A method for constructing idempotents in a unital algebra

A method is proposed for constructing idempotents in a unital algebra 𝒜 using n arbitrary idempotents P 1 , . . . , P n from this algebra. The properties of the resulting idempotents P= P ( P 1 , . . . , 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 ↦→ 𝐴( P 1 , P 2 ), f( P 2 ) = 𝐴( P 1 , P 2 ) и P 3 ↦→ B( P 1 , P 2 , P 3 ), g( P 3 ) = B( P 1 , P 2 , P 3 ) preserve the complements ⊥ and are multiplicative on wide classes of idempotent pairs. For a finite trace 𝜙 on a unital C * -algebra 𝒜, 𝜙( P( P 1 , . . . , P n )) = 𝜙( P n ). For the projections P 1 , . . . , P n from the von Neumann algebra 𝒜, the method yields a new projection and enables the construction of some partial isometries.