Finite dimensional $C^*$ algebra

Question

If $P$ is a finite rank projection, $A$ is an infinite dimensional $C^*$ algebra, can we deduce that $PAP$ is finite dimensional?

Answer 1

Assuming that the situation is that $A\subset B(H)$, $P\in B(H)$, with $P$ of finite-rank, then yes, $PAP$ is finite-dimensional.

One can construct an explicit isomorphism $PB(H)P=M_n(\mathbb C)$, where $n=\operatorname{Tr}(P)$ is the rank of $P$. It then follows that $PAP\subset PB(H)P$ is finite-dimensional.