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POLISH MODELS AND SOFIC ENTROPY
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 2 / April 2018
- Published online by Cambridge University Press:
- 18 January 2016, pp. 241-275
- Print publication:
- April 2018
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- Article
- Export citation
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We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if
$\unicode[STIX]{x1D6E4}$ is a nonamenable group and 
$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to 
$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.
