Compute the center of $PB(H)P$, where $P$ is a projection

Question

Let $H$ be a Hilbert space and $P$ be a projection on $H$. I guess that the center of $PB(H)P$ is $\Bbb C P$.

Is there any method to compute the center of $PB(H)P$?

Answer 1

This is standard basic von Neumann algebra theory. Given a von Neumann algebra $M\subset B(H)$ and a projection $p\in M$, then when we consider $pMp\subset B(pH)$ we have $$ (pMp)'=pM'p,\ \ \ Z(pMp)=pZ(M)p. $$ You can find this, among many places, in Proposition 5.5.6 in Kadison-Ringrose.

Answer 2

Hint: To compute the center of $PB(H)P$, first observe that $PB(H)P$ is isomorphic to $B(PH)$.