Having studied compact extensions in the previous lecture, we now consider the opposite type of extension, namely that of a weakly mixing extension. Just as compact extensions are “relative” versions of compact systems, weakly mixing extensions are “relative” versions of weakly mixing systems, in which the underlying algebra of scalars is replaced by . As in the case of unconditionally weakly mixing systems, we will be able to use the van der Corput lemma to neglect “conditionally weakly mixing” functions, thus allowing us to lift the uniform multiple recurrence property (UMR) from a system to any weakly mixing extension of that system.
To finish the proof of the Furstenberg recurrence theorem requires two more steps. One is a relative version of the dichotomy between mixing and compactness: if a system is not weakly mixing relative to some factor, then that factor has a non-trivial compact extension. This will be accomplished using the theory of conditional Hilbert-Schmidt operators in this lecture. Finally, we need the (easy) result that the UMR property is preserved under limits of chains; this will be accomplished in the next lecture.
— Conditionally weakly mixing functions —
Recall that in a measure-preserving system , a function is said to be weakly mixing if the squared inner products converge in the Cesàro sense, thus
Now let be a factor of X, so that can be viewed as a subspace of . Recall that we have the conditional inner product and the Hilbert module of functions f for which lies in . We shall say that a function is conditionally weakly mixing relative to Y if the norms converge to zero in the Cesàro sense, thus
Example 1. If is a product system of the factor space and another system , then a function of the vertical variable is weakly mixing relative to Y if and only if f(z) is weakly mixing in Z.
Much of the theory of weakly mixing systems extends easily to the conditionally weakly mixing case. For instance:
Exercise 1. By adapting the proof of Corollary 2 from Lecture 12, show that if is conditionally weakly mixing and , then and converge to zero in the Cesàro sense. (Hint: you will need to show that expressions such as converge in in the Cesàro sense. Apply the van der Corput lemma and use the fact that are uniformly bounded in by conditional Cauchy-Schwarz.)
Exercise 2. Show that the space of conditionally weakly mixing functions in is a module over (i.e. it is closed under addition and multiplication by the “scalars” ), which is also shift-invariant and topologically closed in the topology of (see Exercise 2 from Lecture 13).
Let us now see the first link between conditional weak mixing and conditional almost periodicity (cf. Exercise 18 from Lecture 12):
Lemma 1. If is conditionally weakly mixing and is conditionally almost periodic, then a.e.
Proof. Since , it will suffice to show that
Let be arbitrary. As g is conditionally almost periodic, one can find a finitely generated module zonotope with such that all the shifts lie within (in ) of this zonotope. Thus (by conditional Cauchy-Schwarz) we have
for all n and some with norm at most 1. We can pull these constants out of the conditional inner product and bound the left-hand side of (4) by
By Exercise 1, the Cesàro supremum of (5) is at most . Since is arbitrary, the claim (3) follows.
Since all functions in are conditionally almost periodic, we conclude that every conditionally weakly mixing function f is orthogonal to , or equivalently that a.e. Let us say that f has relative mean zero if the latter holds.
Definition 1. A system X is a weakly mixing extension of a factor Y if every with relative mean zero is relatively weakly mixing.
Exercise 3. Show that a product of a system Y with a weakly mixing system Z is always a weakly mixing extension of Y.
Remark 1. If X is regular, then we can disintegrate the measure as an average , see Theorem 4 from Lecture 9. It is then possible to construct a relative product system , which is the product system but with the measure instead of . It can then be shown (cf. Exercise 9 from Lecture 12) that X is a weakly mixing extension of Y if and only if is ergodic; see for instance Furstenberg’s book for details. However, in these notes we shall focus instead on the more abstract operator-algebraic approach which avoids the use of disintegrations.
Theorem 1. Suppose that is a weakly mixing extension of . If Y obeys UMR, then so does X.
The proof of this theorem rests on the following analogue of Proposition 1 from Lecture 12:
Proposition 1. Let be distinct integers for some . Let is a weakly mixing extension of , and let be such that at least one of has relative mean zero. Then
Exercise 4. Prove Proposition 1. (Hint: modify (or “relativise”) the proof of Proposition 1 from Lecture 12.)
Corollary 1. Let be distinct integers for some . Let is a weakly mixing extension of , and let . Then
Exercise 5. Prove Corollary 1. (Hint: adapt the proof of Corollary 2 from Lecture 12.)
Proof of Theorem 1. Let be non-negative with positive mean. Then is also non-negative with positive mean. Since Y obeys UMR, we have
Applying Corollary 1 we see that the same statement holds with replaced by f, and the claim follows.
Remark 2. As the above proof shows, Corollary 1 lets us replace functions in the weakly mixing extension X by their expectations in Y for the purposes of computing k-fold averages. In the notation of Furstenberg and Weiss, Corollary 1 asserts that Y is a characteristic factor of X for the average (7). The deeper structural theory of such characteristic factors (and in particular, on the minimal characteristic factor for any given average) is an active and difficult area of research, with surprising connections with Lie group actions (and in particular with flows on nilmanifolds), as well as the theory of inverse problems in additive combinatorics (and in particular to inverse theorems for the Gowers norms); see for instance the ICM paper of Kra for a survey of recent developments. The concept of a characteristic factor (or more precisely, finitary analogues of this concept) also is fundamental in my work with Ben Green on primes in arithmetic progression.
— The dichotomy between structure and randomness —
The remainder of this lecture is devoted to proving the following “relative” generalisation of Theorem 1 from Lecture 12, and which is a fundamental ingredient in the proof of the Furstenberg recurrence theorem:
Theorem 2. Suppose that is an extension of a system . Then exactly one of the following statements is true:
- (Structure) X has a factor Z which is a non-trivial compact extension of Y.
- (Randomness) X is a weakly mixing extension of Y.
As in Lecture 12, the key to proving this theorem is to show
Proposition 2. Suppose that is an extension of a system . Then a function is relatively weakly mixing if and only if a.e. for all relatively almost periodic g.
The “only if” part of this proposition is Lemma 1; the harder part is the “if” part, which we will prove shortly. But for now, let us see why Proposition 2 implies Theorem 2.
From Lemma 1, we already know that no non-trivial function can be simultaneously conditionally weakly mixing and conditionally almost periodic, which shows that cases 1 and 2 of Theorem 2 cannot simultaneously hold. To finish the proof of Theorem 2, suppose that X is not a weakly mixing extension of Y, thus there exists a function of relative mean zero which is not weakly mixing. By Proposition 2, there must exist a relatively almost periodic such that does not vanish a.e.. Since f is orthogonal to all functions in , we conclude that g is not in , thus we have a single relatively almost periodic function. From Exercise 6 of Lecture 13, this shows that the maximal compact extension of Y is non-trivial, and the claim follows.
It thus suffices to prove the “if” part of Proposition 2; thus we need to show that every non-conditionally-weakly-mixing function correlates with some conditionally almost periodic function. But observe that if is not conditionally weakly mixing, then by definition we have
We can rearrange this as
where is the operator
To prove Proposition 2, it thus suffices (by weak compactness) to show that
Proposition 3. (Dual functions are almost periodic) Suppose that is an extension of a system , and let . Let be any limit point of in the weak operator topology. Then is relatively almost periodic.
Remark 3. By applying the mean ergodic theorem to the dynamical system , one can show that the sequence is in fact convergent in the weak or strong operator topologies (at least when X is regular). But to avoid some technicalities we shall present an argument that does not rely on existence of a strong limit.
As one might expect from the experience with unconditional weak mixing, the proof of Proposition 3 relies on the theory of conditionally Hilbert-Schmidt operators on . We give here a definition of such operators which is suited for our needs.
Definition 2. Let X, Y be as above. A sub-orthonormal set in is any at most countable sequence such that a.e. for all and a.e. for all . A linear operator is said to be a conditionally Hilbert-Schmidt operator if we have the module property
for all (12)
and the bound
for all sub-orthonormal sets , and some constant ; the best such C is called the (uniform) conditional Hilbert-Schmidt norm of A.
Remark 4. As in Lecture 12, one can also set up the concept of a tensor product of two Hilbert modules, and use that to define conditionally Hilbert-Schmidt operators in a way which does not require sub-orthonormal sets. But we will not need to do so here. One can also define a pointwise conditional Hilbert-Schmidt norm for each , but we will not need this concept.
Example 2. Suppose Y is just a finite set (with the discrete -algebra), then X splits into finitely many fibres with the conditional measures , and can be direct sum (with the norm) of the Hilbert spaces . A conditional Hilbert-Schmidt operator is then equivalent to a family of Hilbert-Schmidt operators for each y, with the uniformly bounded in Hilbert-Schmidt norm.
Example 3. In the skew shift example , , one can show that an operator A is conditionally Hilbert-Schmidt if and only if it takes the form a.e. for all , with finite.
Exercise 6. Let with a.e.. Show that the rank one operator is conditionally Hilbert-Schmidt with norm at most 1.
Observe from (11) that the are averages of rank one operators arising from the functions , and so by Exercise 6 and the triangle inequality we see that the are uniformly conditionally Hilbert-Schmidt. Taking weak limits using (13) (and Fatou’s lemma) we conclude that is also conditionally Hilbert-Schmidt.
Next, we observe from the telescoping identity that for every h, converges to zero in the weak operator topology (and even in the operator norm topology) as ; taking limits, we see that commutes with T. To show that is conditionally almost periodic, it thus suffices to show the following analogue of Lemma 2 from Lecture 12:
Lemma 2. Let be a conditionally Hilbert-Schmidt operator. Then the image of the unit ball of under is conditionally precompact.
Proof. We shall prove this lemma by establishing a sort of conditional singular value decomposition for A. We can normalise A to have uniform conditional Hilbert-Schmidt norm 1. We fix , and we will also need an integer k and a small quantity depending on to be chosen later.
We first consider the quantities where ranges over all sub-orthonormal sets of cardinality 1. On the one hand, these quantities are bounded pointwise by 1, thanks to (13). On the other hand, observe that if and are of the above form, then so is the join , as can be seen by taking and , where E is the set where exceeds . By using a maximising sequence for the quantity and applying joins repeatedly, we can thus (on taking limits) find a pair which is near-optimal in the sense that a.e. for all competitors .
Now fix , and consider the quantity , where and are sub-orthonormal sets. By arguing as before we can find an which is near optimal in the sense that a.e. for all competitors .
We continue in this fashion k times to obtain sub-orthonormal sets and with the property that whenever are sub-orthonormal sets. On the other hand, from (13) we know that . From these two facts we soon conclude that a.e. whenever and are sub-orthonormal. If are chosen appropriately we obtain a.e. Thus (by duality) A maps the unit ball of the orthogonal complement of the span of to the -neighbourhood of the span of (with notions such as orthogonality, span, and neighbourhood being defined conditionally of course, using the -Hilbert module structure of ). From this it is not hard to establish the desired precompactness.
[Update, Mar 1: Typo corrected.]
[Update, June 2 2009: Some minor changes in the proof of Lemma 1. Thanks to Jeremy Avigad for corrections.]