On the Mackey-Topology for a von Neumann Algebra.

A New Topology for Von Neumann Algebras

A Note on Faithful Traces on a Von Neumann Algebra

It is known that a semifinite von Neumann algebra always has a faithful semifinite trace. This trace can be used, for instance, to build up a non-commutative integration and, consequently, to define non commutative L-spaces. In this note we give some techniques for constructing, starting from a family F of semifinite traces, a faithful one which is closely related to the family F. Let F = {ηα; ...

A Counterexample in Von Neumann Algebra Dynamics

In this essay we construct a noncommutative von Neumann dynamical system with certain pathological properties from the viewpoint of analogies with classical ergodic theory. These relate to obstructions to the existence of special kinds of subsystem or subextension. We will build our example as an extension of a commutative dynamical system via a unitary cocycle acting on a Hilbert bundle, and w...

On a canonical lattice structure on the effect algebra of a von Neumann algebra

Let R be a von Neumann algebra acting on a Hilbert space H and let Rsa be the set of hermitean (i.e. selfadjoint) elements of R. It is well known that Rsa is a lattice with respect to the usual partial order ≤ if and only if R is abelian. We define and study a new partial order on Rsa, the spectral order ≤s, which extends ≤ on projections, is coarser than the usual one, but agrees with it on ab...

Affine Connections, Duality and Divergences for a Von Neumann Algebra

On the predual of a von Neumann algebra, we define a dif-ferentiable manifold structure and affine connections by embeddings into non-commutative L p –spaces. Using the geometry of uniformly convex Ba-nach spaces and duality of the L p and L q spaces for 1/p + 1/q = 1, we show that we can introduce the α-divergence, for α ∈ (−1, 1), in a similar manner as Amari in the classical case. If restric...