Moore-Penrose Invertibility of Differences and Products of Projections in Rings with Involution

TL;DR

This paper investigates the Moore-Penrose invertibility of differences and products of projections within rings with involution, providing new conditions and generalizations relevant to algebraic structures like $C^*$-algebras.

Contribution

It establishes equivalent conditions for MP invertibility of projections' differences and products, extending known results in $C^*$-algebras to more general rings with involution.

Findings

Characterization of MP invertibility for commutators and anti-commutators of projections

Generalization of known $C^*$-algebra results to rings with involution

New algebraic conditions for projections' differences and products

Abstract

This article concerns the MP inverse of the differences and the products of projections in a ring $R$ with involution. Some equivalent conditions are obtained. As applications, the MP invertibility of the commutator $pq-qp$ and the anti-commutator $pq+qp$ are characterized, where $p$ and $q$ are projections in $R$. Some related known results in $C^*$-algebra are generalized.