BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

A von Neumann algebra is the commutant in B(H) (the algebra of bounded operators on a Hilbert space H) of a selfadjoint subset of B(H), in other words, a *-subalgebra of B(H) that is equal to its own bicommutant. These objects, whose investigation began with the work of Murray and von Neumann in the 30’s and 40’s on “rings of operators”, are vastly numerous and diverse, but there is a good schematic picture of what they look like in the large, and some especially significant parts of the terrain have been mapped out in great detail. The theory of von Neumann algebras impinges on the rest of mathematics in such areas as ergodic theory, group theory, combinatorics, and mathematical physics. Interest in the cohomology von Neumann algebras goes back about thirty years. The ingredients of the definition are a von Neumann algebra (or, for the moment, a Banach algebra)M and a BanachM−bimodule V . Write L(M,V) for the space of bounded n−linear maps fromM to V , and form the Hochschild complex V ∂ −→ L(M,V) ∂ −→ L(M,V)→ . . . , where the map ∂ : V → L is given by (∂v)(x) = xv−vx, and in higher dimensions, (∂φ)(x1, . . . , xn+1) = x1φ(x2, . . . , xn+1) + ∑n j=1(−1)φ(x1, . . . , xj−1, xjxj+1, xj+2, . . . , xn+1) + (−1)φ(x1, . . . , xn)xn+1. Let H∗(M,V) denote the cohomology of this complex. What one mainly wants to know about H(M,V) in a given situation is whether or not it vanishes. For orientation, we mention a theorem of Connes [Co] characterizing injective von Neumann algebras as precisely thoseM such that H∗(M,V) vanishes for all dual normalM−bimodules V . (To say that the bimodule V is dual normal means that it, likeM, is the conjugate space of a Banach space and that the module action is appropriately respectful of the w∗-topologies onM and V . Hyperfinite means having a w∗-dense increasing net of finite-dimensional *-subalgebras.) The oldest result in von Neumann algebra cohomology is the Kadison Sakai theorem [K], [S], which says that all derivations of any von Neumann algebra M into itself are inner; that is, if the linear map d :M→M satisfies d(xy) = xd(y)+d(x)y, then there exists v inM such that d(x) = xv−vx. This says that H(M,M) = (0) for all M (even algebraically, since automatic continuity of derivations is part of the theorem). The central result in Hochschild cohomology of von Neumann algebras asserts that H∗(M,M) vanishes for all M whose type II1 part is stable under tensoring with the hyperfinite II1 factor. Depending on vantage point, this either answers within epsilon the question for general von Neumann algebras posed many years ago by Kadison and Ringrose or shows the way to the last frontier. The book is addressed to readers familiar with the basics of the theory of operator algebras,