Abstract: We will discuss non-commutative analogs of ideas in measure theory and probability theory, and in particular recent work on non-commutative triangular transport of measure.

Von Neumann algebras have long been regarded as a non-commutative generalization of measure spaces.  More precisely, a von Neumann algebra is an algebra of operators on a Hilbert space that has many similar properties to the algebra $L^\infty(\Omega,\mathcal{F})$ for measurable space $(\Omega,\mathcal{F})$.  This leads to the notion of non-commutative probability theory.  In classical probability, independent random variables lead to tensor products of measure spaces and of the associated $L^\infty$ of algebras.  In the non-commutative world, one can also take free products of von Neumann algebras, a generalization of free products of groups.  Voiculescu reframed these free products probabilistically as producing “freely independent” random variables.  This has led to many interesting results such as free central limit theorems, free entropy, free Levy-Khintchin theorems, and so forth, as well as a natural analog of an $n$-dimensional Gaussian vector in free probability is a free semicircular family.

However, von Neumann algebras are much more complicated and mysterious than classical measure spaces, and it is in general very difficult to tell when they are isomorphic to each other.  But things are much easier to understand in the case of free Gibbs laws.  These laws are the non-commutative version of a probability measure on $\mathbb{R}^n$ given by a density of the form $e^{-V}$; although we do not know a direct analog of density in the non-commutative case, we can define the free Gibbs laws associated to $V$ in terms of free entropy.  It was shown by Dabrowski, Guionnet, and Shlyakhtenko that if $V$ is close enough to a quadratic (which would correspond to a free semicircular family), then the free Gibbs law can be transported to the law of a free semicircular family, hence the associated von Neumann algebras are isomorphic.  The present author has shown the existence of transport maps which are triangular functions, meaning functions $f = (f_1,\dots,f_n)$ where $f_j$ only depends on $x_1$, \dots, $x_j$.