In the paper from the link http://arxiv.org/pdf/0906.0139.pdf the author uses a diagonal conditional expectation. We take a seperable Hilbert space $H$ and fix an orthonormal basis $(e_n)_{n \in \mathbb{N}}$.
The diagonal conditional expectation is the idempotent map $E \colon \mathbf{B}(H) \rightarrow D$ which erases the off-diagonal entries,
where $D$ is the set made of diagonal elements of $\mathbf{B}(H)$ (that is, $x$ is a diagonal element of $\mathbf{B}(H)$ if $P_{\{e_i\}}xP_{\{e_j\}}= 0$, whenever $i \neq j$, $P_{\{e_k\}}$ is a rank-one projection onto $span\{e_k\}$), and it is a maximal abelian self-adjoint algebra in $\mathbf{B}(H)$ (it is equal to its commutant).
Now let me introduce bra-ket notation:
I assume that the inner product in $H$ is linear with respect to the 2nd coordinate.
Let $x \in H$, then
- $\left|x\right> \colon \mathbb{C} \rightarrow H$ if defined by $\left|x\right> \alpha = \alpha x$ for all $\alpha \in \mathbb{C}$,
- $\left< x \right| \colon H \rightarrow \mathbb{C}$ is defined by $\left< x \right| y= \left<x, y \right>$ for all $y \in H$.
How does this conditional expectation act on an arbitrary element $T \in \mathbf{B}(H)$, what $E(T)= ?$, can we write it using the bra-ket notation ?