Numerical radius orthogonality in $C^{\ast }$-algebras

Abstract

In this paper we characterize the Birkhoff--James orthogonality with respect to the numerical radius norm $v(\cdot )$ in $C^{\ast }$-algebras. More precisely, for two elements $a,b$ in a $C^{\ast }$-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 ${\phi }_{{}_{\theta }}$ on $\mathfrak{A}$ such that $|{\phi }_{{}_{\theta }}(a)|=v(a)$ and $\mbox{Re}\big(e^{iθ}\overline{φ_{_θ}(a)}φ_{_θ}(b)\big)\geq 0$. 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.

to appear in Annals of Functional Analysis