Spatial numerical ranges of elements of subalgebras of \(C_0(X)\)
When \(A\) is a subalgebra of the commutative Banach algebra \(C_0(X)\) of all continuous complex-valued functions on a locally compact Hausdorff space \(X\), which vanish at infinity, the spatial numerical range of elements of \(A\) can be described in terms of positive measures. If \(A\) is a unital, then \[ V_1(A,f)\equiv \{m(f): m\in A^*\text{ and }\|m\|= m(1)= 1\} \] is called the numerical range of \(f\). When \(A\) is not unital, \textit{A. K. Gaur} and \textit{T. Husain} in their paper [Int. J. Math. Math. Sci. 12, No. 4, 633-640 (1989; Zbl 0719.46025)] defined the numerical range as: \[ V(A,f)= \{m(fg):\exists m\in A^*\text{ and }g\in A\text{ such that }\|m\|=\|g\|_\infty= m(g)= 1\}. \] In this paper with co and \(\overline{\text{co}}\) denoting the convex hull and the closed hull, the author obtains the following generalization of a result in that paper. Theorem: Let \(A\) be a subalgebra of \(C_0(X)\) and \(f\in A\). Then \[ \begin{multlined} V(A,f)= \{\int fd|\mu|: \exists\mu\in M(X)\text{ and }\exists g\in A\text{ such that}\\ \|\mu\|=\|g\|_\infty= \int g d\mu= 1\}\subseteq \overline{\text{co }}R(f),\end{multlined}\tag{i} \] where \(|\mu|\) denotes the total variation of \(\mu\). (ii) If \(A\) has the following property \((\#)\), then \(\text{co }R(f)\subseteq V(A,f)\). \((\#)\) For any finite set \(\{x_1,\dots, x_n\}\) in \(X\), there exists \(g\in A\) such that \(\|g\|_\infty= 1\) and \(g(x_1)=\cdots= g(x_n)= 1\). Furthermore, the author obtained Corollary: Let \(A\) be a *-subalgebra \(C_0(X)\) and \(f\in A\). Then \[ \begin{multlined} V(A,f)= \{\int f d\mu: \exists\mu\in M(X)\text{ and }\exists g\in A\text{ such that}\\ \|\mu\|= 1,\;\mu\geq 0,\;0\leq g\leq 1\text{ and }\int g d\mu= 1\}.\end{multlined}\tag{i} \] (ii) If \(A\) has the following property \((\#\#)\), then \[ V(A, f)= \{\int f d\mu: 0\leq\mu\in M(X),\;\|\mu\|= 1\text{ and supp}(\mu)\text{ is compact}\}. \] \((\#\#)\) For any compact set \(E\subseteq X\), there exists \(g\in A\) such that \(0\leq g\leq 1\) and \(g(x)= 1\) for all \(x\in E\). Here \(\text{supp}(\mu)\) denotes the support of \(\mu\). Remark: If \(A= C_0(X)\), then we have \[ V(C_0(X), f)= \{\int f d\mu: 0\leq \mu\in M(X),\;\|\mu\|= 1\text{ and supp}(\mu)\text{ is compact}\}, \] and \[ \text{co }R(f)\subseteq V(C_0(X), f)\subseteq \overline{\text{co }}R(f). \]
scientific article; zbMATH DE number 1505958