Numerical radius orthogonality in $C^*$-algebras

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 $\mbox{Re}\big(e^{i\theta}\overline{\varphi_{_{\theta}}(a)}\varphi_{_{\theta}}(b)\big)\geq 0$. 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.