A method for constructing idempotents in a unital algebra
https://doi.org/10.26907/2541-7746.2025.2.384-396
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 mappings
P2 ↦ 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.
Supporting agencies: This study was carried out as a part of the Program for Development of the Scientific and Educational Mathematical Center of the Volga Federal District (agreement no. 075-02-2024-1438).
About the Author
M. Khadour
Kazan Federal University
Russian Federation
Russian Federation
Mahmoud Khadour, Postgraduate Student
Kazan
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Review
For citations:
Khadour M. A method for constructing idempotents in a unital algebra. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(2):384-396. (In Russ.) https://doi.org/10.26907/2541-7746.2025.2.384-396
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