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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Stable rank of corner rings


Authors: P. Ara and K. R. Goodearl
Journal: Proc. Amer. Math. Soc. 133 (2005), 379-386
MSC (2000): Primary 19B10; Secondary 16S50
Published electronically: September 20, 2004
MathSciNet review: 2093058
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Abstract: B. Blackadar recently proved that any full corner $pAp$ in a unital C*-algebra $A$ has K-theoretic stable rank greater than or equal to the stable rank of $A$. (Here $p$ is a projection in $A$, and fullness means that $ApA=A$.) This result is extended to arbitrary (unital) rings $A$ in the present paper: If $p$ is a full idempotent in $A$, then $\operatorname{sr} (pAp)\ge \operatorname{sr}(A)$. The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners $pAq$. The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if $B\cong \operatorname{End}_{A}(P)$ where $P_{A}$ is a finitely generated projective generator, and $P$ can be generated by $n$ elements, then $\operatorname{sr}(A)\le n{\cdot }\operatorname{sr}(B)-n+1$.


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Additional Information

P. Ara
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella terra (Barcelona), Spain
Email: para@mat.uab.es

K. R. Goodearl
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: goodearl@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07773-1
PII: S 0002-9939(04)07773-1
Keywords: Stable range, stable rank, corner ring, matrix ring
Received by editor(s): September 5, 2003
Published electronically: September 20, 2004
Additional Notes: The first-named author was partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM2002-01390, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The second-named author was partially supported by an NSF grant. Part of this work was done during his research stay at the Centre de Recerca Matemàtica (Barcelona) in Spring 2003; he thanks the CRM for its hospitality and support
Communicated by: Martin Lorenz
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.