Characterization of a-Birkhoff–James orthogonality in $$C^*$$ -algebras and its applications

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. Suppose that \({\mathcal {S}}({\mathcal {A}})\) is the set of all states on \(\mathcal {{\mathcal {A}}}\) and let $$\begin{aligned} {\mathcal {S}}_a ({\mathcal {A}})=\left\{ \dfrac{f}{f(a)} \, : \, f \in {\mathcal {S}}({\mathcal {A}}), \, f(a)\ne 0\right\} . \end{aligned}$$ The norm \( \Vert x\Vert _a \) for every \( x \in {\mathcal {A}} \) is defined by $$\begin{aligned} \Vert x\Vert _a = \sup _{\varphi \in {\mathcal {S}}_a ({\mathcal {A}}) } \sqrt{\varphi (x^* ax)}. \end{aligned}$$ In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm \(\Vert \cdot \Vert _a\) in \({\mathcal {A}},\) namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in \({\mathcal {A}}\) by means of the elements of generalized state space \({\mathcal {S}}_a({\mathcal {A}})\) is provided. As an application, a characterization for the best approximation to elements of \({\mathcal {A}}\) in a subspace \({\mathcal {B}}\) with respect to \(\Vert \cdot \Vert _a\) is obtained. Moreover, a formula for the distance of an element of \({\mathcal {A}}\) to the subspace \({\mathcal {B}}={\mathbb {C}}1_{{\mathcal {A}}}\) is given. We also study the strong version of a-Birkhoff–James orthogonality in \( {\mathcal {A}} .\) The classes of \(C^*\) -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and \({\mathcal {A}}\) -valued inner product orthogonality in \({\mathcal {A}}\) implies that the center of \({\mathcal {A}}\) is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in \({\mathcal {A}},\) then the center of \({\mathcal {A}}\) is trivial.