Simple corona $C^*$-algebras
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- by Huaxin Lin
- Proc. Amer. Math. Soc. 132 (2004), 3215-3224
- DOI: https://doi.org/10.1090/S0002-9939-04-07607-5
- Published electronically: June 16, 2004
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Abstract:
Let $A$ be a non-unital and $\sigma$-unital simple $C^*$-algebra. We show that if $M(A)/A$ is simple, then $M(A)/A$ is purely infinite. We also show that $M(A)/A$ is simple if and only if $A$ has a continuous scale provided that $A$ is not isomorphic to ${\mathcal K},$ the compact operators.References
- Charles A. Akemann and Gert K. Pedersen, Ideal perturbations of elements in $C^*$-algebras, Math. Scand. 41 (1977), no. 1, 117–139. MR 473848, DOI 10.7146/math.scand.a-11707
- Charles A. Akemann and Frederic W. Shultz, Perfect $C^\ast$-algebras, Mem. Amer. Math. Soc. 55 (1985), no. 326, xiii+117. MR 787540, DOI 10.1090/memo/0326
- Bruce Blackadar and David Handelman, Dimension functions and traces on $C^{\ast }$-algebras, J. Functional Analysis 45 (1982), no. 3, 297–340. MR 650185, DOI 10.1016/0022-1236(82)90009-X
- Joachim Cuntz, The structure of multiplication and addition in simple $C^*$-algebras, Math. Scand. 40 (1977), no. 2, 215–233. MR 500176, DOI 10.7146/math.scand.a-11691
- Joachim Cuntz, Dimension functions on simple $C^*$-algebras, Math. Ann. 233 (1978), no. 2, 145–153. MR 467332, DOI 10.1007/BF01421922
- Joachim Cuntz, The internal structure of simple $C^{\ast }$-algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 85–115. MR 679697
- George A. Elliott, Derivations of matroid $C^{\ast }$-algebras. II, Ann. of Math. (2) 100 (1974), 407–422. MR 352999, DOI 10.2307/1971079
- Hua Xin Lin, Simple $C^*$-algebras with continuous scales and simple corona algebras, Proc. Amer. Math. Soc. 112 (1991), no. 3, 871–880. MR 1079711, DOI 10.1090/S0002-9939-1991-1079711-8
- Huaxin Lin, Extensions of $C(X)$ by simple $C^*$-algebras of real rank zero, Amer. J. Math. 119 (1997), no. 6, 1263–1289. MR 1481815
- Huaxin Lin, An introduction to the classification of amenable $C^*$-algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1884366, DOI 10.1142/9789812799883
- H. Lin, Extensions by simple $C^*$-algebras — quaisdiagonal extensions, preprint.
- Hua Xin Lin and Shuang Zhang, On infinite simple $C^*$-algebras, J. Funct. Anal. 100 (1991), no. 1, 221–231. MR 1124300, DOI 10.1016/0022-1236(91)90109-I
- 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
- Mikael Rørdam, Ideals in the multiplier algebra of a stable $C^\ast$-algebra, J. Operator Theory 25 (1991), no. 2, 283–298. MR 1203034
- Mikael Rørdam, On the structure of simple $C^*$-algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (1992), no. 2, 255–269. MR 1172023, DOI 10.1016/0022-1236(92)90106-S
- Mikael Rørdam, A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. MR 2020420, DOI 10.1007/BF02392697
- Shuang Zhang, On the structure of projections and ideals of corona algebras, Canad. J. Math. 41 (1989), no. 4, 721–742. MR 1012625, DOI 10.4153/CJM-1989-033-4
- Shuang Zhang, A property of purely infinite simple $C^*$-algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 717–720. MR 1010004, DOI 10.1090/S0002-9939-1990-1010004-X
- Shuang Zhang, $C^*$-algebras with real rank zero and the internal structure of their corona and multiplier algebras. III, Canad. J. Math. 42 (1990), no. 1, 159–190. MR 1043517, DOI 10.4153/CJM-1990-010-5
- Shuang Zhang, $C^*$-algebras with real rank zero and their corona and multiplier algebras. IV, Internat. J. Math. 3 (1992), no. 2, 309–330. MR 1146816, DOI 10.1142/S0129167X92000102
Bibliographic Information
- Huaxin Lin
- Affiliation: Department of Mathematics, East China Normal University, Shanghai, China
- Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- Received by editor(s): February 1, 2003
- Published electronically: June 16, 2004
- Additional Notes: This research was partially supported by NSF grant DMS 0097003
- Communicated by: David R. Larson
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3215-3224
- MSC (2000): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-04-07607-5
- MathSciNet review: 2073295