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

For every square matrix $A$ over a field $\mathbb{K}$, we have the equality $\mathrm{rank}(A) + \mathrm{rank}(I-A) = \mathrm{rank}(I) + \mathrm{rank}(A-A^2)$ where $I$ denotes the identity matrix with the same dimensions as $A$. In this article, we start a program to systematically characterize and generalize such 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. In our framework, a field $\mathbb{K}$ may be viewed as a ranked ring with a $\mathbb{Z}^+$-valued rank-system consisting of the usual rank functions on $M_n(\mathbb{K})$ (for $n \in \mathbb{N})$ and serves as the motivating example. We show that a finite von Neumann algebra $\mathscr{R}$ with center $\mathscr{C}$ (and the corresponding Murray-von Neumann algebra $\mathscr{R}_{\textrm{aff}}$) may be endowed with a $\mathscr{C}^{+}$-valued rank-system, considering $\mathscr{C}^{+}$ as a commutative monoid with respect to operator addition. We give an algorithm to generate rank identities in ranked $\mathbb{K}$-algebras via the polynomial function calculus, and in ranked complex Banach algebras via the holomorphic function calculus. As an illustrative application, using these 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 $1 \le i, j \le m$. In contrast, this does not hold in general in the ring of bounded operators on an infinite-dimensional complex Hilbert space.