Partial orders on $\ast$-regular rings

Abstract

In this work we consider some new partial orders on $\ast$-regular rings. Let $\mathcal{A}$ be a $\ast$-regular ring, $P(\mathcal{A})$ be the lattice of all projectors in $\mathcal{A}$ and $\mu$ be a sharp normal normalized measure on $P(\mathcal{A}).$ Suppose that $(\mathcal{A}, \rho)$ is a complete metric $\ast$-ring with respect to the rank metric $\rho$ on $\mathcal{A}$ defined as $\rho(x, y) = \mu(l(x-y))=\mu (r(x-y))$, $x, y \in \mathcal{A}$, where $l(a)$, $r(a)$ is respectively the left and right support of an element $a$. On $\mathcal{A}$ we define the following three partial orders: $a \prec_s b \Longleftrightarrow b = a + c$, $a \perp c;$ $a \prec_l b \Longleftrightarrow l(a) b = a;$ $ a \prec_r b \Longleftrightarrow br (a) = a,$ $a\perp c$ means algebraic orthogonality, that is, $ac = ca = a^\ast c = ac^\ast = 0.$ We prove that the order topologies associated with these partial orders are stronger than the topology generated by the metric $\rho.$ We consider the restrictions of these partial orders on the subsets of projectors, unitary operators and partial isometries of $\ast$-regular algebra $\mathcal{A}.$ In particular, we show that these three orders coincide with the usual order $\le$ on the lattice of the projectors of $\ast$-regular algebra. We also show that the ring isomorphisms of $\ast$-regular rings preserve partial orders $\prec_l$ and $\prec_r$.