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.