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Extremally rich C*-crossed products and the cancellation property

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Ja A Jeong  and
Hiroyuki Osaka
Ja A Jeong
Affiliation:
Department of Mathematics Kyung Hee University Seoul, 130 - 701, Korea
Hiroyuki Osaka
Affiliation:
Mathematics Institute University of CopenhagenUniversitetsparken 5 DK-2100, Copenhagen ØDenmark and Department of Mathematics Sciences Ryukyu UniversityNishihara-cho, Okinawn 903 - 01, Japan
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Abstract

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A unital C*-algebra A is called extremally rich if the set of quasi-invertible elements A-1 ex (A)A-1 (= A-1q) is dense in A, where ex(A) is the set of extreme points in the closed unit ball A1 of A. In [7, 8] Brown and Pedersen introduced this notion and showed that A is extremally rich if and only if conv(ex(A)) = A1. Any unital simple C*-algebra with extremal richness is either purely infinite or has stable rank one (sr(A) = 1). In this note we investigate the extremal richness of C*-crossed products of extremally rich C*-algebras by finite groups. It is shown that if A is purely infinite simple and unital then A xα, G is extremally rich for any finite group G. But this is not true in general when G is an infinite discrete group. If A is simple with sr(A) =, and has the SP-property, then it is shown that any crossed product A xαG by a finite abelian group G has cancellation. Moreover if this crossed product has real rank zero, it has stable rank one and hence is extremally rich.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 3 , June 1998 , pp. 285 - 301
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Blackadar, B., ‘A simple unital projectionless C*-algebras’, J. Operator Theory 5 (1981), 63–71.Google Scholar
[2]Blackadar, B., K-theory for operator algebras (Springer, New York, 1986).CrossRefGoogle Scholar
[3]Blackadar, B., Comparison theory for simple C*-algebras, Operator algebras and applications, London Math. Soc. Lecture Note Ser. 135 (Cambridge University Press, Cambridge, 1988).Google Scholar
[4]Blackadar, B., ‘Symmetries on the CAR algebra’, Ann. of Math. 131 (1990), 589–623.CrossRefGoogle Scholar
[5]Blackadar, B. and Kumjian, A., ‘Skew products of relations and the structure of simple C*-algebras’, Math. Z. 189 (1985), 55–63.CrossRefGoogle Scholar
[6]Brown, L. G. and Pedersen, G. K., ‘C*-algebras of real rank zero’, J. Funct. Anal. 99 (1991), 131–149.CrossRefGoogle Scholar
[7]Brown, L. G. and Pedersen, G. K., ‘On the geometry of the unit ball of a C*-algebra’, J. Reine Angew. Math. 469 (1995), 113–147.Google Scholar
[8]Brown, L. G. and Pedersen, G. K., ‘On the geometry of the unit ball of a C*-algebra, Part II’, preprint, Copenhagen University Mathematics Institute 1993.Google Scholar
[9]Connes, A., ‘Periodic automorphism of the hyperfinite factor of type II1’, Acta Sci. Math. 39 (1977), 39–66.Google Scholar
[10]Cuntz, J., ‘The structure of multiplication and addition in simple C*-algebras’, Math. Scand. 40 (1977), 215–233.CrossRefGoogle Scholar
[11]Cuntz, J., ‘K-theory for certain C*-algebras’, Annals of Math. 113 (1981), 181–197.CrossRefGoogle Scholar
[12]Elliott, G. A., ‘A classification of certain simple C*-algebras’, in: Quantum and non-commutative analysis (eds. Araki, H. et al. ) (Kluwer Academic Publisher, Dordrecht, 1993) pp. 373–385.CrossRefGoogle Scholar
[13]Jeong, J. A., ‘Purely infinite simple crossed products’, Proc. Amer. Math. Soc. 123 (1995), 3075–3078.Google Scholar
[14]Jeong, J. A., Kodaka, K. and Osaka, H., ‘Purely infinite simple C*-crossed products II’, Canad. Math. Bull. 39 (1996), 203–210.CrossRefGoogle Scholar
[15]Kishimoto, A., ‘Outer automorphisms and related crossed products of simple C*-algebras’, Comm. Math. Phys. 81 (1981), 429–435.CrossRefGoogle Scholar
[16]Kishimoto, A. and Kumjian, A., ‘Crossed products of Cuntz algebras by quasi-free automorphisms’, in: Operator algebras and their applications, The Fields Institute Communications 13 (1997) pp. 173–192.Google Scholar
[17]Larsen, N. S. and Osaka, H., ‘Extremal richness of multiplier algebras and corona algebras of simple C*-algebras’, J. Operators Theory 38 (1997), 131–149.Google Scholar
[18]Lin, H., ‘Generalized Weyl-von Neumann theorems’, Internat. J. Math. 2 (1991), 725–739.CrossRefGoogle Scholar
[19]Lin, H. and Zhang, S., ‘Certain simple C*-algebras with non-zero real rank whose corona algebras have real rank zero’, Houston J. Math. 18 (1992), 57–71.Google Scholar
[20]Nagisa, M., Osaka, H. and Phillips, N. C., in preparation.Google Scholar
[21]Olsen, D., Pedersen, G. K. and Størmer, E., ‘Compact abelian groups of automorphisms of simple C*-algebras’, Invent. Math. 39 (1977), 55–64.CrossRefGoogle Scholar
[22]Pedersen, G. K., ‘The λ-function in operator algebras’, J. Operator Theory 26 (1991), 345–381.Google Scholar
[23]Rieffel, M. A., ‘Actions of finite groups on C*-algebras’, Math. Scand. 47 (1980), 157–176.CrossRefGoogle Scholar
[24]Rieffel, M. A., ‘Dimension and stable rank in the K-theory of C*-algebras’, Proc. London Math. Soc. 46 (1983), 301–333.CrossRefGoogle Scholar
[25]Rieffel, M. A., ‘The cancellation theorem for projective modules over irrational rotation C*-algebras’, Proc. London Math. Soc. 47 (1983), 285–302.CrossRefGoogle Scholar
[26]Rørdam, M., ‘On the structure of simple C*-algebras tensored with a UHF-algebra’, J. Funct. Anal. 100 (1991), 1–17.CrossRefGoogle Scholar
[27]Rosenberg, J., ‘Appendix to O. Bratteli's paper on crossed products of UHF-algebras’, Duke Math. J. 46 (1979), 25–26.CrossRefGoogle Scholar
[28]Wegge-Olsen, N. E., K-theory and C*-algebras (Oxford Science Publications, Oxford, 1993).CrossRefGoogle Scholar
[29]Zhang, S., ‘A property of purely infinite simple C*-algebras’, Proc. Amer. Math. Soc. 109 (1990), 717–720.Google Scholar