In Murphy's book,there is a statement:If p is a finite rank projection on $H$,then $pB(H)p$ is finite dimensional.
My question:Given any $S\subset B(H)$,$S$ does not contain $Id_H$.Does there exist a projection $p$ such that the dimension of $pSp$ is finite but not zero?
The question can be reduced to whether given nonzero $T\in B(H)$, there exists a finite-rank projection $P$ such that $PTP\ne0$.
There is a direct construction of such $P$, but one easy way to see that it has to exist is to take an increasing net of finite-rank projections $\{p_j\}$ such that $p_j\nearrow I$ wot. Then, as wot is multiplicative on bounded nets, $$ p_jTp_j\to T\ \text{ wot}. $$ So if $pTp=0$ for all finite-rank projections, then $T=0$.