Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Simple real rank zero algebras with locally Hausdorff spectrum

Author(s): Ping Wong Ng
Journal: Proc. Amer. Math. Soc. 134 (2006), 2223-2228.
MSC (2000): Primary 46L35
Posted: March 14, 2006
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: Let $ \mathcal{A}$ be a unital, simple, separable $ C^*$-algebra with real rank zero, stable rank one, and weakly unperforated ordered $ K_0$ group. Suppose, also, that $ \mathcal{A}$ can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then $ \mathcal{A}$ is tracially approximately finite dimensional (i.e., $ \mathcal{A}$ has tracial rank zero).

Hence, $ \mathcal{A}$ is an $ AH$-algebra with bounded dimension growth and is determined by $ K$-theoretic invariants.

The above result also gives the first proof for the locally $ AH$ case.

References:

1.
N. Brown, Invariant means and finite representation theory of $ C^*$-algebras, preprint. Available online at http://arxiv.org/abs/math.OA/0304009.

2.
M. Dadarlat, Some remarks on the universal coefficient theorem in KK-theory, Constanta, (2003), 65-74. MR 2018224 (2004j:19005)

3.
M. Dadarlat and S. Eilers, Approximate homogeneity is not a local property, J. Reine Angew. Math., 507, (1999), 1-13. MR 1670254 (2000b:46094)

4.
M. Dadarlat and S. Eilers, On the classification of nuclear $ C^*$-algebras, Proc. London Math. Soc. (3), 85, (2002), no. 1, 168-210. MR 1901373 (2003d:19006)

5.
M. Dadarlat and G. Gong, A classification result for approximately homogeneous $ C^*$-algebras of real rank zero, Geom. Funct. Anal., 7, (1997), no. 4, 646-711. MR 1465599 (98j:46062)

6.
J. Dixmier, $ C^*$-algebras, Paris, 1964; North Holland, Amsterdam/New York/Oxford.

7.
G. Elliott and G. Gong, On the classification of $ C^*$-algebras of real rank zero II, Ann. of Math. (2), 144, (1996), no. 3, 497-610. MR 1426886 (98j:46055)

8.
G. Elliott, G. Gong, L. Li, On the classification of simple inductive limit $ C^*$-algebras, II: the isomorphism theorem, preprint.

9.
J. M. G. Fell, The structure of algebras of operator fields, Acta Math., 106, (1961), 233-280. MR 0164248 (29:1547)

10.
E. Kirchberg and W. Winter, Covering dimension and quasidiagonality, Internat. J. Math., 15, (2004), 63-85. MR 2039212 (2005a:46148)

11.
H. Lin, Homomorphisms from $ C^*$-algebras of continuous trace, Math. Scand., 86, (2000), no. 2, 249-272. MR 1754997 (2001b:46088)

12.
H. Lin, The tracial topological rank of $ C^*$-algebras, Proc. London Math. Soc. (3), 83, (2001), no. 1, 199-234. MR 1829565 (2002e:46063)

13.
H. Lin, Classification of simple $ C^*$-algebras of tracial topological rank zero, Duke Math. J., 125, (2004), 91-119. MR 2097358

14.
H. Lin, Classification of simple $ C^*$-algebras with unique traces, Amer. J. Math., 120, (1998), no. 6, 1289-1315. MR 1657178 (2000c:46106)

15.
H. Lin, Locally type I simple TAF $ C^*$-algebras, preprint.

16.
H. Lin, Simple AH-algebras of real rank zero, preprint. To appear in the Proceedings of the AMS.

17.
H. Lin, Traces and simple $ C^*$-algebras with tracial topological rank zero. J. Reine Angew. Math., 568, (2004), 99-137. MR 2034925 (2005b:46122)

18.
I. Raeburn and D. Williams, Morita equivalence and continuous trace $ C^*$-algebras, Mathematical Surveys and Monographs, 60, AMS, Providence, RI, 1998. MR 1634408 (2000c:46108)

19.
N. B. Vasil'ev, $ C^*$-algebras with finite-dimensional irreducible representations, Uspehi Mat. Nauk, 21, (1966), no. 1, (127), 135-154. MR 0201994 (34:1871)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L35

Retrieve articles in all Journals with MSC (2000): 46L35

Additional Information:

Ping Wong Ng
Affiliation: Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Address at time of publication: The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Email: pwn@erdos.math.unb.ca

DOI: 10.1090/S0002-9939-06-07916-0
PII: S 0002-9939(06)07916-0
Received by editor(s): November 21, 2003
Received by editor(s) in revised form: June 23, 2004
Posted: March 14, 2006
Communicated by: David R. Larson
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google