Further results on the a-numerical range in $C^*$-algebras

Abstract

Let ${\mathfrak {A}}$ be a unital $C^*$-algebra with the unit $\mathbf{1}$ and ${\mathcal {S}}({\mathfrak {A}})$ be the set of all states on ${\mathfrak {A}}$. For a positive element a of ${\mathfrak {A}}$, let ${\Vert x\Vert }_{a}$, $v_{a}(x)$, and $V_{a}(x)$ denote the a-operator semi-norm, the a-numerical radius, and the a-numerical range of $x\in {\mathfrak {A}}$, respectively. Let ${\mathfrak {A}}^{a}=\left\{ x\in {\mathfrak {A}}: \, \Vert x\Vert _{a} <\infty \right\}$ and ${\mathcal {S}}_{a}({\mathfrak {A}})=\left\{ \frac{f}{f(a)}: f\in {\mathcal {S}}({\mathfrak {A}}),\, f(a)\ne 0\right\}$. Let also ${\mathfrak {A}}_{a}$ be the set of all elements in ${\mathfrak {A}}$ that admit a-adjoints. In this paper, we first characterize an element $x\in {\mathfrak {A}}$ for which $|f(ax)|=f(a)$ for every pure state f. Next, we prove that if a linear map $f: a{\mathfrak {A}}^{a}\longrightarrow \mathbb {C}$ satisfies $f(a)=1$ and $|f(ay)|\le {\Vert y\Vert }_{a}$ for all $y\in {\mathfrak {A}}^{a}$, then there exists $g\in {\mathcal {S}}_{a}({\mathfrak {A}})$ such that $f(ay)=g(ay)$ for all $y\in {\mathfrak {A}}_{a}$. We also show that $v_{a}(x) = {\Vert x\Vert }_{a}$ if and only if there is $f\in {\mathcal {S}}_{a}({\mathfrak {A}})$ such that $\sqrt{f(x^*ax)} = {\Vert x\Vert }_{a}$ and $|f(ax)| = v_{a}(x)$. In addition, we prove that ${\Vert x\Vert }_{a}<\infty$ if and only if $v_{a}(x)<\infty$. Finally, we show that $V_{a}(x)=\bigcap _{\zeta \in \mathbb {C}}D(\zeta , v_{a}(x-\zeta \mathbf{1} ))$.