# Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 23-48.

Von Neumann algebras of Thompson-like groups from cloning systems

Authors:  Eli Bashwinger (1), Matthew C.B. Zaremsky (2)
Author institution: (1) Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222, U.S.A.
(2) Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222, U.S.A.

Summary: We prove a variety of results about group von Neumann algebras $\mathcal{L}(\mathscr{T}_d(G_*))$ associated to Thompson-like groups $\mathscr{T}_d(G_*)$ arising from so called $d$-ary cloning systems. In particular, we find sufficient conditions to ensure that $\mathcal{L}(\mathscr{T}_d(G_*))$ is a type $\textrm{II}_1$ factor, or even a McDuff factor. For example our results show that, for $bV$ and $bF$ the braided Thompson groups, $\mathcal{L}(bV)$ and $\mathcal{L}(bF)$ are type $\textrm{II}_1$ factors and $\mathcal{L}(bF)$ is McDuff. In particular we get the surprising result that $bF$ is inner amenable.

DOI: http://dx.doi.org/10.7900/jot.2021apr16.2355
Keywords: group von Neumann algebra, type $II_1$ factor, McDuff factor, ICC, inner amenable, Thompson group, cloning system