Simplicity and the stable rank of some free product C*-algebras
HTML articles powered by AMS MathViewer
- by Kenneth J. Dykema PDF
- Trans. Amer. Math. Soc. 351 (1999), 1-40 Request permission
Abstract:
A necessary and sufficient condition for the simplicity of the C$^{*}$–algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C$^{*}$–algebras are proved.References
- Joel Anderson, Bruce Blackadar, and Uffe Haagerup, Minimal projections in the reduced group $C^*$-algebra of $Z_n\ast Z_m$, J. Operator Theory 26 (1991), no. 1, 3–23. MR 1214917
- Daniel Avitzour, Free products of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 271 (1982), no. 2, 423–435. MR 654842, DOI 10.1090/S0002-9947-1982-0654842-1
- Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645, DOI 10.2140/pjm.1977.71.335
- Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928, DOI 10.2140/pjm.1977.71.349
- Ken Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1993), no. 1, 97–119. MR 1201693, DOI 10.1215/S0012-7094-93-06905-0
- Kenneth J. Dykema, Free products of finite-dimensional and other von Neumann algebras with respect to non-tracial states, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 41–88. MR 1426835
- —, Faithfulness of free product states, J. Funct. Anal. 154(1998), 323–329.
- —, Free Probability Theory and Operator Algebras, Seoul National University GARC lecture notes, in preparation.
- K.J. Dykema, U. Haagerup, M. Rørdam, The stable rank of some free product C$^{*}$–algebras, Duke Math. J. 90 (1997), 95–121.
- K.J. Dykema, M. Rørdam, Purely infinite simple $C^{*}$-algebras arising from free product constructions, Can. J. Math. 50 (1998), 323–341.
- Emmanuel Germain, $KK$-theory of reduced free-product $C^*$-algebras, Duke Math. J. 82 (1996), no. 3, 707–723. MR 1387690, DOI 10.1215/S0012-7094-96-08229-0
- Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
- Richard H. Herman and Leonid N. Vaserstein, The stable range of $C^{\ast }$-algebras, Invent. Math. 77 (1984), no. 3, 553–555. MR 759256, DOI 10.1007/BF01388839
- William L. Paschke and Norberto Salinas, $C^{\ast }$-algebras associated with free products of groups, Pacific J. Math. 82 (1979), no. 1, 211–221. MR 549845, DOI 10.2140/pjm.1979.82.211
- Robert T. Powers, Simplicity of the $C^{\ast }$-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151–156. MR 374334
- Marc A. Rieffel, Morita equivalence for operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 285–298. MR 679708
- 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
- Mikael Rørdam, Advances in the theory of unitary rank and regular approximation, Ann. of Math. (2) 128 (1988), no. 1, 153–172. MR 951510, DOI 10.2307/1971465
- Dan Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR 799593, DOI 10.1007/BFb0074909
- Dan Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), no. 2, 223–235. MR 915507
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
Additional Information
- Kenneth J. Dykema
- Affiliation: Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK-5230 Odense M, Denmark
- MR Author ID: 332369
- Email: dykema@imada.ou.dk
- Received by editor(s): January 21, 1997
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 1-40
- MSC (1991): Primary 46L05, 46L35
- DOI: https://doi.org/10.1090/S0002-9947-99-02180-7
- MathSciNet review: 1473439