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

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

$$\begin{aligned} \bigg| f \bigg( \sum_{k=1}^n a_k^*x_kb_k\bigg)\bigg|^{r} & \leq \frac{n^{r-1}}{\sqrt{2}} \bigg| f\bigg( \sum_{k=1}^n \big( (b_k^*|x_k| b_k)^{r}+ i (a_k^*|x_k^*|a_k)^{r} \big) \bigg) \bigg| \quad (i=\sqrt{-1}), \\ \bigg| f\bigg( \sum_{k=1}^n a_k\bigg)\bigg|^{2r} & \leq \frac{n^{2r-1}}{2} f \bigg(\sum_{k=1}^n \textrm{Re} ( |a_k|^r|a_k^*|^r) + \frac{1}{2} \sum_{k=1}^n (|a_k|^{2r}+ |a_k^*|^{2r} )\bigg).\end{aligned}$$

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 \(\mathcal{A}\).