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Inner amenability, property Gamma, McDuff $\text{II}_{1}$ factors and stable equivalence relations

Published online by Cambridge University Press:  14 March 2017

TOBE DEPREZ  and
STEFAAN VAES
TOBE DEPREZ
Affiliation:
KU Leuven, Department of Mathematics, Leuven, Belgium email [email protected], [email protected]
STEFAAN VAES
Affiliation:
KU Leuven, Department of Mathematics, Leuven, Belgium email [email protected], [email protected]
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Abstract

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 7 , October 2018 , pp. 2618 - 2624
Copyright
© Cambridge University Press, 2017 

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