A remark of the numerical ranges of operators on Hilbert spaces. (Q1396052)

Let \({\mathcal B}(H)\) be the Banach algebra of all bounded linear operators on a separable Hilbert space \(H\). The usual numerical range of an operator \(A\in {\mathcal B}(H)\) is defined as \(W(A)=\{(Ax, x) : x\in H, \| x\| =1\}\). The authors consider a von Neumann algebra containing \(A\) and introduce the following another notion of numerical range for \(A\): Let \({\mathcal M}\) be a von Neumann algebra containing \({\mathbf 1}\) and \(A\). Let \(NS({\mathcal M})\) denote the set of all normal states of \({\mathcal M}\) and define \(V(A)=\{ \varphi(A): \varphi\in NS({\mathcal M})\}\). The such defined numerical range \(V(A)\) is a convex subset of \({\mathbb C}\), has several properties as the usual numerical range \(W(A)\), and \(W(A)\subset V(A)\). If \(\pi\) ia s faithful normal \(\ast\)-representation of \({\mathcal M}\) onto a von Neumann algebra \({\mathcal N}\) on a Hilbert space \(K\), then \(V(A)=V(\pi(A))\). The main theorem of this paper states that \(V(A)=W(A)\). As a consequence, the authors obtain their Theorem 6. Let \(A\) be an operator acting on a Hilbert space \(H\) and \({\mathcal M}\) a von Neumann algebra containing \(A\). Let \(\pi\) be a faithful normal representation of \({\mathcal M}\). Then \(W(A)= W(\pi(A))\).