the unitary group of von Neumann algebra

Question

Let $A$ be a von Neumann algebra and $P\in A'\subset B(H)$. Is it true that $U(A_P) =U(A)_p$?

Here $A_p$ is the von Neumann algebra containing all the element in the form of $XP$, $X\in A$, $U(A_p)$ is the unitary group of $A_P$ and $U(A)_p$ contains all the element the unitaries of $A$ times $P$.

Answer 1

see

In induced von Neumann algebra, do we have $P(PMP) =P\cdot P(M)P$ and $U(PMP) =P\cdot U(M)P$?