Defining metric spaces via operators from unital \(C^*\)-algebras
Let \(A\) be a unital \(C^{\star}\)-algebra and \(T\) an operator with \(\text{ Dom}(T)\subset A\), \(\text{ Range}(T)\) in a normed space and \(\text{Ker}(T)= {\mathbb{C}}1.\) The author considers the metric \(d_T\) on the set \({\mathcal S}(A)\) of states of \(A\) which is given by \(d_{T}(\phi , \psi) = \sup \{ |\varphi (a)-\psi (a)|: a\in A \text{ and } \|T(a) \|\leq 1\}\) for \(\varphi , \psi \in {\mathcal S}(A).\) This generalizes the definition of the metrics on \({\mathcal S}(A)\) introduced via unbounded Fredholm modules by A. Connes. The main results in this paper are two different characterizations of those operators for which the topology of \(d_T\) agrees with the week-\(\star \) topology of \({\mathcal S}(A).\)
scientific article; zbMATH DE number 1344258