Differences of idempotents in $C^*$-algebras

Suppose that $P$ and $Q$ are idempotents on a Hilbert space $\mathscr H$, while $Q=Q^*$ and $I$ is the identity operator in $\mathscr H$. If $U=P-Q$ is an isometry then $U=U^*$ is unitary and $Q=I-P$. We establish a double inequality for the infimum and the supremum of $P$ and $Q$ in $\mathscr H$ and $P-Q$. Applications of this inequality are obtained to the characterization of a trace and ideal $F$-pseudonorms on a $W^*$-algebra. Let $\varphi$ be a trace on the unital $C^*$-algebra $\mathscr A$ and let tripotents $P$ and $Q$ belong to $\mathscr A$. If $P-Q$ belongs to the domain of definition of $\varphi$ then $\varphi(P-Q)$ is a real number. The commutativity of some operators is established.