Strongly singular MASAs and mixing actions in finite von Neumann algebras
Résumé
Let Γ be a countable group and let Γ_0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ_0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ_0) is a singular MASA in M=L(Γ) which satisfies a weakly mixing condition. If, moreover, it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, Higman–Neumann–Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.
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