Topological Free Entropy Dimensions in Nuclear C*-algebras and in Full Free Products of Unital C*-algebras

In the paper, we introduce a new concept, topological orbit dimension of an n-tuple of elements in a unital C*-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C*-algebra is less than or equal to 1. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C*-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that Ext(C*r (F2) *C C* r (F2)) is not a group.

ORBITAL APPROACH TO MICROSTATE FREE ENTROPY

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy χ orb for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties, the relation of χorb with the usual free entropy χ is shown. Moreover, the dimension counterpart δ0, orb of χ orb is discussed, and we obtain the relation between δ0, orb and the original free entropy dimension δ0 together with applications to δ0 itself.

FREE ENTROPY DIMENSION OF PROJECTIONS AND FACTORIALITY OF GENERATED VON NEUMANN ALGEBRAS

We examine the free entropy and free entropy dimension for projections, and obtain a sufficient condition for the factoriality of the von Neumann algebra generated by projections in terms of their free entropy dimension. This corresponds to Voiculescu's result for self-adjoint elements.