A framework for rank identities - with a view towards operator algebras
Soumyashant Nayak · Journal of Operator Theory · 2023
In this article, we start a program to systematically characterize rank identities with a view towards applications to operator algebras. We initiate the study of so called ranked rings (unital rings with a ``rank system''), the main examples of interest being finite von Neumann algebras, Murray-von Neumann algebras, and von Neumann rank rings. As an illustrative application, using some abstract rank identities we show that the sum of finitely many idempotents e1,…,em in a finite von Neumann algebra is an idempotent if and only if they are mutually orthogonal, that is, eiej=δijei for 1⩽i,j⩽m.