On conditional expectation on vN subalgebras

Question

Let $M$ be tracial vN algebra with trace $\tau$, $E$ be a conditional expectationonto a subalgebra $B$ preserving the trace $\tau$, if a unitary $u$ in $M$, is it true $E(u)$ is again unitary in some special cases?

Answer 1

"Special cases" meaning $u\in B$.

If $E(u)$ is a unitary, you can check directly that $$ E((u-E(u))(u-E(u))^*)=0. $$ Since $E$ is faithful (from the faithfulness of $\tau$) we get that $u-E(u)=0$. That is, $u\in B$.