A FRAMEWORK FOR RANK IDENTITIES — WITH A VIEW TOWARDS OPERATOR ALGEBRAS

Article Type

Research Article

Publication Title

Journal of Operator Theory

Abstract

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.

First Page

477

Last Page

520

DOI

https://10.7900/jot.2021aug22.2339

Publication Date

1-1-2023

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