The non-commutative law of a tuple X from a von Neumann algebra is often viewed as an analog of the joint probability distribution of random variables. In the model-theoretic framework the law represents the quantifier-free type and specifies the values of quantifier-free formulas on X. The values of all formulas, including those constructed with iterated sup's and inf's over an operator-norm ball are specified by the full type. In classical probability theory there is no distinction between the full type and the quantifier-free type because of quantifier elimination. But in the non-commutative setting, the full type is also a viable candidate for the analog of a probability distribution. We therefore describe how free entropy theory and various aspects of free probability could generalize to the setting of full types rather than only quantifier-free types. We also give several applications to embeddings into matrix ultraproducts.

Bio: David Jekel is a mathematician working on operator algebras, in particular free probability theory and model theory of tracial von Neumann algebras. He finished his Ph.D. in 2020 with Dimitri Shlyakhtenko at UCLA, and then worked as an NSF postdoctoral fellow at UCSD with Todd Kemp. He is now a postdoc at the Fields Institute.