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Proceedings of the American Mathematical Society

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

 

 

Quasi-expectations and amenable von Neumann algebras


Authors: John W. Bunce and William L. Paschke
Journal: Proc. Amer. Math. Soc. 71 (1978), 232-236
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1978-0482252-3
MathSciNet review: 0482252
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Abstract: A quasi-expectation of a $ {C^ \ast }$-algebra A on a $ {C^\ast}$-subalgebra B is a bounded linear projection $ Q:A \to B$ such that $ Q(xay) = xQ(a)y \forall x,y \in B,a \in A$. It is shown that if M is a von Neumann algebra of operators on Hilbert space H for which there exists a quasi-expectation of $ B(H)$ on M, then there exists a projection of norm one of $ B(H)$ on M, i.e. M is injective. Further, if M is an amenable von Neumann subalgebra of a von Neumann algebra N, then there exists a quasi-expectation of N on $ M' \cap N$. These two facts yield as an immediate corollary the recent result of A. Connes that all amenable von Neumann algebras are injective.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0482252-3
Keywords: Amenable, injective, normal and singular parts, projection of norm one, quasi-expectation
Article copyright: © Copyright 1978 American Mathematical Society