Numerical radius orthogonality in $$C^*$$-algebras

Abstract

In this paper we characterize the Birkhoff–James orthogonality with respect to the numerical radius norm $$v(\cdot )$$ in $$C^*$$ -algebras. More precisely, for two elements a, b in a $$C^*$$ -algebra $$\mathfrak {A}$$ , we show that $$a\perp _{B}^{v} b$$ if and only if for each $$\theta \in [0, 2\pi )$$ , there exists a state $$\varphi _{_{\theta }}$$ on $$\mathfrak {A}$$ such that $$|\varphi _{_{\theta }}(a)| = v(a)$$ and $$\text{ Re }\big (e^{i\theta }\overline{\varphi _{_{\theta }}(a)}\varphi _{_{\theta }}(b)\big )\ge 0$$ . Moreover, we compute the numerical radius derivatives in $$\mathfrak {A}$$ . In addition, we characterize when the numerical radius norm of the sum of two (or three) elements in $$\mathfrak {A}$$ equals the sum of their numerical radius norms.