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On the Structure of Projections and Ideals of Corona Algebras

Published online by Cambridge University Press:  20 November 2018

Shuang Zhang
Shuang Zhang*
Affiliation:
University of Kansas, Lawrence, Kansas
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If K is the set of all compact bounded operators and L(H) is the set of all bounded operators on a separable Hilbert space H, then L(H) is the multiplier algebra of K. In general we denote the multiplier algebra of a C*-algebra A by M(A). For more information about M(A), readers are referred to the articles [1], [3],[7], [9], [14], [18],[20], [23], [26], [27], among others. It is well known that in the Calkin algebra L(H)/K every nonzero projection is infinite. If we assume that A is a-unital (nonunital) and regard the corona algebra M(A)/A as a generalized case of the Calkin algebra, is every nonzero projection in M(A)/A still infinite? Another basic question can be raised: How does the (closed) ideal structure of A relate to the (closed) ideal structure of M(A)/A?

In the first part of this note (Sections 1 and 2) we shall give an affirmative answer for the first question if A is a simple a-unital (nonunital) C*-algebra with FS.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 4 , 01 August 1989 , pp. 721 - 742
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Akemann, C. A., Pedersen, G. K. and Tomiyama, J., Multipliers of C*-algebras, J. Funct. Anal. 13 (1973), 277–301.Google Scholar
2. J, Bunce and Deddens, J., A family of simple C*-algebras related to weighted shift operators, J. Funct. Anal. 19 (1975), 13–24.Google Scholar
3. Blackadar, B., K-theory for operator algebras (Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo, 1987).Google Scholar
4. . Blackadar, B., Notes on the structure of projections in simple C*-algebras, Semesterbericht Funktionalanalysis, W82 (Tübingen, 1983).Google Scholar
5. Blackadar, B.and Kumjian, A., Skew products of relations and the structure of simple C*-algebras, Math. Z. 189 (1985), 55–63.Google Scholar
6. Bratteli, O., Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234.Google Scholar
7. Brown, L. G., Extensions of AF algebras: the projection lifting problem, Operator Algebras and Applications, Proc. Symp. Pure Math. 38 (A.M.S., Providence, 1981), Part I, 175–176.Google Scholar
8. Brown, L. G., Stable isomorphism of hereditary subalgebras of C* -algebras, Pacific J. Math. 71 (1977), 335-348. 9. Brown, L. G., Semicontinuity and multipliers of C*-algebras, Can. J. Math., to appear.Google Scholar
10. Brown, L. G., P. Green and Rieffel, M. A., Stable isomorphism and strong Morita equivalence of C*-algebras, Pacific J. Math. 71 (1977), 349–363.Google Scholar
11. Brown, L. G. and Pedersen, G., C*-algebras of real rank zero and their multiplier algebras. In preparation.Google Scholar
12. Busby, R., Double centralizers and extensions of C*-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99.Google Scholar
13. Cuntz, J., K-theoryfor certain C*-algebras, Ann. of Math. 113 (1981), 181–197.Google Scholar
14. Cuntz, J.and Higson, N., Kuiper's theorem for Hilbert modules, Operator Algebras and Mathematical Physics, Proceedings of a Summer Conference June 17–21, 1985, Contemporary Math. 62 (Amer. Math. Soc, 1987), 429–435.Google Scholar
15. Dixmier, J., C*-algebras (North-Holland, 1977).Google Scholar
16. Dixmier, J., On some C*-algebras considered by Glimm, J. Funct. Anal. 1 (1967), 182–203.Google Scholar
17. Effros, E. G., Dimensions and C*-algebras, CBMS Regional Conference Series in Mathematics 46 (A.M.S., Providence, 1981).Google Scholar
18. Elliott, G. A., Derivations of matroid C*-algebras II, Ann. of Math. 100 (1974), 407–422.Google Scholar
19. Elliott, G. A., Automorphisms determined by multipliers on ideals of a C*-algebra, J. Funct. Anal. 23 (1976), 1–10.Google Scholar
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. Elliott, G. A. and Olesen, D., A simple proof of the Dauns-Hofmann theorem, Math. Scand. 34 (1914), 231–234.Google Scholar
22. Elliott, G. A. and Handelman, D. E., Addition of C*-algebra extensions, Pacific J. Math., to appear.Google Scholar
23. Lin, H., On ideals of multiplier algebras of simple AF C*-algebras, Proc. Amer. Math. Soc. 104 (1988), 239–244.Google Scholar
24. Grove, K.and Pedersen, G. K., Sub-stonean spaces and corona sets, J. Funct. Anal. 56 (1984), 124–143.Google Scholar
25. Pedersen, G. K., The linear span of projections in simple C*-algebras, J. Operator Theory 4 (1980), 289–296.Google Scholar
26. Pedersen, G. K., SAW *-algebras and corona C*-algebras, contributions to noncommutative topology, J. Operator Theory 75 (1986), 15–32.Google Scholar
27. Zhang, Shuang, A Riesz decomposition property and ideal structure of multiplier algebras, preprint.Google Scholar
28. Shuang, Zhang, K1-Regroups, quasidiagonality and interpolation by multiplier projections, preprint.Google Scholar