Kadison's transitivity for locally \(C^ *\)-algebras
This article gives a generalization to locally \(C^*\)-algebras of Kadison's transitivity theorem: a topologically irreducible representation of a \(C^*\)-algebra on a Hilbert space is algebraically irreducible. A locally \(C^*\)-algebra is a complete topological vector space with involution whose topology is given by a family of submultiplicative and *-preserving seminorms \(p_{\alpha}\) with the property \(p_{\alpha}(x*x)=p_{\alpha}(x)^ 2\) for all \(x\in E\) and all \(p_{\alpha}\). \textit{K. Schmüdgen} [Math. Nachr. 68, 167-182 (1975; Zbl 0315.46042)] showed that \(E_{\alpha}=E/\ker p_{\alpha}\) with norm \(p_{\alpha}\) is a \(C^*\)-algebra provided E has an identity. The author has shown [Pac. J. Math. 95, 61-73 (1981; Zbl 0438.46036)] that every continuous representation of a locally \(C^*\)-algebra filters through an algebra \(E_{\alpha}\). In the present paper the author shows that adjunction of the identity commutes with the operation \(E\to E_{\alpha}\) so that Kadison's theorem is applicable. Now let \({\mathcal P}(E)\) be the set of all continuous positive (i.e., f(x*x)\(\geq 0)\) linear forms on the locally \(C^*\)-algebra E. Then there is an \(\alpha\) and a positive linear functional \(f_{\alpha}\) on \(E_{\alpha}\) such that \(f_{\alpha}(x+\ker p_{\alpha})=f(x)\) for all \(x\in E\). Let \({\mathcal B}(E)\) be the set of all extreme points of all n in \({\mathcal P}(E)\) such that \(\| f_{\alpha}\| \leq 1\) for some \(f_{\alpha}\) corresponding to f. Let \(\phi_ f\) denote the GNS representation induced by f and let \([\phi_ f]\) denote the equivalence class of the representation. The author shows that \(f\to [\phi_ f]\) is a continuous open map on \({\mathcal B}(E)\). Here the equivalence classes have the Jacobson topology.
topologically irreducible representation of a \(C^ *\)-algebra on a Hilbert space
scientific article; zbMATH DE number 3996417
scientific article