In this paper we characterize the Birkhoff–James orthogonality with respect to the numerical radius norm v(·)\documentclass[12pt]{minimal}
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\begin{document}$$v(\cdot )$$\end{document} in C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebras. More precisely, for two elements a, b in a C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra A\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {A}$$\end{document}, we show that a⊥Bvb\documentclass[12pt]{minimal}
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\begin{document}$$a\perp _{B}^{v} b$$\end{document} if and only if for each θ∈[0,2π)\documentclass[12pt]{minimal}
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\begin{document}$$\theta \in [0, 2\pi )$$\end{document}, there exists a state φθ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _{_{\theta }}$$\end{document} on A\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {A}$$\end{document} such that |φθ(a)|=v(a)\documentclass[12pt]{minimal}
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\begin{document}$$|\varphi _{_{\theta }}(a)| = v(a)$$\end{document} and Re(eiθφθ(a)¯φθ(b))≥0\documentclass[12pt]{minimal}
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\begin{document}$$\text{ Re }\big (e^{i\theta }\overline{\varphi _{_{\theta }}(a)}\varphi _{_{\theta }}(b)\big )\ge 0$$\end{document}. Moreover, we compute the numerical radius derivatives in A\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {A}$$\end{document}. In addition, we characterize when the numerical radius norm of the sum of two (or three) elements in A\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {A}$$\end{document} equals the sum of their numerical radius norms.
