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

Abstract

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 $v(x)=\frac{1}{2}\Vert x\Vert$ if and only if $\|x\| = \|\mbox{Re}(e^{iθ}x)\| + \|\mbox{Im}(e^{iθ}x)\|$ 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 $v(x+\lambda x)=v(x)+v(y)$ for some complex unit $\lambda$. We show that this relation can be characterized in terms of pure states acting on $\mathfrak{A}$.

15 pages