Simplicity and the stable rank
of some free product C-algebras
Author:
Kenneth J. Dykema
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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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.
- 1.
J. Anderson, B. Blackadar and U. Haagerup, Minimal projections in the reduced group C
-algebra of
, J. Operator Theory 26 (1991), 3-23. MR 94c:46110
- 2.
D. Avitzour, Free products of C
-algebras, Trans. Amer. Math. Soc. 271 (1982), 423-465. MR 83h:46070
- 3.
L.G. Brown, Stable isomorphism of hereditary subalgebras of C
-algebras, Pacific J. Math. 71 (1977), 335-348. MR 56:12894
- 4.
L.G. Brown, P. Green, M.A. Rieffel, Stable isomorphism and strong Morita equivalence of C
-algebras, Pacific J. Math. 71 (1977), 349-363. MR 57:3866
- 5. K.J. Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1993), 97-119. MR 93m:46071
- 6. -, Free products of finite dimensional and other von Neumann algebras with respect to non-tracial states, Fields Institute Communications, (D. Voiculescu, editor), vol. 12, 1997, pp. 41-88. MR 98c:46131
- 7. -, Faithfulness of free product states, J. Funct. Anal. 154(1998), 323-329. CMP 98:11
- 8. -, Free Probability Theory and Operator Algebras, Seoul National University GARC lecture notes, in preparation.
- 9.
K.J. Dykema, U. Haagerup, M. Rørdam, The stable rank of some free product C
-algebras, Duke Math. J. 90 (1997), 95-121. CMP 98:03
- 10.
K.J. Dykema, M. Rørdam, Purely infinite simple
-algebras arising from free product constructions, Can. J. Math. 50 (1998), 323-341.
- 11.
E. Germain,
-theory of reduced free product C
-algebras, Duke Math. J. 82 (1996), 707-723. MR 97f:46111
- 12.
E. Germain,
-theory of the full free product of unital C
-algebras, J. reine angew. Math. 485 (1997), 1-10. MR 98b:46148
- 13.
R.H. Herman, L.N. Vaserstein, The stable range of C
-algebras, Invent. Math. 77 (1984), 553-555. MR 86a:46074
- 14.
W.L. Paschke, N. Salinas, C
-algebras associated with free products of groups, Pacific J. Math. 82 (1979), 211-221. MR 82c:22010
- 15.
R.T. Powers, Simplicity of the reduced C
-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151-156. MR 51:10534
- 16. M.A. Rieffel, Morita equivalence for operator algebras, Proc. Symp. Pure Math. 38 (1982), 285-298. MR 84k:46045
- 17.
-, Dimension and stable rank in the K-theory of C
-algebras, Proc. London Math. Soc. (3) 46 (1983), 301-333. MR 84g:46085
- 18. M. Rørdam, Advances in the theory of unitary rank and regular approximation, Ann. Math. 128 (1988), 153-172. MR 90c:46072
- 19.
D. Voiculescu, Symmetries of some reduced free product C
-algebras, Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics, Volume 1132, Springer-Verlag, 1985, pp. 556-588. MR 87d:46075
- 20. -, Multiplication of certain non-commuting random variables, J. Operator Theory 18 (1987), 223-235. MR 89b:46076
- 21. D. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series vol. 1, American Mathematical Society, 1992. MR 94c:46133
Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46L05, 46L35
Retrieve articles in all journals with MSC (1991): 46L05, 46L35
Additional Information
Kenneth J. Dykema
Affiliation:
Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK-5230 Odense M, Denmark
Email:
dykema@imada.ou.dk
DOI:
https://doi.org/10.1090/S0002-9947-99-02180-7
Received by editor(s):
January 21, 1997
Article copyright:
© Copyright 1999
American Mathematical Society