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Background/Definitions
Let $(X, \mc X, \mu, T)$ be an invertible measure preserving system. A function $f\in L^2$ is said to be weakly-mixing if $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=0}^{N-1} |\ab{f, T^n f}|^2 = 0 $$
This definition can be given for a more abstract setting. Let $H$ be a Hilbert space equipped with a unitary operator $U:H\to H$. A point $x\in H$ is said to be weakly-mixing if $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=0}^{N-1} |\ab{x, T^nx}|^2 = 0 $$
Now suppose $\mc D$ and $\mc A$ are $T$-invariant sub-$\sigma$-algebras of $\mc X$ with $\mc A\subseteq \mc D$. Then we have a factor map $(X, \mc D, \mu, T)\to (X, \mc A, \mu, T)$ mapping $x$ to $x$. Thus $(X, \mc D)$ is an extension of $(X, \mc A)$. We define a conditional inner product on $L^2(\mc D)$ as $$ \ab{f, g}_{(\mc D|\mc A)} = \E[f\bar g| \mc A] $$ for all $f, g\in L^2(\mc D)$. Note that the "inner product" is valued in $L^1(\mc A)$ and not in $\C$. Let $L^2(\mc D|\mc A)$ be the subspace of $L^2(\mc D)$ consisting of all $f\in L^2(\mc D)$ such that $$ \norm{f}_{(\mc D|\mc A)} := \sqrt{\E[|f|^2|\mc A]} $$ is in $L^\infty(\mc A)$.
Definition. A function $f\in L^2(\mc D| \mc A)$ is said to be conditionally weakly-mixing if for each $g\in L^2(\mc D|\mc A)$ we have $$ \lim_{N\to \infty} \frac{1}{N}\sum_{n=0}^{N-1} \|\ab{T^nf, g}_{(\mc D|\mc A)}\|_{L^2(\mc A)}^2 = 0 $$
This is a natural conditional version of the non-conditional definition given above.
Question
Question. Is there a Hilbert space version of definition of conditionally weakly-mixing.
The first thing that comes to mind is to consider a Hilbert space $K$ equipped with a unitary operator $U$, and let $H$ be a $U$-invariant closed linear subspace of $K$. Thus $K$ mimics $L^2(\mc D)$ and $H$ mimics $L^2(\mc A)$.
Since the adjoint of the inclusion $L^2(\mc A)\to L^2(\mc D)$ is the conditional expectation map $\E[\cdot|\mc A]:L^2(\mc D)\to L^2(\mc A)$, we may define the conditional inner product of $x$ and $y$ in $K$ as $\ab{x, y}_{(K|H)} = \ab{i^*x, i^*y}_H$, where $i:H\to K$ is the inclusion map and $i^*$ is the adjoint of $i$. However, $\ab{x, y}_{(K|H)}$ is a complex number, unlike in the case of the conditional inner product. So perhaps we need more structure on the Hilbert space to come up with a proper definition.