Characterization of numerical radius parallelism in $C^{\ast }$-algebras

Let $v(x)$ be the numerical radius of an element $x$ in a $C^{\ast }$-algebra $\mathfrak{A}$. First, we prove several numerical radius inequalities in $\mathfrak{A}$. Particularly, we present a refinement of the triangle inequality for the numerical radius in $C^{\ast }$-algebras. In addition, we show that if $x\in \mathfrak{A}$, then $v(x)=\frac{1}{2}\mathrm{\parallel }x\mathrm{\parallel }$ if and only if $\mathrm{\parallel }x\mathrm{\parallel }=\mathrm{\parallel }Re(e^{i\theta }x)\mathrm{\parallel }+\mathrm{\parallel }Im(e^{i\theta }x)\mathrm{\parallel }$ for all $\theta \in \mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^{\ast }$-algebras based on numerical radius. More precisely, we consider elements $x$ and $y$ of $\mathfrak{A}$ which satisfy $v(x + \lambda x) = v(x) + v(y)$forsomecomplexunit$\lambda$. We show that this relation can be characterized in terms of pure states acting on $\mathfrak{A}$.