On one-sided star partial orders on a Rickart *-ring

TL;DR

This paper explores and compares various approaches to extending star partial orders from matrices to operators and *-rings, focusing on a specific right-star partial order on Rickart *-rings without assuming regularity.

Contribution

It introduces and analyzes a version of right-star partial order on Rickart *-rings, broadening the understanding of partial orders in algebraic structures.

Findings

Comparison of approaches to star partial orders

Extension of right-star partial order to Rickart *-rings

Analysis without assuming regularity

Abstract

We compare some recent approaches to extending the notions of left- and right-star partial order, introduced for complex matrices in early 90-ies, to bounded linear Hilbert space operators and to certain *-rings, and discuss in more detail a version of right-star partial order on a Rickart *-ring R. Regularity of R is not presumed.