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An example of a solid von Neumann algebra. (English) Zbl 1187.46048
Let \(SL(2,\mathbb{Z})=\left\{\left[\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\right]: a,b,c,d \in \mathbb{Z},\;ad-bc = 1\right\}\) act by linear transformations on the \(2-\) torus \(\mathbb{T}^2\) with the Haar measure, and \(L^{\infty}(\mathbb{T}^2)\rtimes SL(2,\mathbb{Z})\) be the crossed product von Neumann algebra. Recall that a finite von Neumann algebra is called solid if every diffuse subalgebra has an amenable relative commutant.
The main result of the paper says that the group measure space von Neumann algebra \(L^{\infty}(\mathbb{T}^2)\rtimes SL(2,\mathbb{Z})\) is solid. The proof uses topological amenability of the action of \(SL(2,\mathbb{Z})\) on the Higson corona of \(\mathbb{Z}^2.\)

MSC:
46L10 General theory of von Neumann algebras
46L55 Noncommutative dynamical systems
43A07 Means on groups, semigroups, etc.; amenable groups
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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