Free Probability, Free Entropy and Applications to von Neumann Algebras

1. Non-commutative probability spaces In general, a non-commutative probability space is a pair (A, τ), where A is a unital algebra (over the field of complex numbers C) and τ a linear functional with τ(I) = 1, where I is the identity of A. Elements of A are called random variables. Since positivity is a key concept in (classical) probability theory, this can be captured by assuming that A is a * algebra and τ is positive (i.e., a state). Elements of the form A∗A are called positive (random variables). A state τ is a trace if τ(AB) = τ(BA). We often require that τ be a faithful trace (τ corresponds to the classical probability measure, or the integral given by the measure). In this talk, we always assume that A is a unital * algebra over C and τ a faithful state on A. Subalgebras of A are always assumed unital * subalgebras. Examples of noncommutative probability spaces often come from operator algebras on a Hilbert space and the states used here are usually vector states. A C*-probability space is a pair (A, τ), where A is a unital C*-algebra (norm closed subalgebra of B(H)) and τ is a state on A. A W*-probability space is a pair (M, τ) consisting of a von Neumann algebra M (strong-operator closed C*-subalgebra of B(H)) and a normal (i.e., countably additive) state τ on M. The following are some more basic concepts: Independence: In a noncommutative probability space (A, τ), a family {Aj} of subalgebrasAj of A is independent if the subalgebras commute with each other and,