We extend Voiculescu’s microstates-free definitions of free Fisher information and free entropy to the non-tracial framework. We explain the connection between these quantities and free entropy with respect to certain completely positive maps acting on the core of the non-tracial non-commutative probability space. We give a condition on free Fisher information of an infinite family of variables...

We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold. 1. Factors, extensions and relative mixing 1.1. Factors, extensions and Rokhlin cocycle. We are interested in dynamical systems (X,A , μ, T ), where T is an ergodic automorphism of the Lebesgue space (X,A , μ). We will often designate such a system...

For a Cantor set X, let Homeo(X) denote the group of all homeo-morphisms of X. The main result of this note is the following theorem. Several corollaries of this result are given. In particular, it is proved that for any aperiodic T ∈ Homeo(X) the set of all homeomorphisms conjugate to T is dense in the set of aperiodic homeomorphisms. 0. Introduction. One of the most useful results in ergodic ...

We prove that every positive trace on a countably generated ∗-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial ∗-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct...

1. Topologies on B(H) and the double commutant theorem 2 2. Examples of von Neumann algebras 5 2.1. Abelian von Neumann algebras 6 2.2. Group von Neumann algebras and representation theory 6 2.3. The Hyperfinite II1 factor R 9 3. Projections, Type Classification, and Traces 10 3.1. Projections and the spectral theorem 10 3.2. Type classification of factors 12 3.3. Traces 13 4. Tracial ultrapowe...