Let $M$ be a finite von Neumann algebra equipped with a faithful finite normal trace $\tau$. Let $N$ be a von Neumann sub-algebra of $M$.
Let $E_\tau: M \to M$ be the faithful normal conditional expectation preserving the trace $\tau$ on $N$.
1) Let $y \in L^1(M)_+$. Let $E': M \to M$ be the faithful normal conditional expectation on $N$ preserving the normal state $\varphi : x \to \tau(yx)$. Does there exist a link between these two conditional expectations ?
2) Is it possible to characterize the states $\varphi : x \to \tau(yx)$ with $y \in L^1(M)_+$ preserved by $E_\tau$ ?
Best Answer
The second question admit (I think) a straightforward solution. Pick $x \in M_+$ and notice that, if $E$ preserves $\varphi$, we have that $$ \tau(y \, x) = \varphi(x) = \varphi(E(x)) = \tau( y \, E(x) ) = \tau( E(y) \, x) $$ since that holds for every positive $x \in M$ we have that $y = E(y)$ and so $y \in L^1(N)$. The other direction is trivial.
Question 1 has a very clear answer when $\varphi(x) = \tau(\delta \, x)$ is a trace, ie $\delta$ is central, and satisfies that $\delta$ is invertible, ie $0 < \kappa 1 \leq \delta$. In that case, the natural choice is
$$ E_\varphi(x) = {E}_\tau(\delta)^{-\frac12} \, {E}_\tau( \delta^\frac12 \, x \, \delta^\frac12 ) \, {E}_\tau(\delta)^{-\frac12}. $$ the map is ucp and, by the centrality of $\delta$ it satisfies that $E \circ E = E$. It also holds that $\varphi$ is preserved. Removing the invertibility is just a technicality that can be solved by taking $\delta'_\epsilon = \delta + \epsilon 1$ and normalize it to $\delta_\epsilon = \delta_\epsilon'/\| \delta_\epsilon' \|_1$. Then, you can take the limit of $E_\epsilon$ as $\epsilon \to 0$ in the pointwise weak-$\ast$ topology.
I do not have a formula for the general case of $\delta$ noncentral.