Consider a unital $C^*$-algebra $\mathcal{A}$. Let $n\geq 2$ and let $P_1, \ldots , P_n$ be projections in $\mathcal{A}$ such that $P_1 + \ldots +P_n=I$. We costruct $\mathcal{P}_n\colon \mathcal{A}\to \mathcal{A}$ ...

We establish similarity between some tripotents and idempotents on a Hilbert space $\mathcal{H}$ andobtain new results on differences and commutators of idempotents P and Q.In the unital case, the difference $P-Q$ is ...