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