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.