections in a simple σ-unital C*-algebra with the FS property, and in the associated multiplier and corona algebras. We shall also consider the closed ideal structure of the corona algebra. Most of results appear to be new even for separable simple AF algebras, and are technically independent of the previous parts I and II ([37] and [38]). The whole work develops after finding a new property of a σ-unital (nonunital) simple C*-algebra with FS, which was not known even for a separable simple AF algebra. We relate this new property to the structure of the multiplier and corona algebras from vairous points of view.

Détail

DOI : 10.4153/CJM-1990-010-5

Bibliographie

[1] 1. Akemann, C.A., Pedersen, G.K. and Tomiyama, J., Multipliers of C*-algebras, J. Funct. Anal. 13 (1973), 277–301. Google Scholar

[2] 2. Bunce, J. and Deddens, J., A family of simple C*-algebras related to weighted shift operators, J. Funct. Anal. 19 (1975), 13–24. Google Scholar

[3] 3. Blackadar, B., K-theory for operator algebras (Springer-Verlag, New York, Berlin, Heidelberg, London, Paris Tokyo, 1987). Google Scholar

[4] 4. Blackadar, B., Notes on the structure of projections in simple algebras (Semester-bericht Funktionalanalysis, Tubingen, Wintersemester, 1982/83). Google Scholar

[5] 5. Blackadar, B. and Cuntz, J., The structure of stable algebraically simple C*-algebras, Amer. J. Math. 104 (1982), 813–822. Google Scholar

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

[7] 7. Brown, L.G., Stable isomorphism of hereditary subalgebras of C*-algebras, Pacific J. Math. 71 (1977), 335–348. Google Scholar

[8] 8. Brown, L.G., Semicontinuity and multipliers of C*-algebras, Can. J. Math. 40 (1989), 769–887. Google Scholar

[9] 9. Brown, L.G. and Pedersen, G.K., C*-algebras of real rank zero and their multiplier algebras, in preparation. Google Scholar

[10] 10. Busby, R., Double centralizers and extensions of C*-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99. Google Scholar

[11] 11. Choi, M-D., A simple C*-algebra generated by two finite-order unitaries, Can. J. Math. 31 (1979), 867–880. Google Scholar

[12] 12. Cuntz, J., Murray-von Neumann equivalence of projections in infinite simple C*-algebras, Rev. Roum. Math. Pures et Appl. 23 (1978), 1011–1014. Google Scholar

[13] 13. Cuntz, J., K-theory for certain C*-algebras, Ann. of Math. 131 (1981), 181-197, Google Scholar

[14] 14. Cuntz, J., Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185. Google Scholar

[15] 15. Cuntz, J. and Higson, N., Kuiper's theorem for Hilbert modules, in Operator algebras and mathematical physics, Amer. Math. Soc. 62, Proceedings of a Summer conference, June, 17-21 (1985). Google Scholar

[16] 16. Cuntz, J. and Krieger, W., A class of C*-algebras and topological Markov chains, Inventiones Math. 56 (1980), 251–268. Google Scholar

[17] 17. Dixmier, J., On some C*-algebras considered by Glimm, J. Funct. Anal. 1 (1967), 182–203. Google Scholar

[18] 18. Effros, E.G., Dimensions and C*-algebras, CBMS Regional Conference Series in Mathematics 46 (A.M.S., Providence, 1981). Google Scholar

[19] 19. Elliott, G.A., Derivations of matroid C*-algebras, II, Ann. of Math. 100 (1974), 407-22. Google Scholar

[20] 20. Elliott, G.A., The ideal structure of the multiplier algebra of an AF algebra, C.R. Math. Rep. Acad. Sci. Canada 9 (1987), 225–230. Google Scholar

[21] 21. Elliott, G.A., Automorphisms determined by multipliers on ideals of a C*-algebra, J. Funct. Anal. 23 (1976), 1–10. Google Scholar

[22] 22. Elliott, G.A. and Choi, M-D., Density of the self-adjoint elements with finite spectrum in an irrational rotation C*-algebra, preprint. Google Scholar

[23] 23. Lin, H., Fundamental approximate identities and quasi-multipliers of simple AF C*-algebras, J. Funct. Anal. 79 (1988), 32–43. Google Scholar

[24] 24. Lin, H., On ideals of multiplier algebras of simple AF C*-algebras, Proc. Amer. Math. Soc. 104 (1988), 239–244. Google Scholar

[25] 25. Mingo, J.A., K-theory and multipliers of stable C*-algebras, Trans. Amer. Math. Soc. 299 (1987), 225–230. Google Scholar

[26] 26. Pedersen, G.K., The linear span of projections in simple C*-algebras, J. Operator Theory 4 (1980), 289–296. Google Scholar

[27] 27. Pedersen, G.K., SAW*-algebras and corona C*-algebras, contributions to non-commutative topology, J. Operator Theory 75 (1986), 15–32. Google Scholar

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

[29] 29. Olsen, C.L. and Pedersen, G.K., Corona C*-algebras and their applications to lifting problems, preprint. Google Scholar

[30] 30. Voiculescu, D., A non-commutative Weyl-von Neumann theorem, Rev. Roum. Math. Pour et Appl. 21 (1976), 97–113. Google Scholar

[31] 31. Weyl, H., Uber beschrankte quadratischen formen deren different! vollstetigist, Rend. Circ. Mat. Palermo 27 (1909), 373–392. Google Scholar

[32] 32. Shuang, Zhang, Trivial K -flow of AF algebras and finite von Neumann algebras, J. Funct. Anal., to appear. Google Scholar

[33] 33. Shuang, Zhang, K -groups, quasidiagonality and interpolation by multiplier projections, Trans. Amer. Math., to appear. Google Scholar

[34] 34. Zhang, Shuang,On the structure of projections and ideals of corona algebras, Can. J. Math., to appear. Google Scholar

[35] 35. Zhang, Shuang,Diagonalizing projections in multiplier algebras and matrices over a C*-algebra, Pacific J. Math., revised. Google Scholar

[36] 36. Zhang, Shuang, A Riesz decomposition property and ideal structure of multiplier algebras, submitted. Google Scholar

[37] 37. Zhang, Shuang, C* -algebras with real rank zero and the internal structure of their corona and multiplier algebras, Part I, submitted. Google Scholar

[38] 38. Zhang, Shuang, C* -algebras with real rank zero and the internal structure of their corona and multiplier algebras, Part II, submitted. Google Scholar

[39] 39. Zhang, Shuang, Stable isomorphism of hereditary C*-subalgebras and stable equivalence of open projections, Proc. Amer. Math. Soc. 105 (1989), 677–682. Google Scholar

[40] 40. Zhang, Shuang, C* -algebras with real rank zero and the internal structure of their corona and multiplier algebras, Part IV, submitted. Google Scholar

[41] 41. Zhang, Shuang, A property of purely infinite simple C*-algebras, Proc. Amer. Math. Soc, to appear. Google Scholar