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In Lecture 11, we studied compact measure-preserving systems – those systems $(X, {\mathcal X}, \mu, T)$ in which every function $f \in L^2(X, {\mathcal X}, \mu)$ was almost periodic, which meant that their orbit $\{ T^n f: n \in {\Bbb Z}\}$ was precompact in the $L^2(X, {\mathcal X}, \mu)$ topology. Among other things, we were able to easily establish the Furstenberg recurrence theorem (Theorem 1 from Lecture 11) for such systems.

In this lecture, we generalise these results to a “relative” or “conditional” setting, in which we study systems which are compact relative to some factor $(Y, {\mathcal Y}, \nu, S)$ of $(X, {\mathcal X}, \mu, T)$. Such systems are to compact systems as isometric extensions are to isometric systems in topological dynamics. The main result we establish here is that the Furstenberg recurrence theorem holds for such compact extensions whenever the theorem holds for the base. The proof is essentially the same as in the compact case; the main new trick is to not to work in the Hilbert spaces $L^2(X,{\mathcal X},\mu)$ over the complex numbers, but rather in the Hilbert module $L^2(X,{\mathcal X},\mu|Y, {\mathcal Y}, \nu)$ over the (commutative) von Neumann algebra $L^\infty(Y,{\mathcal Y},\nu)$. (Modules are to rings as vector spaces are to fields.) Because of the compact nature of the extension, it turns out that results from topological dynamics (and in particular, van der Waerden’s theorem) can be exploited to good effect in this argument.

[Note: this operator-algebraic approach is not the only way to understand these extensions; one can also proceed by disintegrating $\mu$ into fibre measures $\mu_y$ for almost every $y \in Y$ and working fibre by fibre. We will discuss the connection between the two approaches below.]