Partial orders on $\ast$-regular rings
Abstract
In this work we consider some new partial orders on *-regular rings. Let be a *-regular ring, () be the lattice of all projectors in and be a sharp normal normalized measure on (). Suppose that (, ) is a complete metric *-ring with respect to the rank metric on defined as (, ) = (( -)) = (( -)), , , where (), () is respectively the left and right support of an element . On we define the following three partial orders: = + , ; () = ; () = , means algebraic orthogonality, that is, = = * = * = 0. We prove that the order topologies associated with these partial orders are stronger than the topology generated by the metric . We consider the restrictions of these partial orders on the subsets of projectors, unitary operators and partial isometries of *-regular algebra . In particular, we show that these three orders coincide with the usual order on the lattice of the projectors of *-regular algebra. We also show that the ring isomorphisms of *-regular rings preserve partial orders and .