A simple separable C*-algebra not isomorphic to its opposite algebra
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Abstract:
We give an example of a simple separable C*-algebra that is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial $K_1$, and its $K_0$-group is order isomorphic to a countable subgroup of ${\mathbf R}$.References
- Charles A. Akemann and Gert K. Pedersen, Central sequences and inner derivations of separable $C^{\ast }$-algebras, Amer. J. Math. 101 (1979), no. 5, 1047–1061. MR 546302, DOI 10.2307/2374125
- Bruce E. Blackadar, Weak expectations and nuclear $C^{\ast }$-algebras, Indiana Univ. Math. J. 27 (1978), no. 6, 1021–1026. MR 511256, DOI 10.1512/iumj.1978.27.27070
- Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
- A. Connes, A factor not anti-isomorphic to itself, Ann. of Math. (2) 101 (1975), 536–554. MR 370209, DOI 10.2307/1970940
- Alain Connes, Sur la classification des facteurs de type $\textrm {II}$, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 1, Aii, A13–A15 (French, with English summary). MR 377534
- Kenneth J. Dykema and Mikael Rørdam, Purely infinite, simple $C^*$-algebras arising from free product constructions, Canad. J. Math. 50 (1998), no. 2, 323–341. MR 1618318, DOI 10.4153/CJM-1998-017-x
- G. A. Elliott, G. Gong, and L. Li, On the classification of simple inductive limit C*-algebras, II: The isomorphism theorem, preprint.
- L. Ge and N. C. Phillips, Nonisomorphic simple exact C*-algebras with the same Elliott and Haagerup invariants, preprint.
- K. R. Goodearl, Notes on real and complex $C^{\ast }$-algebras, Shiva Mathematics Series, vol. 5, Shiva Publishing Ltd., Nantwich, 1982. MR 677280
- E. Kirchberg, The classification of purely infinite C*-algebras using Kasparov’s theory, preliminary preprint (3rd draft).
- H. Lin, Classification of simple C*-algebras with tracial topological rank zero, preprint.
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
- N. Christopher Phillips, Continuous-trace $C^*$-algebras not isomorphic to their opposite algebras, Internat. J. Math. 12 (2001), no. 3, 263–275. MR 1841515, DOI 10.1142/S0129167X01000642
- N. C. Phillips, Examples of nonisomorphic nonnuclear simple stably finite C*-algebras with the same Elliott invariants, in preparation.
- Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
- Şerban Strătilă and László Zsidó, Lectures on von Neumann algebras, Editura Academiei, Bucharest; Abacus Press, Tunbridge Wells, 1979. Revision of the 1975 original; Translated from the Romanian by Silviu Teleman. MR 526399
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
Additional Information
- N. Christopher Phillips
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- Received by editor(s): July 25, 2002
- Received by editor(s) in revised form: February 21, 2003
- Published electronically: June 2, 2004
- Additional Notes: Research partially supported by NSF grant DMS 0070776.
- Communicated by: David R. Larson
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2997-3005
- MSC (2000): Primary 46L35
- DOI: https://doi.org/10.1090/S0002-9939-04-07330-7
- MathSciNet review: 2063121