Simple -algebras of real rank zero

Author:
Huaxin Lin

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3813-3819

MSC (2000):
Primary 46L05, 46L35

DOI:
https://doi.org/10.1090/S0002-9939-03-06995-8

Published electronically:
March 25, 2003

MathSciNet review:
1999928

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a unital simple -algebra with real rank zero. It is shown that if satisfies a so-called fundamental comparison property, then has tracial topological rank zero. Combining some previous results, it is shown that a unital simple -algebra with real rank zero, stable rank one and weakly unperforated must have slow dimension growth.

**1.**B. Blackadar,*-theory for Operator Algebras*, MSRI Monographs, vol. 5, Springer-Verlag, Berlin and New York, 1986. MR**88g:46082****2.**B. Blackadar, M. Dadarlat and M. Rørdam,*The real rank of inductive limit**-algebras*, Math. Scand.**69**, (1992), 211-216. MR**93e:46067****3.**B. Blackadar and D. Handelman,*Dimension functions and traces on**-algebras*, J. Funct. Anal.**45**(1982), 297-340. MR**83g:46050****4.**B. Blackadar, A. Kumjian and M. Rørdam,*Approximately central matrix units and the structure of non-commutative tori*, -theory**6**(1992), 267-284. MR**93i:46129****5.**L. G. Brown,*Interpolation by projections in**-algebras of real rank zero*, J. Operator Theory**26**(1991), 383-387. MR**94j:46054****6.**M. Dadarlat,*Reduction to dimension three of local spectra of real rank zero**-algebras*, J. Reine Angew. Math.**460**(1995), 189-212. MR**95m:46116****7.**G. A. Elliott and G. Gong,*On the classification of**-algebras of real rank zero, II*, Ann. Math.**144**(1996), 497-610. MR**98j:46055****8.**G. A. Elliott, G. Gong, H. Lin and C. Pasnicu,*Abelian**-subalgebras of**-algebras of real rank zero and inductive limit**-algebras*, Duke Math. J.**85**(1996), 511-554. MR**98a:46076****9.**G. Gong,*On the inductive limits of matrix algebras over higher dimensional spaces, Part I & II*, Math. Scand.**80**(1997) 40-55 & 56-100. MR**98j:46061****10.**K. Goodearl,*Notes on a class of simple**-algebras with real rank zero*, Publications Math.**36**(1992), 637-654. MR**94f:46092****11.**D. Husemoller,*Fibre Bundles*, McGraw-Hill, New York, 1966, reprinted in Springer-Verlag Graduate Texts in Mathematics. MR**94k:55001****12.**H. Lin,*Homomorphisms from**-algebras of continuous trace*, Math. Scand.**86**(2000), 249-272. MR**2001b:46088****13.**H. Lin,*Tracially AF**-algebra*, Trans. Amer. Math. Soc.**353**(2001), 693-722. MR**2001j:46089****14.**H. Lin,*Classification of simple tracially AF C*-Algebras*Canad. J. Math.**53**(2001), 161-194. MR**2002h:46102****15.**H. Lin,*The tracial topological rank of**-algebras*, Proc. London Math. Soc.**83**(2001), 199-234. MR**2002e:46063****16.**H. Lin,*Classification of simple**-algebras and higher dimensional non-commutative tori*, Ann. of Math.**157**(2003), to appear.**17.**H. Lin,*Classification of simple**-algebras of tracial topological rank zero*, MSRI preprint.**18.**J. Villadsen,*Simple**-algebras with perforation*, J. Funct. Anal.**154**(1998), 110-116. MR**99j:46069****19.**S. Zhang,*A Riesz decomposition property and ideal structure of multiplier algebras*, J. Operator Theory**24**(1990), 209-225. MR**93b:46116**

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Additional Information

**Huaxin Lin**

Affiliation:
Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China

Address at time of publication:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Email:
hxlin@noether.uoregon.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-06995-8

Keywords:
$AH$-algebras,
tracial topological rank zero

Received by editor(s):
May 7, 2001

Received by editor(s) in revised form:
July 16, 2002

Published electronically:
March 25, 2003

Additional Notes:
This research was partially supported by NSF grant DMS 009790

Communicated by:
David R. Larson

Article copyright:
© Copyright 2003
American Mathematical Society