Rigid actions need not be strongly ergodic

Ioana, Adrian; Vaes, Stefaan

doi:10.1090/S0002-9939-2012-11261-4

Rigid actions need not be strongly ergodic
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by Adrian Ioana and Stefaan Vaes PDF

Proc. Amer. Math. Soc. 140 (2012), 3283-3288 Request permission

Abstract:

A probability measure preserving action $\Gamma \curvearrowright (X,\mu )$ is called rigid if the inclusion of $\mathrm {L}^\infty (X)$ into the crossed product $\mathrm {L}^\infty (X) \rtimes \Gamma$ has the relative property (T) in the sense of Popa. We give examples of rigid, free, probability measure preserving actions that are ergodic but not strongly ergodic. The same examples show that rigid actions may admit non-rigid quotients.

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