Characterization of Approximate a-Birkhoff-James Orthogonality in $C^*$-Algebras

Document Type : Original Scientific Paper

Authors

1 ‎Department of Mathematical Science, ‎Yazd University, ‎Yazd‎, ‎Iran

2 ‎Department of Pure Mathematics, ‎University of Kashan, Kashan‎, ‎Iran

10.22052/mir.2025.256282.1501

Abstract

‎Assume that $ \mathcal{A} $ is a unital C*-algebra and $ a\in\mathcal{A} $ is 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 state space of $ \mathcal{A} $‎.
 
‎The main aim of this paper is to introduce and study the notions of approximate a-orthogonality and approximate a-Birkhoff-James orthogonality associated 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 $\mathcal{A}$‎.
‎First‎, ‎by providing some examples‎, ‎we show that these approximate orthogonalities are generally incomparable in non-commutative C*-algebras‎. ‎Next‎, ‎we will see that under what conditions‎, ‎these orthogonality relationships are related‎. ‎Also‎, ‎two different characterizations of approximate a-Birkhoff-James orthogonality in terms of the elements of $ \mathcal{S}_a (\mathcal{A}) $ are obtained‎.
‎Moreover‎, ‎the strong version of approximate a-Birkhoff-James orthogonality is studied‎. ‎Finally‎, ‎we prove that if approximate a-Birkhoff-James orthogonality and its strong version coincide on $ \mathcal{A} $‎, ‎then $ \mathcal{A} $ is commutative‎.

Keywords

Main Subjects

References

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Mathematics Interdisciplinary Research
Volume 11, Issue 1
March 2026
Pages 15-30

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