A-Roberts orthogonality in C*-algebras and its characterization via a-numerical ranges

Dehghani, Mahdi; Jalali Ghamsari, Hooriye Sadat

doi:10.22080/cjms.2025.28593.1742

A-Roberts orthogonality in C*-algebras and its characterization via a-numerical ranges

Document Type : Research Articles

Authors

¹ Department of Mathematical Sciences‎, ‎Yazd University‎, ‎Yazd‎, ‎Iran‎.

² Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Kashan‎, ‎Kashan‎, ‎Iran

10.22080/cjms.2025.28593.1742

Abstract

‎Let $ \mathcal{A} $ be a unital $ C^{*} $-algebra with unit $1_{\mathcal{A}}$ and let $ a\in\mathcal{A} $ be a positive and invertible element‎. ‎Set

\[ \mathcal{S}_a (\mathcal{A})=\{ \dfrac{f}{f(a)} \‎, : ‎\‎, ‎f \in \mathcal{S}(\mathcal{A})‎, ‎\‎, ‎f(a)\neq 0\}‎, ‎\]

where $ \mathcal{S}(\mathcal{A}) $ is the set of all states on $ \mathcal{A} $‎.

‎In this paper‎, ‎by using the concept of algebraic $a$-Davies-Wielandt shell of elements of $\mathcal{A}$‎, ‎we obtain a characterization of Roberts orthogonality with respect to the norm‎:
‎\[ \|x\|_a = \sup_{\varphi \in \mathcal{S}_a(\mathcal{A})} \sqrt{\varphi(x^* ax)}\quad (x\in \mathcal{A}),\]‎
‎in $C^*$-algebra $\mathcal{A}$‎, ‎so called‎, ‎$a$-Roberts orthogonality‎.
‎More precisely‎, ‎for any $a$-isometry $x\in\mathcal{A}$‎, ‎we prove that $x$ is $a$-Roberts orthogonal to $1_{\mathcal{A}}$ if and only if algebraic $a$-numerical range of $x$ is symmetric with respect to the origin.

Keywords