Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-02-05T15:05:44.682Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction of Exponential Rank in Direct Limits of C*-Algebras

Published online by Cambridge University Press:  20 November 2018

N. Christopher Phillips
N. Christopher Phillips*
Affiliation:
Department of Mathematics, University of Oregon Eugene, Oregon 97403-1222U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

We prove the following result. Let A be a direct limit of direct sums of C*-algebras of the form C(X) ⊗ Mn, with the spaces X being compact metric. Suppose that there is a finite upper bound on the dimensions of the spaces involved, and suppose that A is simple. Then the C* exponential rank of A is at most 1 + ε, that is, every element of the identity component of the unitary group of A is a limit of exponentials. This is true regardless of whether the real rank of A is 0 or 1.

Keywords

46L05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 4 , 01 August 1994 , pp. 818 - 853
Copyright
Copyright © Canadian Mathematical Society 1994

References

Beggs, E. J. and Evans, D. E., The real rank of algebras of matrix valued functions, Internat. J. Math. 2(1991), 131–138.Google Scholar
2. Bhatia, R., Perturbation Bounds for Matrix Eigenvalues, Pitman Research Notes in Math. 162, Longman Scientific and Technical, Harlow, Britain, 1987.Google Scholar
3. Blackadar, B., Bratteli, O., Elliott, G. A. and Kumjian, A., Reduction of real rank in inductive limits of C* -algebras, Math. Ann. 292(1992), 111–126.Google Scholar
4. Blackadar, B., D∅rlat, M. and Rłrdam, M., The real rank of inductive limit C* -algebras, Math. Scand. 69(1991),211–216.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. Brown, L. G., Stable isomorphism of hereditary subalgebras of C* -algebras, Pacific J. Math. 71(1977), 335–348.Google Scholar
7. Choi, M.-D., Lifting projections from quotient C* -algebras, J. Operator Theory 10(1983), 21–30.Google Scholar
8. D∅rlat, M., Nagy, G., Némethi, A. and Pasnicu, C., Reduction of stable rank in inductive limits ofC* -algebras, Pacific J. Math. 153(1992), 267–276.Google Scholar
9. Elliott, G. A. and Evans, D. E., The structure of the irrational rotation C* -algebra, Ann. of Math. 138(1993), 477–501.Google Scholar
10. Engelking, R., Dimension Theory, North-Holland, Amsterdam, Oxford, New York, 1978.Google Scholar
11. Gong, G. and Lin, H., The exponential rank of inductive limit C* -algebras, Math. Scand. 71(1992), 301–319.Google Scholar
12. Goodearl, K. R., Notes on a class of simple C* -algebras with real rank 0, Publ. Sec. Mat. Univ. Autónoma Barcelona 36(1992), 637–654.Google Scholar
13. Goodearl, K. R., Riesz decomposition in inductive limit C* -algebras, Rocky Mountain J. Math., to appear.Google Scholar
14. Guillemin, V. and Pollack, A., Differential Topology, Prentice-Hall, Englewood Cliffs, NJ, 1974.Google Scholar
15. Husemoller, D., Fibre Bundles, McGraw-Hill, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1966.Google Scholar
16. Lin, H., Exponential rank of C* -algebras with real rank 0 and Brown-Pedersen's conjectures, J. Funct. Anal. 114(1993), 1–11.Google Scholar
17. Lin, H. and Rørdam, M., Extensions of inductive limits of circle algebras, J. London Math. Soc, to appear.Google Scholar
18. Mumford, D., Algebraic Geometry I: Complex Projective Varieties, Grundlehren der mathematischen Wissenschaften, 221, Springer-Verlag, Berlin, Heidelberg, New York, 1976.Google Scholar
19. Paschke, W., K-theoryfor actions of the circle group on C* -algebras, J. Operator Theory 6(1981), 125–133.Google Scholar
20. Phillips, N. C., Simple C* -algebras with the property weak (FU), Math. Scand. 69(1991), 127–151.Google Scholar
21. Phillips, N. C., Approximation by unitaries with finite spectrum in purely infinite C* -algebras, J. Funct. Anal. 120(1994), 98–106.Google Scholar
22. Phillips, N. C., How many exponentials?, Amer. J. Math., to appear.Google Scholar
23. Phillips, N. C., The rectifiable metric on the space of projections in a C* -algebra, Internat. J. Math. 3(1992), 679–698.Google Scholar
24. Phillips, N. C., The C* projective length of n-homogeneous C* -algebras, J. Operator Theory, to appear.Google Scholar
25. Phillips, N. C., Exponential length and traces, Proc. Roy. Soc. Edinburgh Sect. A, to appear.Google Scholar
26. Rieffel, M. A., Dimension and stable rank in the K-theory of C* -algebras, Proc. London Math. Soc. (3) 46(1983), 301–333.Google Scholar
27. Rieffel, M. A., The homotopy groups of the unitary groups of noncommutative tori, J. Operator Theory 17(1987), 237ߝ254.Google Scholar
28. Ringrose, J. R., Exponential length and exponential rank in C* -algebras, Proc. Royal Soc. Edinburgh Sect. A 121(1992), 55–71.Google Scholar
29. Thomsen, K., Finite sums and products of commutators in inductive limit C* -algebras, Ann. Inst. Fourier (1)43(1993), 225–249.Google Scholar
30. van der Waerden, B. L., Algebra (vol. 1), Trans, from 7th edition, by F. Blum and J. R. Schulenberger, Frederick Ungar, New York, 1970.Google Scholar
31. Zhang, S., Rectifiable diameters of the Grassman spaces of von Neumann algebras and certain C* -algebras, preprint.Google Scholar