Characterization of numerical radius parallelism in $$C^*$$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 $$\mathfrak {A}$$.