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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quasi-expectations and amenable von Neumann algebras
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by John W. Bunce and William L. Paschke PDF
Proc. Amer. Math. Soc. 71 (1978), 232-236 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • 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