## Banach Journal of Mathematical Analysis

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

Janko Marovt

#### Abstract

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

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 495-508.

Dates
Accepted: 21 October 2015
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.bjma/1465230960

Digital Object Identifier
doi:10.1215/17358787-3607090

Mathematical Reviews number (MathSciNet)
MR3509881

Zentralblatt MATH identifier
1351.16038

#### Citation

Marovt, Janko. On star, sharp, core, and minus partial orders in Rickart rings. Banach J. Math. Anal. 10 (2016), no. 3, 495--508. doi:10.1215/17358787-3607090. https://projecteuclid.org/euclid.bjma/1465230960

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