Proof. The assertions in (i) and (ii) follow by standard properties of conditional expectations.

As for (iii), we follow the proof of the classical Cauchy–Schwarz inequality with some necessary modifications. First suppose that $f,g\in \mathtt {L^2(X|Y)}$ satisfy $\|f\|_{L^2(X|Y)}, \|g\|_{L^2(X|Y)},\|f-ag\|_{L^2(X|Y)}>0$ for all $a\in L^0(Y)$ . Then we have

$$ \begin{align*} 0<\|f-ag\|_{L^2(X|Y)} =\|f\|_{L^2(X|Y)}^2 - a\langle f,g\rangle_{L^2(X|Y)} - \bar{a}\langle f,g\rangle_{L^2(X|Y)} + a\bar a\|g\|_{L^2(X|Y)}^2 \end{align*} $$

for all $a\in L^0(Y)$ . Setting

$$ \begin{align*}a=\frac{|\langle f,g\rangle_{L^2(X|Y)}|^2}{\|f'\|_{L^2(X|Y)}^2\langle f,f'\rangle_{L^2(X|Y)}}\end{align*} $$

and after some elementary algebraic manipulations, we obtain equation (7) in this case.

Second suppose that $f,g\in \mathtt {L^2(X|Y)}$ satisfy only $\|f\|_{L^2(X|Y)},\|g\|_{L^2(X|Y)}>0$ . Let

$$ \begin{align*} \mathcal{E}=\{E\in \mathcal{B}a(\tilde{Y}): \text{ there exists } a\in L^0(Y), f=ag \text{ on } E\}. \end{align*} $$

By completeness of Y (see Remark 2.2), there exists a least upper bound $E_{\max }$ for $\mathcal {E}$ (with respect to almost sure inclusion). By the countable chain condition, there is $\mathcal {E}'\subset \mathcal {E}$ countable such that $E_{\max }=\bigcup \mathcal {E}'$ . We can assume that $\mathcal {E}'$ is a Y-partition of $E_{\max }$ . Choose $a\in L^0(Y)$ such that $a=a_E$ on E with $a_E$ such that $f=a_E g$ for all $E\in \mathcal {E}'$ . Then $f=ag$ on $E_{\max }$ . It holds that $\|f\|_{L^2(X|Y)},\|g\|_{L^2(X|Y)},\|f-ag\|_{L^2(X|Y)}>0$ on $E_{\max }^c$ for all $a\in L^0(Y)$ . Thus, equation (7) is satisfied on $E_{\max }^c$ by repeating the first step on $E_{\max }^c$ . Since equation (7) is also trivially satisfied on $E_{\max }$ , we obtain equation (7) also in this case.

Finally, suppose that $\|f\|_{L^2(X|Y)}$ or $\|g\|_{L^2(X|Y)}$ are equal to $0$ on a set of positive measure. Let $E=\{\|f\|_{L^2(X|Y)}=0\}\cup \{ \|g\|_{L^2(X|Y)}=0\}$ . Then equation (7) is satisfied on $E^c$ by repeating the second step on $E^c$ . Since equation (7) is also trivially satisfied on E, we obtain equation (7) also in this case, and this completes the proof of (iii).

It is enough to prove (iv) for $K\in \mathtt {HS(X|Y)}$ with $K\geq 0$ . By the Stone–Weierstraß theorem, the algebra of finite disjoint unions of rectangles $E\times F$ with $E,F\in \mathcal {B}a(\tilde {X})$ generates $\mathcal {B}a(\tilde {X}\times \tilde {X})$ . Hence there is a sequence $(K_n)$ in $\mathcal {D}$ such that $K_n\uparrow K \tilde \mu \times _{\tilde Y}\tilde \mu $ -almost surely as $n\to \infty $ . By monotone convergence, $\|K_n- K\|_{\mathtt {HS}(X|Y)}\to 0\ \tilde \nu $ -almost surely. Hence $\|K_n-K\|_{\mathtt {HS}(X|Y)}\to 0$ in convergence in $\tilde \nu $ -measure, and this proves (iv).

As for (v), let $K\in \mathtt {HS(X|Y)}$ and $f\in \mathtt {L^2(X|Y)}$ . By the conditional Cauchy–Schwarz inequality and the Fubini–Tonelli theorem, one obtains

$$ \begin{align*} \int_{\tilde{X}}| K\ast_Y f |^2(x) \,d\mu_y(x) &= \int_{\tilde{X}}|\langle K(x,\cdot),\bar{f}\rangle_{L^2(X|Y)}(\tilde{\pi}(x)) |^2 \,d\mu_y(x) \\ &\leq \|f\|_{L^2(X|Y)}^2\int_{\tilde{X}} \|K(x,\cdot)\|_{L^2(X|Y)}^2(\tilde{\pi}(x)) \,d\mu_y(x) \\ &=\|K\|_{\mathtt{HS}(X|Y)}^2 \|f\|_{L^2(X|Y)}^2 <\infty. \end{align*} $$

This shows that $K\ast _Y f\in \mathtt {L^2(X|Y)}$ , equation (8), and continuity.

As for (vi), we only prove the claim for $\mathtt {L^2(X|Y)}$ (the arguments for $\mathtt {HS(X|Y)}$ are identical). For $f\in L^2(X)$ , it holds

(9) $$ \begin{align} \int_{\tilde{X}} f \,d\tilde\mu=\int_{\tilde{Y}}\int_{\tilde{X}} f \,d\mu_y \,d\tilde\nu. \end{align} $$

In particular, we have

(10) $$ \begin{align} \|f\|^2_{L^2(X)}=\int_{Y}\|f\|_{L^2(X|Y)}^2 \,d\nu. \end{align} $$

The assertion follows from the fact that convergence in $L^2$ is equivalent to convergence in measure.

Finally, as for (vii), let $f\in \mathtt {L^2(X|Y)}$ . Define $f_m=f 1_{\|f\|_{L^2(X|Y)}\leq m}$ for $m\geq 1$ . Then $\mathtt {d}_{\mathtt {L^2(X|Y)}}(f,f_m)\to 0$ as $m\to \infty $ . By (ii) and equation (9), $f_m\in L^2(X)$ . As $L^\infty (X)$ is dense in $L^2(X)$ in $L^2$ -topology, we can approximate each $f_m$ by a sequence $(f_{m,n})$ in $L^\infty (X)$ in $L^2$ -metric. The claim now follows from equation (9), (vi), and triangle inequality.

Proof. Let $f_1,\ldots ,f_n\in \mathtt {L^2(X|Y)}$ be a collection of generators of $\mathcal {M}$ . Define

$$ \begin{align*}h_1=\frac{f_1}{\|f_1\|_{L^2(X|Y)}} 1_{\|f_1\|_{L^2(X|Y)}>0}.\end{align*} $$

Next suppose we defined $h_1,\ldots ,h_k$ with $k\leq n-1$ . Set

$$ \begin{align*}g_{k+1}=f_{k+1}-\sum_{i=1}^k \langle f_{k+1},h_i\rangle_{L^2(X|Y)} h_i\end{align*} $$

and define

$$ \begin{align*} h_{k+1}=\frac{g_{k+1}}{\|g_{k+1}\|_{L^2(X|Y)}} 1_{ \|g_{k+1}\|_{L^2(X|Y)}>0}. \end{align*} $$

Denote by $F_i=\{ \|g_{i}\|_{L^2(X|Y)}>0\}$ and $F_{i+n}=F_i^c$ for all $i=1,\ldots , n$ . Form all finite intersections $F_{i_1}\cap F_{i_2}\cap \cdots \cap F_{i_k}$ with $1\kern-1pt\leq i_1\kern-1pt<i_2<\cdots <i_k\kern-1pt\leq 2n$ for some $1\kern-1pt\leq k\kern-1pt\leq 2n$ . Let $\mathcal {P}_0$ denote the set of all such finite intersections whose measure is positive. Then $\mathcal {P}_0$ forms a Y-partition. For $E=F_{i_1}\cap F_{i_2}\cap \cdots \cap F_{i_k}\in \mathcal {P}_0$ , let $\mathcal {F}_{E}=\{h_{i_j}: i_j\leq n, j=1,\ldots ,k\}$ . Let $\mathcal {P}$ consist of those $E\in \mathcal {P}_0$ such that $\mathcal {F}_E$ is not empty. Set $E_0=(\bigcup _{E\in \mathcal {P}} E)^c$ . By construction, $\mathcal {P}$ , $(\mathcal {F}_E)_{E\in \mathcal {P}}$ , and $E_0$ satisfy properties (i)–(iv).

Proof. Let $\mathcal {P}$ and $(\mathcal {F}_E)_{E\in \mathcal {P}}$ satisfy properties (i)–(iv) in Lemma 3.7 for $\mathcal {M}$ . Let $(f_n)$ be a sequence in $\mathcal {M}$ such that $\mathtt {d}_{\mathtt {L^2(X|Y)}}(f_n,f)\to 0$ as $n\to \infty $ for some $f\in \mathtt {L^2(X|Y)}$ . Each $f_n$ is of the form

$$ \begin{align*}f_n=\sum_{E\in \mathcal{P}}\bigg(\sum_{f_E\in \mathcal{F}_E}a_{f_E}^n f_E \bigg)|E\end{align*} $$

for some $(a_{f_E}^n)_{f\in \mathcal {F}_E}$ for each $E\in \mathcal {P}$ . For arbitrary $n,m$ , by Proposition 3.4(ii) and Proposition 3.7(ii) and (iii),

(11) $$ \begin{align} \|f_n-f_m\|_{L^2(X|Y)}=\sum_{E\in \mathcal{P}}\bigg(\sum_{f_E\in \mathcal{F}_E}|a_{f_E}^n-a_{f_E}^m|\bigg)|E. \end{align} $$

Since $\mathtt {d}_{\mathtt {L^2(X|Y)}}(f_n,f_m)\to 0$ as $n,m\to \infty $ by hypothesis, the claim follows by applying [Reference Kallenberg35, Lemma 4.6] to (11).

Proof. To prove (ii)′ from (i)′, it suffices to show that the union of the closed and finitely generated $\Gamma $ -invariant $L^\infty (Y)$ -submodules of the range of $K\ast _Y\colon L^2(X)\to L^2(X)$ is dense in the range of $K\ast _Y$ for any $\Gamma $ -invariant $K\in L^\infty (X\times _Y X)$ . (We thank the anonymous referee for pointing out a gap in a previous version of this argument. The current arguments are borrowed from a forthcoming work joint with Pieter Spaas, where we establish a non-commutative analog of several results in (uncountable) Furstenberg–Zimmer structure theory.) By splitting into real and imaginary parts and then further into positive and negative parts, we can assume that $K\in L^\infty (X\times _Y X)$ is non-negative. Note that $K\ast _Y\colon L^2(X)\to L^2(X)$ is then a bounded and positive operator (cf. Lemma A.5) and thus its spectrum is compact and contained in $[0,+\infty )$ . For $\varepsilon>0$ , consider the spectral projection $P_\varepsilon := 1_{[\varepsilon , \|K\|_{L^\infty (X\times _YX)}]}(K\ast _Y)$ , where we apply Borel functional calculus to $K\ast _Y$ as an operator from $L^2(X)$ to $L^2(X)$ to define $P_\varepsilon $ . Since $P_\varepsilon $ arises as a limit of polynomials in $K\ast _Y$ in the strong operator topology, $P_\varepsilon $ is $\Gamma $ -invariant. Thus its range $\mathcal {H}_\varepsilon := P_\varepsilon (L^2(X))$ is a $\Gamma $ -invariant $L^\infty (Y)$ -submodule of $L^2(X)$ (this follows from that $L^\infty (Y)$ -linearity is preserved when passing to strong operator limits as well). By standard properties of the Borel functional calculus, $P_\varepsilon \circ K = K\circ P_\varepsilon = K\vert _{\mathcal {H}_\varepsilon } \geq \varepsilon \mathrm {Id}_{\mathcal {H}_\varepsilon }$ (where the inequality is understood in the sense of operators on the Hilbert space $\mathcal {H}_\varepsilon $ ). Now we can apply Proposition A.6 and obtain that $\mathcal {H}_\varepsilon $ is necessarily finitely generated as an $L^\infty (Y)$ -module. Letting $\varepsilon \to 0$ , we get that the increasing union of the finitely generated $\Gamma $ -invariant $L^\infty (Y)$ -submodules of the range of K is dense in the range of K. This concludes the proof of (i)′ $\Rightarrow $ (ii)′.

Proof. Let $\mathfrak {A}$ be the union of finitely generated and $\Gamma $ -invariant $L^0(Y)$ -submodules and let $\mathcal {G}=\{f\in \mathfrak {A}: \text {there exists } M>0 \text {such that} \|f\|_{L^2(X|Y)}< M\}$ . By assumption and Lemma 3.8, $\mathcal {G}$ is dense in $\mathtt {L^2(X|Y)}$ with respect to the metric $\mathtt {d}_{\mathtt {L^2(X|Y)}}$ .

Fix $f\in \mathcal {G}$ and $\varepsilon>0$ . Then there is a finitely generated and $\Gamma $ -invariant $L^0(Y)$ -submodule $\mathcal {M}$ such that $f\in \mathcal {M}$ . By Proposition 3.7, there are a finite Y-partition $\mathcal {P}$ and for each $E\in \mathcal {P}$ , a finite family $\mathcal {F}_E$ such that for all $f\in \mathcal {M}$ , we have the representation $f=\sum _{E\in \mathcal {P}} f_E1_E$ , where $f_E=\sum _{g\in \mathcal {F}_E} a_{g} g$ is an $L^0(Y)$ -linear combination for each $E\in \mathcal {P}$ . Since $\|(T^\gamma )^*f\|_{L^2(X|Y)}=(T^\gamma )^*\|f\|_{L^2(X|Y)} <M$ for all $\gamma \in \Gamma $ , the orbit $\mathtt {Orb}_\Gamma (f)$ is bounded with respect to the conditional norm $\|\cdot \|_{L^2(X|Y)}$ . It follows from Proposition 3.4(ii) and Proposition 3.7 that for all $h\in \mathtt {Orb}_\Gamma (f)$ ,

$$ \begin{align*} \|h\|_{L^2(X|Y)}&=\bigg\|\sum_{E\in\mathcal{P}} h_E 1_E\bigg\|_{L^2(X|Y)}=\sum_{E\in\mathcal{P}}\|h_E\|_{L^2(X|Y)} 1_E\\ &=\sum_{E\in\mathcal{P}} \bigg(\sum_{g\in\mathcal{F}_E} |a^h_{g}|^{2}\bigg)^{1/2} 1_E <M. \end{align*} $$

For each $E\in \mathcal {P}$ , choose $c_1^E,\ldots , c_{j_E}^E\in \mathbb {C}^{\#(\mathcal {F}_{E})}$ such that the closed ball in $\mathbb {C}^{\#(\mathcal {F}_{E})}$ with radius M and center the origin is covered by the union of the open balls of radius $\varepsilon $ and center $c_k^E$ , $k=1,\ldots , j_E$ . Denote by $C_k^E$ the constant measurable vector in $L^0(Y)^{\#(\mathcal {F}_{E})}$ with value $c_k^E$ . For $f\in \mathcal {F}_{E}$ , let $C_k^E(f)$ denote a component of the vector $C_k^E$ . Put

$$ \begin{align*} \mathcal{L}_E=\bigg\{\sum_{f\in\mathcal{F}_{E}} C_k^E(f) f: k=1,\ldots, j_E\bigg\} \end{align*} $$

and define

$$ \begin{align*} \mathcal{L}= \bigg\{\sum_{E\in\mathcal{P}} g_E|E: g_E\in\mathcal{L}_E \text{ for each } E\in\mathcal{P} \bigg\}. \end{align*} $$

Clearly, $\mathcal {L}$ is finite and by construction, it holds that

$$ \begin{align*} \min_{g\in \mathcal{L}} \|g-f\|_{L^2(X|Y)}<\varepsilon.\\[-3pc] \end{align*} $$

Proof. The equivalence is a consequence of Proposition 3.4(v)–(vii).

Proof. We first show that (ii) implies (ii)′. Let $\mathcal {M}$ be an finitely generated and $\Gamma $ -invariant $L^0(Y)$ -submodule of $\mathtt {L^2(X|Y)}$ . For $\mathcal {M}$ , choose a finite Y-partition $\mathcal {P}$ and for each $E\in \mathcal {P}$ , a finite set $\mathcal {F}_E$ satisfying the properties in Proposition 3.7. Let

$$ \begin{align*} \mathcal{L}=\bigg\{\sum_{E\in\mathcal{P}} g_E|E: g_E\in \mathcal{F}_E \text{ each } E\in\mathcal{P}\bigg\}. \end{align*} $$

By property (ii) in Proposition 3.7, $\|g\|_{L^2(X|Y)}\leq 1$ for each $g\in \mathcal {L}$ . Thus by equation (10), $\|g\|_{L^2(X)}\leq 1$ for all $g\in \mathcal {L}$ . Let $\mathcal {M}'$ be the $L^\infty (Y)$ -submodule of $L^2(X)$ generated by $\mathcal {L}$ . Then $\mathcal {M}'$ is $\Gamma $ -invariant since $\mathcal {M}$ is $\Gamma $ -invariant, and by Proposition 3.4(vi), $\mathcal {M}'$ is also $L^2$ -closed. It remains to show that the union of all $\mathcal {M}'$ , where $\mathcal {M}$ is a finitely generated and $\Gamma $ -invariant $L^0(Y)$ -submodule of $\mathtt {L^2(X|Y)}$ , is dense in $L^2(X)$ . However, this follows from Proposition 3.4(v) and (vii).

Let us show that (ii)′ implies (ii). Let $\mathcal {M}'$ be an $L^\infty (Y)$ -finitely generated, $\Gamma $ -invariant, and closed submodule of $L^2(X)$ . Let $\mathcal {L}$ be a finite set of generators of $\mathcal {M}'$ . Let $\mathcal {M}=\{\sum _{g\in \mathcal {L}} a_g g: a_g\in L^0(Y) \text { for all } g \in \mathcal {L}\}$ . By construction and Proposition 3.4(ii), $\mathcal {M}$ is a finitely generated and $\Gamma $ -invariant $L^0(Y)$ -submodule of $\mathtt {L^2(X|Y)}$ . By Proposition 3.4(vii), the union of all such $\mathcal {M}$ is dense in $\mathtt {L^2(X|Y)}$ with respect to $\mathtt {d}_{\mathtt {L^2(X|Y)}}$ .

Proof. Let $\mathcal {G}$ be a dense set in $\mathtt {L^2(X|Y)}$ satisfying the property in (iii). Let $f\in \mathcal {G}$ and $M>0$ . By Proposition 3.4(ii), $f 1_{\|f\|_{L^2(X|Y)}\leq M}$ is also an element of $\mathcal {G}$ . Then $\mathcal {R}=\{f 1_{\|f\|_{L^2(X|Y)}\leq M}: f\in \mathcal {S}, M>0\}$ is a subset of $L^2(X)$ by Proposition 3.4(v) and dense in $\mathtt {L^2(X|Y)}$ . Now (iii)′ follows from Proposition 3.4(vi). The converse direction is an immediate consequence of Proposition 3.4(vii).

Proof. We show that if $f\in \mathcal {H}$ is orthogonal to all functions of the form $K\ast _Y g$ , where K ranges over $\Gamma $ -invariant functions in $L^\infty (X\times _Y X)$ and g ranges over $L^2(X)$ , then $f=0$ . Let K be the unique element of minimal norm in the closed convex hull of $\mathtt {Orb}_\Gamma (f\otimes \bar f)$ in $L^2(X\times _Y X)$ , which is $\Gamma $ -invariant by Theorem 2.9. Set $K_M=K1_{|K|\leq M}$ for $M\geq 0$ . By hypothesis, f is orthogonal to $K_M\ast _Y \bar f$ . We can rewrite this, using the disintegration $(\mu _y)_{y\in \tilde {Y}}$ of $\tilde {\mu }$ , as

$$ \begin{align*} 0&=\int_{\tilde{X}} f(x) \bigg(\int_{\tilde{X}} K_M(x,x') \bar f(x') \,d\mu_{\tilde{\pi}(x)}(x')\bigg) \,d\tilde{\mu}(x) \\[3pt] &= \int_{\tilde{Y}} \int_{\tilde{X}} f(x) \bigg(\int_{\tilde{X}} K_M(x,x') \bar f(x') \,d\mu_y(x')\bigg) \,d\mu_y(x) \,d\tilde{\nu}(y) \\[3pt] &= \int_{\tilde{X}\times\tilde{X}} f\otimes \bar f K_M \,d\tilde\mu\times_{\tilde{Y}} \tilde\mu. \end{align*} $$

Thus $ f\otimes \bar f$ is orthogonal to $K_M$ in $L^2(X\times _Y X)$ . As $K_M$ is $\Gamma $ -invariant, $(T^\gamma \times T^\gamma )^* f\otimes \bar f$ is orthogonal to $K_M$ for all $\gamma \in \Gamma $ as well. Thus K must be orthogonal to all $K_M$ which implies that $K_M=0$ for all $M\geq 0$ , and therefore also $K=0$ .

Now choose a finite set $\mathcal {F}$ such that the property in (iii)′ is satisfied for f and an arbitrary $\varepsilon>0$ . Let $(K_n)$ be a sequence in the convex hull of $\mathtt {Orb}_\Gamma (f\otimes \bar f)$ such that $\lim \|K_n\|_{L^2(X\times _Y X)}\to 0$ . By Cauchy–Schwarz, we also have $\langle K_n, g\otimes \bar g\rangle _{L^2(X\times _Y X)}\to 0$ for all $g\in \mathcal {F}$ . Expanding the inner product and choosing n large enough, we can always obtain a $\gamma \in \Gamma $ such that

$$ \begin{align*} \sum_{g\in\mathcal{F}} \|\langle (T^{\gamma})^*f,g\rangle_{L^2(X|Y)}\|_{L^2(Y)} \end{align*} $$

is arbitrarily small. Thus for any $\delta>0$ , we can find a set $E\in \mathcal {B}a(\tilde {Y})$ with $\tilde \nu (E^c)<\delta $ such that

$$ \begin{align*} |\langle (T^{\gamma})^*f,g\rangle_{L^2(X|Y)}| <\varepsilon \quad \text{on } E \end{align*} $$

for all $g\in \mathcal {F}$ for some $\gamma $ . By (iii)′, we also have for all $\gamma \in \Gamma $ ,

$$ \begin{align*} \min_{g\in\mathcal{F}} \|(T^{\gamma})^*f -g\|_{L^2(X|Y)} <\varepsilon. \end{align*} $$

By triangle inequality,

$$ \begin{align*} \|f\|_{L^2(X|Y)} 1_{T^{-\gamma}(E)}=\|(T^\gamma)^*f\|_{L^2(X|Y)} 1_{E}< 3 \varepsilon \end{align*} $$

for some $\gamma $ . Since $\varepsilon ,\delta $ were chosen arbitrarily, it follows that $f=0$ , and this finishes the proof of Theorem 4.1.

Proof. The first identity is proved in [Reference Jamneshan and Tao33, Corollary 3.5]. Routine verification shows that

$$ \begin{align*}\prod_{\alpha\in A} (K_\alpha/L_\alpha)\equiv\prod_{\alpha\in A} K_\alpha / \prod_{\alpha\in A} L_\alpha\end{align*} $$

is a $\mathbf {CH}$ -isomorphism. This proves the second identity.

Proof of Theorem 5.3

Let $\pi :(X,\mu ,T)\to (Y,\nu ,S)$ be a relatively compact ${\mathbf {PrbAlg}_\Gamma }$ - extension of ergodic systems. Let $(\mathcal {M}_\alpha )_{\alpha \in A}$ be the family of all $\Gamma $ -invariant closed finitely generated $L^\infty (Y)$ -submodules of $L^2(X)$ . By Theorem 4.1(ii)′, $L^2(X)$ is the $L^2$ closure of $\bigcup _{\alpha \in A} \mathcal {M}_\alpha $ . Fix $\alpha \in A$ . Since $\mathcal {M}_\alpha $ is $\Gamma $ -invariant, the map $\mathtt {cdim}(\mathcal {M}_\alpha )$ is also $\Gamma $ -invariant and by ergodicity, $\mathtt {cdim}(\mathcal {M}_\alpha )\equiv d_\alpha $ for some integer $d_\alpha \geq 1$ . By Proposition 3.7, we find $f^\alpha _1,\ldots ,f^\alpha _{d_\alpha }\in \mathcal {M}_\alpha $ such that $\|f^\alpha _i\|_{L^2(X|Y)}=1$ for all $i=1,\ldots ,d_\alpha $ and $\langle f^\alpha _i,f^\alpha _j\rangle _{L^2(X|Y)}=0$ whenever $i\neq j$ . In fact, we can choose $f^\alpha _1,\ldots ,f^\alpha _{d_\alpha }$ such that all $f^\alpha _i$ take values in the unit circle $\mathbb {S}$ . For any $\gamma \in \Gamma $ , then $(\tilde T^\gamma )^*f^\alpha _1,\ldots , (\tilde T^\gamma )^*f^\alpha _{d_\alpha }$ also satisfies $\|(\tilde T^\gamma )^*f^\alpha _i\|_{L^2(X|Y)}=1$ for all $i=1,\ldots ,d_\alpha $ and $\langle (\tilde T^\gamma )^*f^\alpha _i,(\tilde T^\gamma )^*f^\alpha _j\rangle _{L^2(X|Y)}=0$ whenever $i\neq j$ . Let

$$ \begin{align*}\Lambda_\gamma^\alpha= \begin{bmatrix} a_{11}^\gamma & \cdots & a_{1d_{\alpha}}^\gamma \\ \vdots & \ddots & \vdots \\ a_{d_{\alpha}1}^\gamma & \cdots & a_{d_{\alpha}d_{\alpha}}^\gamma \end{bmatrix} \end{align*} $$

where $a^\gamma _{ij}\in L^\infty (Y)$ is such that $f^\alpha _i=\sum _{j=1}^{d_\alpha } a_{ij}^\gamma (\tilde T^\gamma )^*f^\alpha _j$ for all $i=1,\ldots ,d_\alpha $ . We can identify $\Lambda _\gamma ^\alpha $ with an element of the function space $L^0(Y\to \mathbb {U}(d_{\alpha }))$ of equivalence classes of $\mathbb {U}(d_{\alpha })$ -valued random variables on $\tilde {Y}$ , where $\mathbb {U}(d_{\alpha })$ is the group of $d_{\alpha }\times d_{\alpha }$ unitary matrices. We obtain the identity $F^{\alpha }=\Lambda _\gamma ^\alpha F^{\alpha }_\gamma $ , where $F^{\alpha }=(f^{\alpha }_1,\ldots ,f^{\alpha }_{d_{\alpha }})$ and $F^{\alpha }_\gamma =((\tilde T^\gamma )^*f^{\alpha }_1,\ldots , (\tilde T^\gamma )^*f^{\alpha }_{d_{\alpha }})$ . Let $\rho ^\alpha _\gamma := ((\Lambda _\gamma ^\alpha )^*)^{\operatorname {op}}$ , which is an element of $\mathtt {Cond}_Y(\mathbb {U}(d_{\alpha }))$ . We thus obtain a $\mathbb {U}(d_{\alpha })$ -valued ${\mathbf {PrbAlg}_\Gamma }$ -cocycle $\rho ^\alpha =(\rho ^\alpha _\gamma )_{\gamma \in \Gamma }$ on $(Y,\nu ,S)$ . Define the $\mathbf {AbsMbl}$ -morphism $\theta _\alpha :Y\to \mathcal {B}a(\mathbb {S}^{2d_{\alpha }-1})$ by $\theta _\alpha := ((({1}/{\sqrt {d_{\alpha }}}) F^\alpha )^*)^{\operatorname {op}}$ .

Now define $\rho =(\rho _\gamma )_{\gamma \in \Gamma }$ by $\rho _\gamma =(\rho _\gamma ^\alpha )_{\alpha \in A}$ , $\theta =(\theta _\alpha )_{\alpha \in A}$ , $K=\prod _{\alpha \in A} \mathbb {U}(d_\alpha )$ , and $L=\prod _{\alpha \in A} \mathbb {U}(d_\alpha -1)$ . By Lemma 5.4, $\theta \in \mathtt {Cond}_Y(K/L)$ is a vertical coordinate such that $\pi ,\theta $ jointly generate the ${\mathbf {Bool}_\sigma }$ -algebra X, and $\rho =(\rho _\gamma )_{\gamma \in \Gamma }$ is a ${\mathbf {PrbAlg}_\Gamma }$ -cocycle such that $\theta \circ T^\gamma = (\rho _\gamma \circ \pi )\theta $ . The claim follows from Theorem 5.2.

Proof. If (i) is false, there are $f\in L^2(X)$ with $\mathbb {E}(f|Y)=0$ and $\varepsilon>0$ such that for all $\gamma \in \Gamma $ ,

(14) $$ \begin{align} \langle (T^\gamma)^*f\otimes \bar f, f\otimes \bar f\rangle_{L^2(X\times_Y X)}=\|\langle (T^\gamma)^*f,f\rangle_{L^2(X|Y)}\|^2_{L^2(Y)}\geq \varepsilon. \end{align} $$

Let K be the unique element of minimal norm in the closed convex hull of $\mathtt {Orb}_\Gamma (f\otimes \bar f)$ . By Theorem 2.9, K is $\Gamma $ -invariant, and by equation (14), K is non-trivial. Since $\mathbb {E}(f|Y)=0$ , there must exist $g\in L^2(X)$ such that $K\ast _Y g\in L^2(X)\backslash L^2(Y)$ . Let $\mathcal {A}=\{K\ast _Y g\in L^\infty (X): g\in L^2(X)\}$ . By Theorem 4.1, every $K\ast _Y g\in \mathcal {A}$ is contained in a closed $\Gamma $ -invariant finitely generated $L^\infty (Y)$ -submodule of $L^2(X)$ . It is not difficult to check that $\mathcal {A}$ is a $\Gamma $ -invariant vector subspace of $L^\infty (X)$ closed under complex conjugation and multiplication. Thus $\mathcal {A}$ has the structure of a ${\mathbf {CvNAlg}^\tau _{\Gamma ^{\operatorname {op}}}}$ -system such that $\Phi :\mathcal {A}\to L^\infty (X,\mu ,T)$ and $\Psi :L^\infty (Y,\nu ,S)\to \mathcal {A}$ are ${\mathbf {CvNAlg}^\tau _{\Gamma ^{\operatorname {op}}}}$ -factors with $\Psi $ non-trivial. Then $(Z,\unicode{x3bb} , R):= \mathtt {Proj}(\mathcal {A})$ , $\phi := \mathtt {Proj}(\Phi )$ , and $\psi := \mathtt {Proj}(\Psi )$ satisfy (ii).

Now let f be relatively weakly mixing and $g\in \mathtt {AP}_{X|Y}$ . Without loss of generality, we may assume that f is bounded and g is an element of an invariant finitely generated $L^\infty (Y)$ -submodule $\mathcal {M}$ of $L^2(X)$ . For any $\gamma \in \Gamma $ and $h\in \mathcal {M}$ , we have

$$ \begin{align*} &\|\langle f,g\rangle_{L^2(X|Y)}\|_{L^2(Y)}\\[3pt] &\quad=\|\langle (T^\gamma)^*f,(T^\gamma)^*g\rangle_{L^2(X|Y)}\|_{L^2(Y)}\\[3pt] &\quad\leq \|\langle (T^\gamma)^*f,(T^\gamma)^*g - h\rangle_{L^2(X|Y)}\|_{L^2(Y)}+\|\langle (T^\gamma)^*f,h\rangle_{L^2(X|Y)}\|_{L^2(Y)} \\[3pt] &\quad\leq \|\|(T^\gamma)^*f\|_{L^2(X|Y)}\|(T^\gamma)^*g - h\|_{L^2(X|Y)}\|_{L^2(Y)} +\|\langle (T^\gamma)^*f,h\rangle_{L^2(X|Y)}\|_{L^2(Y)} \\[3pt] &\quad\leq \|f\|_{L^\infty(X)}\|\|(T^\gamma)^*g - h\|_{L^2(X|Y)}\|_{L^2(Y)} + \|\langle (T^\gamma)^*f,h\rangle_{L^2(X|Y)}\|_{L^2(Y)}, \end{align*} $$

where the second inequality follows from the conditional Cauchy–Schwarz inequality. For any $\delta>0$ , we can first choose h and then $\gamma $ such that the last term is $<\delta $ uniformly in $\gamma $ . Hence $\|\langle f,g\rangle _{L^2(X|Y)}\|_{L^2(Y)}=0$ , and thus $\langle f,g\rangle _{L^2(X)}=0$ . It remains to show that if $f\not \in \mathtt {WM}_{X|Y}$ , then f must correlate with some $g\in \mathtt {AP}_{X|Y}$ , but this follows by a similar argument as in the first part of the proof.

Proof. Let $\mathcal {Y}_0$ be the trivial ${\mathbf {PrbAlg}_\Gamma }$ -system, and let $\pi _0\colon \mathcal {X}\to \mathcal {Y}_0$ be the corresponding ${\mathbf {PrbAlg}_\Gamma }$ -factor map. By Theorem 6.2, $\pi _0\colon \mathcal {X}\to \mathcal {Y}_0$ is either a relatively weakly mixing ${\mathbf {PrbAlg}_\Gamma }$ -extension, in which case we set $\beta =0$ , or there are a ${\mathbf {PrbAlg}_\Gamma }$ -system $\mathcal {Z}$ and ${\mathbf {PrbAlg}_\Gamma }$ -extensions $\phi \colon \mathcal {X}\to \mathcal {Z}$ and $\psi \colon \mathcal {Z}\to \mathcal {Y}_0$ such that $\psi \colon \mathcal {Z} \to \mathcal {Y}_0$ is a non-trivial relatively compact ${\mathbf {PrbAlg}_\Gamma }$ -extension of $\mathcal {Y}_0$ . Then set $\mathcal {Y}_1:= \mathtt {AP_{X|Y_0}}$ , let $\pi _1\colon \mathcal {X}\to \mathcal {Y}_1$ and $\pi _{1,0}\colon \mathcal {Y}_1\to \mathcal {Y}_0$ be the corresponding ${\mathbf {PrbAlg}_\Gamma }$ -factors, and repeat the previous step with $\mathcal {X}$ and $\mathcal {Y}_1$ instead of $\mathcal {Y}_0$ using Theorem 6.2. This process may end here if $\pi _1\colon \mathcal {X}\to \mathcal {Y}_1$ is a relatively weakly mixing ${\mathbf {PrbAlg}_\Gamma }$ -extension and we set $\beta =1$ , or we may need to continue and repeat the previous step again. Now repeating the previous steps in a transfinite recursion, while passing to inverse limits at limit ordinals, this process will eventually terminate at an ordinal number $\beta $ .