Probabilistic Operator Algebra Seminar: Pinsker algebras for 1-bounded entropy

Seminar | March 13 | 3-5 p.m. | 736 Evans Hall

 Benjamin R. Hayes, Vanderbilt University

 Department of Mathematics

I will discuss joint work in progress with Thomas Sinclair. The Pinsker algebra for a probability measure-preserving action of a group G is a classical object of study in ergodic theory — it is the largest quotient of this action with entropy zero. There is a natural analogue of a Pinsker algebra in free probability corresponding to 1-bounded entropy (a quantity related to the notion of being strongly 1-bounded of Kenley Jung). Here, a crucial difference is that a diffuse von Neumann algebra has many Pinsker algebras. I will give conditions in terms of measures on the microstates spaces which guarantee that a given subalgebra is a Pinsker algebra. Using this, we can give free probability reproofs of some interesting structural properties of free group factors.