Let $T$ be a measure preserving bijection of a probability space $(X,\nu)$. Consider the Koopman representation of $\mathbb{Z}$ on $L^2(X,\nu)$ given by $[z.f](x) = f(T^{-z}(x))$. The question is: can I tell from the representation whether $(X,\nu,T)$ has zero entropy?



Zero entropy and the Koopman representation
There is an example (due to [Newton - Parry](https://mathscinet.ams.org/mathscinet-getitem?mr=206209) and also attributed by [Rokhlin](http://www.mathnet.ru/links/8b6eec06952e8e0c9d023e3caa73e40b/rm5788.pdf) to Girsanov) of a zero entropy measure preserving transformation with a countable Lebesgue spectrum (and mixing of all orders). Therefore, the associated Koopman operator is unitarily equivalent to that of any Bernoulli shift.


