We prove that the normal cohomology groups H w(M, K(H)) of a von Neumann algebra M with coefficients in the algebra of compact operators are zero if M is atomic of type Ifin. In addition, the completely bounded normal cohomology groups H wcb(B(H), K(H)) are shown to be 0 as well.

Suppose M is a von Neumann algebra with normal, tracial state ϕ and {a 1 ,. .. , a n } is a set of self-adjoint elements of M. We provide an alternative uniform packing description of δ 0

We prove a necessary and sufficient condition for the states which satisfy strong subadditivity of von Neumann entropy with equality on CAR algebra and we show an example when the equality holds but the state is not separable.

We prove that the group-measure-space von Neumann algebra L(T) ⋊ SL(2,Z) is solid. The proof uses topological amenability of the action of SL(2,Z) on the Higson corona of Z.

Let A be a factor von Neumann algebra with dimA ? 2. In this paper, it is proved that map : nonlinear mixed Jordan triple ?-derivation if and only an additive ?-derivation.

We prove that finiteness of the index of the intersection of a finite set of finite index subalgebras in a von Neumann algebra (with small centre) is equivalent to the finite dimensionality of the algebra generated by the conditional expectations onto the subalgebras. 2000 Mathematics Subject Classification. 46S99, 81R10.

We prove a basic result about tensor products of a II j factor with a finite von Neumann algebra and use it to answer, affirmatively, a question asked by S. Popa about maximal injective factors.

A completely bounded bilinear operator φ: M×M → M on a von Neumann algebra M is said to have a factorization in M if there exist completely bounded linear operators ψj , θj : M → M such that

We prove that finiteness of the index of the intersection of a finite set of finite index subalgebras in a von Neumann algebra (with small centre) is equivalent to the finite dimensionality of the algebra generated by the conditional expectations onto the subalgebras. Supported in part by NSF gramt DMS-9322675 and Marsden grant UOA520. Supported in part by NSF grant DMS-0200770.

We prove that a finite von Neumann algebra A is semisimple if the algebra of affiliated operators U of A is semisimple. When A is not semisimple, we give the upper and lower bounds for the global dimensions of A and U . This last result requires the use of the Continuum Hypothesis.