Preliminaries Free Probability

Let (M, ϕ) be a von Neumann algebra with a faithful state: non-commutative probability space. Elements X ∈ M are non-commutative random variables. Law of X, ϕX: C[t] 3 p(t) 7 → ϕ(p(X)). For an N-tuple X = (X1,...,XN), ϕX: C 〈t1,..., tN 〉 3 p(t1,..., tN) 7 → ϕ(p(X1,...,XN)). All random variables in this talk will be self-adjoint and non-commutative. Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 2 / 38 Preliminaries Free Probability Let (M, ϕ) be a von Neumann algebra with a faithful state: non-commutative probability space. Elements X ∈ M are non-commutative random variables. Law of X, ϕX: C[t] 3 p(t) 7 → ϕ(p(X)). For an N-tuple X = (X1,...,XN), ϕX