On star, sharp, core, and minus partial orders in Rickart rings

Abstract

Let $\mathcal{A}$ be a Rickart $*$-ring and let ${\le }^{*},{\le }^{♯},{\le }^{\oplus }$, and ${\le }_{\oplus }$ denote the star, the sharp, the core, and the dual core partial orders in $\mathcal{A}$, respectively. The sets of all $b\in \mathcal{A}$ such that $a\le b$, along with the sets of all $b\in \mathcal{A}$ such that $b\le a$, are characterized, where $a\in \mathcal{A}$ is given and where $\le$ is one of the partial orders: ${\le }^{*}$, or ${\le }^{♯}$, or ${\le }^{\oplus }$, or ${\le }_{\oplus }$. Such sets of elements that are above or below a given element under the minus partial order ${\le }^{-}$ in a Rickart ring $\mathcal{A}$ are also studied. Some recent results of Cvetković-Ilić et al. on partial orders in $\mathcal{B}\left(H\right)$, the algebra of all bounded linear operators on a Hilbert space $H$, are thus generalized.