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Journal of Operator Theory

Volume 89, Issue 2, Spring 2023  pp. 477-520.

A framework for rank identities - with a view towards operator algebras

Authors:  Soumyashant Nayak
Author institution: Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bengaluru - 560 059, Karnataka, India

Summary:  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 $e_1, \ldots, e_m$ in a finite von Neumann algebra is an idempotent if and only if they are mutually orthogonal, that is, $e_i e_j = \delta_{ij} e_i$ for $1 \leqslant i, j \leqslant m$.

Keywords:  center valued rank, rank identities, ranked ring, Murray--von Neumann algebras

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