# 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

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