Semidiscrete Hilbert spaces
Extract from the paper: ``In 1938 Sz. Nagy characterized \(L^ 2\)-spaces of commutative \(W^*\)-algebras in the following way: A Hilbert space H, which is ordered by a selfdual cone \(H^+\) having the Riesz interpolation property, is isomorphic to \(L^ 2(X,\mu)\) for some measure space (X,\(\mu)\). Grothendieck has shown that the identity on \(L^ 2(X,\mu)\) can be approximated in the strong operator topology by factorisation through \(C^ m\) with positive linear maps. As it can be seen from a result of \textit{G. Wittstock} [Lect. Notes Phys. 29, 67-84 (1974 Zbl 0329.46007); Theorem 3.1] Grothendieck's property imples the Riesz interpolation property for any Hilbert space with selfdual cone \(H^+\). Hence it characterizes as well the spaces \(L^ 2(X,\mu)\). In this paper an analogous theory is formulated for \(L^ 2\)-spaces of semidiscrete \(W^*\)-algebras.''
\(L^ 2\)-spaces of commutative \(W^*\)-algebras
scientific article; zbMATH DE number 4142787