Related papers: Free probability, free entropy and applications to…
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.
This in an introduction to the theory of non-commutative distributions of non-commuting operators or random matrices. Starting from the basic problem to find a good approach to the meaning of "non-commutative distribution" we will, in…
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.
Given any $1 < q \le 2$, we use new free probability techniques to construct a completely isomorphic embedding of $\ell_q$ (equipped with its natural operator space structure) into the predual of a sufficiently large QWEP von Neumann…
This is an introduction to some of the most probabilistic aspects of free probability theory.
These lecture notes provide an introduction to free probability theory, with a focus on tools and techniques useful in the study of large random matrices. Topics include freeness, free cumulants, additive and multiplicative free…
We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free…
We consider the analysis of probability distributions through their associated covariance operators from reproducing kernel Hilbert spaces. We show that the von Neumann entropy and relative entropy of these operators are intimately related…
This in an introduction to free probability theory, covering the basic combinatorial and analytic theory, as well as the relations to random matrices and operator algebras. The material is mainly based on the two books of the lecturer, one…
Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its…
The paper is devoted to the investigation of Segal's entropy in semifinite von Neumann algebras. The following questions are dealt with: semicontinuity, the 'ideal-like' structure of the linear span of the set of operators with finite…
The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von…
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…
Free probability theory was created by Dan Voiculescu around 1985, motivated by his efforts to understand special classes of von Neumann algebras. His discovery in 1991 that also random matrices satisfy asymptotically the freeness relation…
We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed.…
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the…
We review origins and developments of Noncommutative Potential theory as underpinned by the notion of energy form. Recent and new applications are shown to approximation properties of von Neumann algebras.
One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges…
Free independence is an important tool for studying the structure of operator algebras. It is natural to ask from the model-theoretic standpoint whether free independence is captured well in first-order model theory via the notion of a…