A method for constructing idempotents in a unital algebra

A method is proposed for constructing idempotents in a unital algebra $\mathcal{A}$ using $n$ arbitrary idempotents $P_1, \ldots , P_n$ from this algebra. The properties of the resulting idempotents $P=P(P_1, \ldots , P_n)$ are investigated; for $n=2$ and $n=3$, explicit forms of the idempotents are obtained: $A(P_1,P_2)$ and $B(P_1,P_2,P_3)$, respectively. It is shown that the mappings
$$ P_2 \mapsto A(P_1,P_2), f(P_2)=A(P_1,P_2) \text{\rm and} P_3 \mapsto B(P_1,P_2,P_3), g(P_3)=B(P_1,P_2,P_3) $$
preserve the complements $^{\perp}$ and are multiplicative on wide classes of idempotent pairs. For a finite trace $\varphi$ on a unital $C^*$-algebra $\mathcal{A}$, $\varphi (P(P_1, \ldots , P_n))=\varphi (P_n)$. For the projections $P_1, \ldots , P_n$ from the von Neumann algebra $\mathcal{A}$, the method yields a new projection and enables the construction of some partial isometries.