A method for constructing idempotents in a unital algebra

Abstract

A method is proposed for constructing idempotents in a unital algebra 𝒜 using n arbitrary idempotents P1,...,Pn from this algebra. The properties of the resulting idempotents P = P(P1,...,Pn) are investigated; for n = 2 and n = 3, explicit forms of the idempotents are obtained: A(P1,P2) and B(P1,P2,P3), respectively. It is shown that the mappingsP2 ↦ A(P1,P2), f(P2) = A(P1,P2) and P3 ↦ B(P1,P2,P3), g(P3) = B(P1,P2,P3)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(P1,...,Pn)) = 𝜑(Pn). For the projections P1,...,Pn from the von Neumann algebra 𝒜, the method yields a new projection and enables the construction of some partial isometries.

Keywords

• unital algebra
• c*-algebra
• idempotent
• symmetry
• projection
• partial isometry
• trace