"Uncertainty principle" for self-adjoint operators in a finite von Neumann algebra

Question

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a projection), and let $T\in M$ be another self-adjoint operator.

The question will be formulated in the case of $P$ being a projection. Let $\mathcal H_0$ and $\mathcal H_1$ be the eigenspaces of $P$.

Question: Is there a well-studied condition on $P$ and $T$ which would imply that there exists $\epsilon$ such that whenever $U\subset \mathcal H$ is a "generalized eigenspace" of $T$ (i.e. the image of a spectral projection of $T$) then there exists $u\in U$ of norm $1$ such that $u=h_1+h_2$, $h_1\in \mathcal H_1$, $h_2\in \mathcal H_2$, and the norm of $h_2$ is at least $\epsilon$?

In particular (EDIT: the previous version had a weeker ergodicity assumption, in which case the answer is negative, by the answer of Steven Deprez below)

Question: If a discrete group $\Gamma$ acts freely and in a measure preserving way on a probablity measure space $X$, each element of $\Gamma$ acts ergodically, $M=L^\infty(X) \rtimes \Gamma$, $P$ is the characteristic function of some subset of $X$ of positive measure, and $T = \sum_{\gamma\in \Gamma} f_\gamma \gamma$, where $f_\gamma$ are real-valued positive functions on $X$ such that $\sum f_\gamma=1$; do $T$ and $P$ fulfill the property from the question?

Answer 1

I don't know about your first question, but i can show that your second question fails in general.

Take any free ergodic pmp action of $\Gamma$ on $(X,\mu)$, but assume that there is a group element $g\in\Gamma$ that globally preserves some non-trivial subset $Y\subset X$. Set $P=\chi_Y$ and $T=(1-\chi_Y)+\frac12\chi_Y(u_g+u_g^\ast)$. For any subset $A$ of the spectrum of $T$, denote the corresponding spectral projection by $p_{T,A}$. Whenever $1\not\in A$, then $p_{T,A}\leq P$. Hence the $h_2$ from the first question is always $0$.

Probably the most easy example of such an action is the following: Consider the finite group $\mathbb{Z}/2\times\mathbb{Z}/2$ acting on itself by translation. For example, $(1,0)$ preserves the non-trivial subset $\{(0,0),(1,0)\}$.

The infinite tensor product gives an example where $M$ is the hyperfinite type II$_1$ factor.

Answer 2

A refinement of the argument from my previous answer:

For any free, pmp, ergodic action of any non-trivial countable group, $\Gamma\curvearrowright (X,\mu)$, there is a projection $P\in L^\infty(X,\mu)$ and a $T=\sum_{\gamma\in\Gamma}f_\gamma\gamma$ such that the $f_\gamma$ are positive functions adding to $1$, and such that $T$ and $P$ do not fulfill the property from the question.

We can assume that every nontrivial element of $\Gamma$ acts ergodically on $X$. In particular, $\Gamma$ does not have any finite subgroups and sice the action is free, $X$ is non-atomic. Let $e\not=g\in \Gamma$ be any element. Take any non-trivial subset $U_0\subset X$. Since $gU_0\not=U_0$, we see that $U_1=U_0\setminus gU_0$ is a non-null set. Now we know that $U_1\cap g^{-1}U_1=\emptyset$. Choose a subset $U\subset U_1$ with measure $\mu(U)<\frac14$. This set still satisfies $U\cap g^{-1}U=\emptyset$. Denote $V=X\setminus (U\cup g^{-1}U)$ and $W=V\cap gV$. Observe that $\mu(W)\geq 1-4\mu(U)>0$.

Now, set $T=\chi_V + \chi_U u_g + \chi_{g^{-1}}u_g^\ast$ and $P=1-\chi_W$. Observe that $T\chi_W=\chi_W$. It follows that every vector $\xi\in Ran(\chi_W)$ is an eigenvector of $T$ with eigenvalue $1$. For any subset $A$ of the spectrum of $T$, denote the corresponding spectral projection by $p_{T,A}$. It follows that $p_{T,A}\leq 1-\chi_W=P$ for every subset $1\not\in A\subset \sigma(T)$ of the spectrum, not containing $1$. This contradicts the property from the question.