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

 

Representing conditional expectations as elementary operators


Author: Rajesh Pereira
Journal: Proc. Amer. Math. Soc. 134 (2006), 253-258
MSC (2000): Primary 46L05, 47B47
Published electronically: June 14, 2005
MathSciNet review: 2170565
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Abstract: Let $\mathcal{A}$ be a $C^{*}$-algebra and let $\mathcal{B}$ be a $C^{*}$-subalgebra of $\mathcal{A}$. We call a linear operator from $\mathcal{A}$ to $\mathcal{B}$ an elementary conditional expectation if it is simultaneously an elementary operator and a conditional expectation of $\mathcal{A}$ onto $\mathcal{B}$. We give necessary and sufficient conditions for the existence of a faithful elementary conditional expectation of a prime unital $C^{*}$-algebra onto a subalgebra containing the identity element. We give a description of all faithful elementary conditional expectations. We then use these results to give necessary and sufficient conditions for certain conditional expectations to be index-finite (in the sense of Watatani) and we derive an inequality for the index.


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Additional Information

Rajesh Pereira
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: rjxpereira@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08031-7
PII: S 0002-9939(05)08031-7
Keywords: Prime $C^{*}$-algebras, conditional expectations, elementary operators, index finite type, minimal conditional expectation
Received by editor(s): September 1, 2004
Published electronically: June 14, 2005
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.