arXiv subjects
Publications and source records attributed to Dan Voiculescu.
A duality transform for the coalgebra of the free difference quotient derivation-multiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.
Notions of topological free entropy and of free capacity are introduced in the $C^*$-algebra context. Basic properties, basic problems and connections to potential theory and random matrix theory are discussed.
Using simple operator-valued analytic function considerations several general analytic subordination results for a freely Markovian triple of operator algebras are derived from the analytic subordination result for generalized resolvents in the case of a free increment with amalgamation. This includes a generalization of Biane's subordination result in the case of a unitary free multiplicative increment.
A main result is that, roughly, a dense set of the infinitesimal trace-preserving deformations of a semicircular system $s_1,..., s_n$ arise from one-parameter groups of automorphisms of the free-group factor $L(F(n))$ generated by $s_1,..., s_n$. More generally the paper studies cyclic gradients in von Neumann algebras, Lie algebras of noncommutative trace-preserving vector fields and the class of cyclomorphic maps which preserve the orthogonals of spaces of cyclic gradients.
Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras, connections to random matrix theory and a discussion of open problems.
We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semi-circle distribution is majorized by a modified free entropy quantity.
Simple necessary and sufficient conditions for a $n$-tuple of noncommutative polynomials to be a cyclic gradient are given and similarly for a noncommutative polynomial to have a vanishing cyclic gradient. Connections with free probability are explained.