The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has

in the strong topology, where is the -invariant subspace of , and is the orthogonal projection to . (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if is a countable amenable group acting on a Hilbert space by unitary transformations , and is a vector in that Hilbert space, then one has

for any Folner sequence of , where is the -invariant subspace. Thus one can interpret as a certain average of elements of the orbit of .

I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem)Let be an arbitrary group acting unitarily on a Hilbert space , and let be a vector in . Then is the element in the closed convex hull of of minimal norm, and is also the unique element of in this closed convex hull.

*Proof:* As the closed convex hull of is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element of minimal norm. If for some , then the midpoint of and would be in the closed convex hull and be of smaller norm, a contradiction; thus is -invariant. To finish the first claim, it suffices to show that is orthogonal to every element of . But if this were not the case for some such , we would have for all , and thus on taking convex hulls , a contradiction.

Finally, since is orthogonal to , the same is true for for any in the closed convex hull of , and this gives the second claim.

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any , Theorem 1 implies that there is a finite convex combination of shifts of which lies within (in the norm) to . By the triangle inequality, all the averages also lie within of , but by the Folner property this implies that the averages are eventually within (say) of , giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality)Let be a group acting unitarily on a Hilbert space , and let be a -invariant subspace of . Then and are relatively orthogonal over their common subspace , that is to say the restrictions of and to the orthogonal complement of are orthogonal to each other.

*Proof:* By Theorem 1, we have for all , and the claim follows. (Thanks to Gergely Harcos for this short argument.)

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups . Define a *-system* to be a probability space together with a measure-preserving action of on ; this gives an action of on , which by abuse of notation we also call :

(In this post we follow the usual convention of defining the spaces by quotienting out by almost everywhere equivalence.) We say that a -system is *ergodic* if consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space and quotienting by the -ideal of null sets, and considering maps such as only on this quotient -algebra (or on the associated von Neumann algebra or Hilbert space ). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A *factor* of a -system is another -system together with a *factor map* which commutes with the -action (thus for all ) and respects the measure in the sense that for all . For instance, the *-invariant factor* , formed by restricting to the invariant algebra , is a factor of . (This factor is the first factor in an important hierachy, the next element of which is the *Kronecker factor* , but we will not discuss higher elements of this hierarchy further here.) If is a factor of , we refer to as an *extension* of .

From Corollary 2 we have

Corollary 3 (Relative independence)Let be a -system for a group , and let be a factor of . Then and are relatively independent over their common factor , in the sense that the spaces and are relatively orthogonal over when all these spaces are embedded into .

This has a simple consequence regarding the product of two -systems and , in the case when the action is trivial:

Lemma 4If are two -systems, with the action of on trivial, then is isomorphic to in the obvious fashion.

This lemma is immediate for countable , since for a -invariant function , one can ensure that holds simultaneously for all outside of a null set, but is a little trickier for uncountable .

*Proof:* It is clear that is a factor of . To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero which is orthogonal to . In particular, we have orthogonal to for any . Since lies in , we conclude from Corollary 3 (viewing as a factor of ) that is also orthogonal to . Since is an arbitrary element of , we conclude that is orthogonal to and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.)

Now we discuss the notion of a group extension.

Definition 5 (Group extension)Let be an arbitrary group, let be a -system, and let be a compact metrisable group. A-extensionof is an extension whose underlying space is (with the product of and the Borel -algebra on ), the factor map is , and the shift maps are given bywhere for each , is a measurable map (known as the

cocycleassociated to the -extension ).

An important special case of a -extension arises when the measure is the product of with the Haar measure on . In this case, also has a -action that commutes with the -action, making a -system. More generally, could be the product of with the Haar measure of some closed subgroup of , with taking values in ; then is now a system. In this latter case we will call *-uniform*.

If is a -extension of and is a measurable map, we can define the *gauge transform* of to be the -extension of whose measure is the pushforward of under the map , and whose cocycles are given by the formula

It is easy to see that is a -extension that is isomorphic to as a -extension of ; we will refer to and as *equivalent* systems, and as *cohomologous* to . We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem)Let be an arbitrary group, let be an ergodic -system, and let be a compact metrisable group. Then every ergodic -extension of is equivalent to an -uniform extension of for some closed subgroup of .

This theorem is usually stated for amenable groups , but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, and are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions via the ergodic decomposition, but the result becomes more complicated to state, and we will not do so here.

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