Characterization of numerical radius parallelism in $C^*$-algebras

Abstract

Let v(x) denote the numerical radius of an element x in a $C^*$-algebra $\mathfrak {A}$. First, we prove several numerical radius inequalities in $\mathfrak {A}$. Particularly, we show that if $x\in \mathfrak {A}$, then $v(x) = \frac{1}{2}\Vert x\Vert$ if and only if $\Vert x\Vert = \Vert \text{ Re }(e^{i\theta }x)\Vert + \Vert \text{ Im }(e^{i\theta }x)\Vert$ for all $\theta \in \mathbb {R}$. In addition, we present a refinement of the triangle inequality for the numerical radius in $C^*$-algebras. Among other things, we introduce a new type of parallelism in the setting of $C^*$-algebras based on the notion of numerical radius. More precisely, an element $x\in \mathfrak {A}$ is called the numerical radius parallel to another element $y \in \mathfrak {A}$, denoted by $x\,{\parallel }_v \,y$, if $v(x + \lambda x) = v(x) + v(y)$ for some complex unit $\lambda$. We show that this relation can be characterized in terms of pure states acting on $\mathfrak {A}$.