Related papers: Concavification of free entropy
We give another definition of orbital free entropy introduced by Hiai, Miyamoto and us, which does not need the hyperfiniteness assumption for each given random multi-variable. The present definition is somehow related to one of its several…
Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of…
We define and study a relative free entropy quantity, analogous in its properties to Voiculescu's relative free entropy Chi^*(...:B). Our definition uses matricial microstates, unlike his definition, which involves non-commutative Hilbert…
A lower estimate of the orbital free entropy $\chi_\mathrm{orb}$ under unitary conjugation is proved, and it together with Voiculescu's observation shows that the conjectural exact formula relating $\chi_\mathrm{orb}$ to the free entropy…
We prove a technical result, showing that the existence of a closable unbounded dual system in the sense of Voiculescu is equivalent to the finiteness of free Fisher information. This approach allows one to give a purely operator-algebraic…
The first aims of this work are to endorse the advent of finitely additive set functions as equilibrium states and the possibility to replace the metric entropy by an upper semi-continuous map associated to a general variational principle.…
The free analog of the pressure is introduced for multivariate noncommutative random variables and its Legendre transform is compared with Voiculescu's microstate free entropy.
We study Voiculescu's microstate free entropy for a single non-selfadjoint random variable. The main result is that certain additional constraints on eigenvalues of microstates do not change the free entropy. Our tool is the method of…
The extension $k \mapsto \mu^{\boxplus k}$ of the concept of a free convolution power to the case of non-integer $k \geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory.…
There are given sufficient conditions under which mixtures of dilations of L\'evy spectral measures, on a Hilbert space, are L\'evy measures again. We introduce some random integrals with respect to infinite dimensional L\'evy processes,…
D. Voiculescu [2] proved that a standard family of independent random unitary k by k matrices and a constant k by k unitary matrix is asymtotically free as k goes to infinity. This result was a key ingredient in Voiculescu's proof [3] that…
We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive…
We study two ways (levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms. For free-selfdecomposable measures we found the formula (a differential…
Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical…
This paper is part of a series aiming at proving that the $\limsup$ and $\liminf$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a large deviation principle) for hermitian brownian motion on…
In this paper we introduce the concept of the upper free orbit-dimension of a finite von Neumann algebra, and we derive some of its basic properties. Using this concept, we are able to improve most of the applications of free entropy to…
In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are…
A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and…
We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility…
Inspired by Katok's intermediate entropy property [Inst. Hautes \'Etudes Sci. Publ. Math. 51 (1980), 137-173], we introduce and study the notion of entropy flexibility for discrete-time and continuous-time dynamical systems. By using…