On compact extensions

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This is again a note on my talk in the Szemerédi’s theorem seminar, going through Furstenberg’s book. In this round, my part is to introduce compact extension.
Let \Gamma be an abelian group of measure preserving transformations on (X, \mathcal{B}, \mu), \alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma') be an extension map.
i.e. \alpha: X \rightarrow Y s.t. \alpha^{-1} sends \nu-0 sets to \mu-0 sets;

\gamma'\circ \alpha (x) = \alpha \circ \gamma (x)

Definition: A sequence of subsets (I_k) of \Gamma is a Folner sequence if |I_k| \rightarrow \infty and for any \gamma \in \Gamma,

\frac{| \gamma I_k \Delta I_k|}{|I_k|} \rightarrow 0

Proposition: For any Folner sequence I = (I_k) of \Gamma, for any f \in L^1(X), \displaystyle \frac{1}{|I_k|} \sum_{\gamma \in I_k} \gamma f converges weakly to the orthogonal projection of f onto the subspace of \Gamma-invariant functions. (Denoted P(f) where P: L^2(X) \rightarrow L^2_{inv}(X).

Proof: Let \mathcal{H}_0 = P^{-1}(\bar{0}) = (L^2_{inv}(X))^\bot
For all \gamma \in \Gamma,

\gamma (L^2_{inv}(X)) \subseteq L^2_{inv}(X)

Since \Gamma is \mu-preserving, \gamma is unitary on L^2(X). Therefore we also have \gamma( \mathcal{H}_0) \subseteq \mathcal{H}_0.

For f \in \mathcal{H}_0, suppose there is subsequence (n_k) where \displaystyle \frac{1}{|I_{n_k}|} \sum_{\gamma \in I_{n_k}} \gamma (f) converges weakly to some g \in L^2(X).

By the property that \frac{| \gamma I_k \Delta I_k|}{|I_k|} \rightarrow 0, we have for each \gamma \in \Gamma, \gamma(g) = g, \ g is \Gamma-invariant. i.e. g \in (\mathcal{H}_0)^\bot

However, since f \in \mathcal{H}_0 hence all \gamma(f) are in \mathcal{H}_0 hence g \in \mathcal{H}_0. Therefore g \in \mathcal{H}_0 \cap (\mathcal{H}_0)^\bot, g=\bar{0}

Recall: 1)X \times_Y X := \{ (x_1, x_2) \ | \ \alpha(x_1) = \alpha(x_2) \}.

i.e. fibred product w.r.t. the extension map \alpha: X \rightarrow Y.

2)For H \in L^2(X \times_Y X), \ f \in L^2(X),

(H \ast f)(x) = \int H(x_1, x_2) f(x_2) d \mu_{\alpha(x_1)}(x_2)

Definition: A function f \in L^2(X) is said to be almost periodic if for all \varepsilon > 0, there exists g_1, \cdots g_k \in L^2(X) s.t. for all \gamma \in \Gamma and almost every y \in Y,

\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon

Proposition: Linear combination of almost periodic functions are almost periodic.

Proof: Immediate by taking all possible tuples of g_i for each almost periodic function in the linear combination corresponding to smaller \varepsilonl.

Definition: \alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma') is a compact extension if:

C1: \{ H \ast f \ | \ H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X), f \in L^2(X) \} contains a basis of L^2(X).

C2: The set of almost periodic functions is dense in L^2(X)

C3: For all f \in L^2(X), \ \varepsilon, \delta > 0, there exists D \subseteq Y, \ \nu(D) > 1- \delta, \ g_1, \cdots, g_k \in L^2(X) s.t. for any \gamma \in \Gamma and almost every y \in Y, we have

\displaystyle \min_{1 \leq i \leq k} || \gamma (f)|_{f^{-1}(D)} - g_i||_y < \varepsilon

C4: For all f \in L^2(X), \ \varepsilon, \delta > 0, there exists g_1, \cdots, g_k \in L^2(X) s.t. for any \gamma \in \Gamma, there is a set D \subseteq Y, \ \nu(D) > 1- \delta, for all y \in D

\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon

C5: For all f \in L^2(X), let \bar{f} \in L^1(X \times_Y X) where

\bar{f}: (x_1, x_2) \mapsto f(x_1) \cdot f(x_2)

Let I=(I_k) be a Folner sequence, then \bar{f}=\bar{0} iff P \bar{f} = \bar{0}.

Theorem: All five definitions are equivalent.

Proof: “C1 \Rightarrow C2″

Since almost periodic functions are closed under linear combination, it suffice to show any element in a set of basis is approximated arbitrarily well by almost periodic functions.

Let our basis be as given in C1.

For all H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X), the associated linear operator \varphi_H: L^2(X) \rightarrow L^2(X) where \varphi_H: f \mapsto H \ast f is bounded. Hence it suffice to check H \ast f for a dense set of f \in L^2(X). We consider the set of all fiberwise bounded f i.e. for all y \in Y, ||f||_y \leq M_y.

For all \delta > 0, we perturb H \ast f by multiplying it by the characteristic function of a set of measure at least 1- \delta to get an almost periodic function.

“C2 \Rightarrow C3″:

For any f \in L^2(X), there exists f' almost periodic, with ||f-f'||< \frac{\epsilon \sqrt{\delta}}{2} . Let \{ g_1, g_2, \cdots, g_{k-1} \} be the functions obtained from the almost periodicity of f' with constant \varepsilon/2, g_k = \bar{0}.

Let D = \{ y \ | \ ||f-f'||_y < \varepsilon/2 \}, since

|| f - f'||^2 = \int ||f-f'||_y^2 d \nu(y)

Hence ||f-f'||< \frac{\varepsilon \sqrt{\delta}}{2} \ \Rightarrow \ ||f-f'||^2 < \frac{\varepsilon^2 \delta}{4}, \{ y \ | \ ||f-f'||_y > \varepsilon/2 \} has measure at most \delta/2, therefore \nu(D) > 1- \delta.

For all \gamma \in \Gamma, ify \in \gamma^{-1}(D) then

|| \gamma f|_{\alpha^{-1}(D)} - \gamma f'||_y = ||f|_{\alpha^{-1}(D)} - f'||_{\gamma(y)} < \varepsilon /2

Hence \displaystyle \min_{1 \leq i \leq k-1} ||\gamma f|_{\alpha^{-1}(D)} - g_i||_y < \varepsilon /2 + \varepsilon /2 = \varepsilon

If y \notin \gamma^{-1}(D) then f|_{\alpha^{-1}(D)} vanishes on \alpha^{-1}(\gamma y) so that || \gamma f|_{\alpha^{-1}(D)} - g_i||_y = 0 < \varepsilon.

Hence \alpha satisfies C3.

“C3 \Rightarrow C4″:

This is immediate since for all y \in \gamma^{-1}(D), we have \gamma f = \gamma f|_{\alpha^{-1}(D)} on \alpha^{-1}(y) hence

\displaystyle \min_{1 \leq i \leq k} ||\gamma f - g_i||_y < \min_{1 \leq i \leq k-1} ||\gamma f_{\alpha^{-1}(D)} - g_i||_y < \varepsilon

\nu(\gamma^{-1}(D)) = \nu(D) > 1-\delta. Hence \alpha satisfies C4.

“C4 \Rightarrow C5″:

For all f \in L^2(X), \ \varepsilon, \delta > 0, by C4, there exists g_1, \cdots, g_k \in L^2(X) s.t. for any \gamma \in \Gamma, there is a set D \subseteq Y, \ \nu(D) > 1- \delta, for all y \in D

\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon

W.L.O.G. we may suppose all g_i are bounded since by making \delta slighter larger we can modify the unbounded parts to be bounded.

\bar{g_j} \otimes g_j \in L^\infty(X \times_Y X), suppose P(\bar{f}) = 0.

Recall in C5 we have \bar{f}: (x_1, x_2) \mapsto f(x_1) \cdot f(x_2), and \displaystyle P_I \bar{f}(x_1, x_2) = \lim_{k \rightarrow \infty} \frac{1}{|I_k|} \sum_{\gamma \in I+k} f(\gamma x_1) \bar{ f(\gamma x_2)}.

For each 1 \leq j \leq k, we have \int (\bar{g_j} \otimes g_j) \cdot P \bar{f} d(\mu \times_Y \mu) = 0

Hence we have \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \int (\bar{g_j(x_1)} g_j(x_2)) \cdot \gamma f(x_1) \bar{\gamma f(x_2)} d\mu \times_Y \mu(x_1, x_2) = 0

\Rightarrow \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) = 0

\Rightarrow \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \{ \sum_{j=1}^k \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) \} = 0

Hence for large enough i, there exists \gamma \in I_i s.t. \sum_{j=1}^k \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) is as small as we want.

We may find D' \subseteq Y with \nu(D) > 1-\delta s.t. for all y \in D' and for all j, we have

| \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 < \varepsilon^2

On the other hand, by construction there is j with || \gamma f - g_j||^2_y < \varepsilon^2 for all y \in D, with \nu(D) > 1-\delta .

Hence for y \in D \cap D', \ ||f||_{\gamma'^{-1}(y)}^2 = || \gamma f||_y^2 < 3 \varepsilon^2.

Let \varepsilon \rightarrow 0, \ \delta \rightarrow 0 we get f = \bar{0}. Hence C5 holds.

“C5 \Rightarrow C1″

Let f \in L^2(X) orthogonal to all of such functions. Let (I_k) be a Folner sequence.

Define \displaystyle H(x_1, x_2) := \lim_{i \rightarrow \infty} \frac{1}{|I_i|}\sum_{\gamma \in I_i} \gamma f(x_1) \cdot \gamma f(x_2) = P \bar{f}(x_1, x_2)

Let H_M(x_1, x_2) be equal to H whenever H(x_1, x_2) \leq M and 0 o.w.

H is \Gamma-invariant \Rightarrow \ H_M is \Gamma-invariant and bounded.

Therefore f \bot H_M \ast f, i.e.

\int \bar{f(x_1)} \{ \int H_M(x_1, x_2) d \mu_{\alpha(x_1)}(x_2) \} d \mu(x_1) = 0 <\p>

Since \mu = \int \mu_y d \nu(y), we get

\int \bar{f} \otimes f \cdot H_M d \mu \times_Y \mu = 0 <\p>

Hence H_M \bot (\bar{f} \otimes f). For all \gamma \in \Gamma, \ \gamma (\bar{f} \otimes f) \bot \gamma H_M = H_M.

Since H = P \bar{f} is an average of \gamma (\bar{f} \otimes f), \ \Rightarrow \ H \bot H_M.
0 = \int \bar{H} \cdot H_M = \int |H_M|^2 \ \Rightarrow \ H_M = \bar{0} for all M

Hence H = \bar{0}. By C5, we obtain f = \bar{0}. Hence \{ H \ast f \ | \ H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X), f \in L^2(X) \} contain a basis for L^2(X).

Definition: Let H be a subgroup of \Gamma, \alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma') is said to be compact relative to H if the extension \alpha: (X, \mathcal{B}, \mu, H) \rightarrow ( Y, \mathcal{D}, \nu, H') is compact.

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