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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Approximate innerness of positive linear maps of finite von Neumann algebras


Author: Hideo Takemoto
Journal: Proc. Amer. Math. Soc. 94 (1985), 463-466
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1985-0787895-9
MathSciNet review: 787895
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Abstract: Let $ M$ be a $ \sigma $-finite, finite von Neumann algebra with a faithful, normalized, normal trace Tr and a positive linear map $ \rho $ of $ M$ into itself. If $ \rho $ is approximately inner with respect to the norm $ \vert\vert.\vert{\vert _2}$ induced by Tr, then $ \rho $ is closely related to a $ * $-homomorphism. In particular, if $ \rho $ is unital and approximately inner, then $ \rho $ is a $ * $-homomorphism of $ M$ into itself.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0787895-9
Keywords: Approximate innerness, trace, completely positive map, finite von Neumann algebra
Article copyright: © Copyright 1985 American Mathematical Society