Absolute continuity for linear forms on \(B^*\)-algebras and a Radon- Nikodým type theorem (quadratic version)
In this paper a Radon-Nikodym type theorem for positive linear functionals on \(C^*\)-algebras is presented which is based on the following notion of absolute continuity. Let A be a \(C^*\)-algebra with positive cone \(A_+\), closed unit ball S, and let f be a positive functional on A. A bounded linear functional g on A is said to be absolutely continuous with respect to f if, for each \(\epsilon>0\), there exists \(\delta>0\) such that \(| g(x)|<\epsilon\) for all \(x\in A_+\cap S\) with \(f(x)<\delta\). If \(y\in A\), define \(f^ y\) by \(f^ y(x)=f(y^*xy)\) for \(x\in A.\) The main result of the paper is the following theorem: If A is a \(C^*\)- algebra and g, f are positive functionals on A such that g is absolutely continuous with respect to f, then there exists a sequence \(\{y_ n\}\) in A such that \(\lim_{n\to \infty}\| g-f^{y_ n}\| =0.\) Applications are made to vector states on von Neumann algebras.
Radon-Nikodym type theorem for positive linear functionals on \(C^*\)- algebras
scientific article; zbMATH DE number 3848037