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Journal of Operator Theory

Volume 32, Issue 2, Fall 1994  pp. 381-398.

A description of spatially projective von Neumann algebras

Authors A. Ya. Helemskii
Author institution: Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119899 GSP, RUSSIA

Summary:  Let $\mathcal R$ be a von Neumann algebra on the Hilbert space H. Then H, as a Banach left module over $\mathcal R$ with the multiplication $a \cdot x = a(x)$, is projective if and only if the following conditions are satisfied: 1) $\mathcal R$ is of type I; 2) the center of $\mathcal R$ is the weak-operator-closed linear span of its minimal projections; and 3) in the standard decomposition $\mathcal R = \sum\limits_{m,n} {\mathcal R_{m,n} }$, where $\mathcal R_{m,n}$ is a von Neumann algebra of type I_m with the commutant of type I_n, there is no non-zero summand for which both m and n are finite. The most difficult part of the proof is to show that H is not projective in the case of an infinite type I factor in the standard form.As an application, it is shown that the indicated conditions on $\mathcal R$ characterize the class of von Neumann algebras with the property of vanishing their cohomology groups with coefficients in certain “operator” $\mathcal R$-bimodules.

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