Partial Isometries: Characterization

Question

Given a C*-algebra $\mathcal{A}$

Consider an element: $$J\in\mathcal{A}:\quad P:=J^*J$$

Then the equivalence holds: $$JJ^*J=J\iff P^2=P=P^*$$

How can I prove this?

Answer 1

Note: This proof works also for distinct Hilbert spaces!

On the one hand: $$P^2=J^*(JJ^*J)=J^*J=P$$ $$P^*=(J^*J)^*=J^*J=P$$

By the C*-property: $$A\in\mathcal{A}:\quad A=0\iff A^*A=0$$

So on the other hand: $$(JJ^*J-J)^*(JJ^*J-J)=P^3-P^2-P^2+P=0$$

Concluding the equivalence.