Inequalities for linear functionals and numerical radii on $\mathbf{C}^*$-algebras

TL;DR

This paper establishes new inequalities for positive linear functionals and numerical radii in unital C*-algebras, providing bounds and characterizations that deepen understanding of operator behavior and spectral properties.

Contribution

It introduces novel inequalities for linear functionals and numerical radii, along with characterizations of when certain equalities hold in C*-algebras.

Findings

Derived bounds for numerical radius v(a) in C*-algebras.

Characterized conditions for v(a) = ||a||/2 and related equalities.

Studied inequalities for (,)-normal elements.

Abstract

Let $\mathcal{A}$ be a unital $\mathbf{C}^*$-algebra with unit $e$. We develop several inequalities for a positive linear functional $f$ on $\mathcal{A}$ and obtain several bounds for the numerical radius $v(a)$ of an element $a\in \mathcal{A}$. Among other inequalities, we show that if $ a_k, b_k, x_k\in \mathcal{A}$, $r\in \mathbb{N}$ and $f(e)=1$, then \begin{eqnarray*} \left| f \left( \sum_{k=1}^n a_k^*x_kb_k\right)\right|^{r} &\leq& \frac{n^{r-1}}{\sqrt{2}} \left| f\left( \sum_{k=1}^n \big( (b_k^*|x_k| b_k)^{r}+ i (a_k^*|x_k^*|a_k)^{r} \big) \right) \right| \quad (i=\sqrt{-1}), \end{eqnarray*} \begin{eqnarray*} \left| f\left( \sum_{k=1}^n a_k\right)\right|^{2r} &\leq& \frac{n^{2r-1}}{2} f \left( \sum_{k=1}^n Re(|a_k|^r|a_k^*|^r) + \frac12 \sum_{k=1}^n (|a_k|^{2r}+ |a_k^*|^{2r} ) \right). \end{eqnarray*} We find several equivalent conditions for $v(a)=\frac{\|a\|}{2}$ and $v^2(a)={\frac{1}{4}\|a^*a+aa^*\|}$. We prove that $v^2(a)={\frac{1}{4}\|a^*a+aa^*\|}$ (resp., $v(a)=\frac{\|a\|}{2}$) if and only if $\mathbb{S}_{\frac12{ \| a^*a+aa^*\|}^{1/2}} \subseteq V(a) \subseteq \mathbb{D}_{\frac12 {\| a^*a+aa^*\|}^{1/2}}$ (resp., $\mathbb{S}_{\frac12 \| a\|} \subseteq V(a) \subseteq \mathbb{D}_{\frac12 \| a\|}$), where $V(a)$ is the numerical range of $a$ and $\mathbb{D}_k$ (resp., $\mathbb{S}_k$) denotes the circular disk (resp., semi-circular disk) with center at the origin and radius $k$. We also study inequalities for the $(\alpha,\beta)$-normal elements in $\mathcal{A}.$