there is a deep analogy between, on the one hand, matrices and their trace, and on the other hand, random variables and their expectation.  this idea motivates "quantum" or non-commutative probability theory. tracial von neumann algebras are infinite-dimensional analogs of matrix algebras and the normalized trace, and there are several ways to construct von neumann algebras that represent suitable "limits" of matrix algebras, either through inductive limits, random matrix models, or ultraproducts.  i will give an introduction to this topic and discuss the ultraproduct of matrix algebras and its automorphisms or symmetries. this study incorporates ideas from model theory as well as probability and optimal transport theory.