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

TL;DR

This paper investigates numerical radius inequalities in $C^*$-algebras, refines the triangle inequality, characterizes when $v(x) = rac{1}{2} orm{x}$, and introduces a new concept of parallelism based on numerical radius.

Contribution

It introduces a new type of numerical radius parallelism in $C^*$-algebras and characterizes it using pure states, along with refined inequalities and conditions for numerical radius equality.

Findings

Refined triangle inequality for numerical radius.

Characterization of elements with $v(x) = rac{1}{2} orm{x}$.

New parallelism concept based on numerical radius and pure states.

Abstract

Let $v(x)$ be 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 present a refinement of the triangle inequality for the numerical radius in $C^*$-algebras. In addition, we show that if $x\in\mathfrak{A}$, then $v(x) = \frac{1}{2}\|x\|$ if and only if $\|x\| = \|\mbox{Re}(e^{i\theta}x)\| + \|\mbox{Im}(e^{i\theta}x)\|$ for all $\theta \in \mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^*$-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 $\mathfrak{A}$.