## Mathematical Physics and Operator Algebras

•  Patrick DeBonis, Purdue University
•  Properties of the Actions and von Neumann algebras of Thompson-Like Groups from Cloning Systems
•  03/21/2023
•  10:30 AM - 11:20 AM
•  C304 Wells Hall
•  Brent Nelson (banelson@msu.edu)

Cloning systems are a method for generalizing Thompson's groups, for example $V_d$, that result in a family of groups, $\mathcal{T}_d(G_*)$, whose group von Neumann algebras have been intensely studied by Bashwinger and Zarmesky in recent years. We consider the group actions of a large class of $\mathcal{T}_d(G_*)$ and show they are stable, that is, $G \sim_{OE} G \times \mathbb{Z}.$ As a corollary, we answer Bashwinger and Zaremsky question about when $\mathcal{T}_d(G_*)$ is a McDuff Group in the sense of Deprez and Vaes. As a contrasting result, we show $L(V_d)$ is a prime II$_1$ factor. This is joint work with Rolando de Santiago and Krishnendu Khan.

## Contact

Department of Mathematics
Michigan State University