International Congress of Mathematicians

ICAl 2002 • Vol. II • 787^794Free Probability, Free Entropy andApplications to von Neumann AlgebrasLiming Ge*This talk is organized as follows: First we explain some basic concepts in noncommutativeprobability theory in the frame of operator algebras. In Section 2, wediscuss related topics in von Neumann algebras. Sections 3 and 4 contain some ofthe key ideas and results in free probability theory. Last section states some of theimportant applications of free probability theory.1. Non-commutative probability spacesIn general, a non-commutative probability space is a pair (A,T), where A isa unital algebra (over the field of complex numbers C) and r a linear functionalwith T(I) = 1, where J is the identity of A. Elements of A are called randomvariables. Since positivity is a key concept in (classical) probability theory, thiscan be captured by assuming that A is a * algebra and r is positive (i.e., a state).Elements of the form A* A are called positive (random variables).A state r is a trace if T(AB) = T(BA). We often require that r be a faithfultrace (r corresponds to the classical probability measure, or the integral given bythe measure). In this talk, we always assume that A is a unital * algebra over C andr a faithful state on A. Subalgebras of A are always assumed unital * subalgebras.Examples of noneommutative probability spaces often come from operatoralgebras on a Hilbert space and the states used here are usually vector states.A C*-probability space is a pair (A,T), where A is a unital C*-algebra (normclosed subalgebra of B(H)) and r is a state on A.A W*-probability space is a pair (M,T) consisting of a von Neumann algebraM (strong-operator closed C*-subalgebra of B(lfij) and a normal (i.e., countablyadditive) state r on ;M.The following are some more basic concepts:Independence: In a noneommutative probability space (A,T), a family {Aj} ofsubalgebras Aj of A is independent if the subalgebras commute with each other and,for n £ N, T(AI • • • A n ) = T(A{) • • • r(A n ) for all Ak in Aj k and jk fi 1 ji wheneverfc#l.* Academy of Mathematics and System Science, CAS, Beijing 100080, China. Department ofMathematics UNH, Durham, NH 03824, USA. E-mail: liming@math.unh.edu