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

 

Positive combinations and sums of projections in purely infinite simple $ C^*$-algebras and their multiplier algebras


Authors: Victor Kaftal, P. W. Ng and Shuang Zhang
Journal: Proc. Amer. Math. Soc. 139 (2011), 2735-2746
MSC (2010): Primary 46L05; Secondary 47C15
Published electronically: March 24, 2011
MathSciNet review: 2801613
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Abstract: Every positive element in a purely infinite simple $ \sigma$-unital $ C^*$-algebra $ \mathscr{A}$ is a finite linear combination of projections with positive coefficients. Also, every positive $ a$ in the multiplier algebra $ \mathscr M(\mathscr{A})$ of a purely infinite simple $ \sigma$-unital $ C^*$-algebra $ \mathscr{A}$ is a finite linear combination of projections with positive coefficients. Furthermore, if the essential norm $ \Vert a\Vert _{ess} > 1$, then $ a$ is a finite sum of projections in $ \mathscr M(\mathscr{A})$. As a consequence, any positive element in the generalized Calkin Algebra $ \mathscr M(\mathscr{A})/\mathscr{A}$ or in $ \mathscr M(\mathscr{A})$ but not in $ \mathscr{A}$ is a positive scalar multiple of a finite sum of projections.


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Additional Information

Victor Kaftal
Affiliation: Department of Mathematics, University of Cincinnati, P. O. Box 210025, Cincinnati, Ohio 45221-0025
Email: victor.kaftal@math.uc.edu

P. W. Ng
Affiliation: Department of Mathematics, University of Louisiana, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, Louisiana 70504-1010
Email: png@louisiana.edu

Shuang Zhang
Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025
Email: zhangs@math.uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10995-X
PII: S 0002-9939(2011)10995-X
Received by editor(s): July 1, 2010
Published electronically: March 24, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.