Let $M$ be a finite von Neumann algebra equipped with a faithful normal tracial state $\tau:M\to\mathbb C$. Let $N\subset M$ be a von Neumann subalgebra with $1_N=1_M$. Then there exists a unique normal [conditional expectation](/page/Conditional%20Expectation) $E_N:M\to N$, meaning a normal positive unital idempotent $N$-bimodule map with range $N$, such that

\begin{align*} \tau(E_N(x))=\tau(x) \end{align*}

for every $x\in M$.