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 $\rm I$; 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 $\rm I_m$ with the commutant of type $\rm I_n$, 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 $\rm I$ 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.

Keywords: Projective Banach module, von Neumann algebra, spatial projectivity

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