Quasi-expectations and amenable von Neumann algebras

Bunce, John W.; Paschke, William L.

doi:10.1090/S0002-9939-1978-0482252-3

Quasi-expectations and amenable von Neumann algebras
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by John W. Bunce and William L. Paschke

Proc. Amer. Math. Soc. 71 (1978), 232-236

DOI: https://doi.org/10.1090/S0002-9939-1978-0482252-3

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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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