Operator Algebras and Positive Mappings

Abstract

In various parts of quantum physics positive mappings play a fundamental role. These mappings are defined on some involutive algebra (often of operators on some Hilbert space). By definition a positive mapping sends positive elements of its domain to positive elements of the target space. Thus first several characterizations of positive elements in an involutive normed algebra are given. Two types of positive mappings are considered: Positive linear functionals which have values in \(\mathbb{C}\) and completely positive mappings which have values in some other involutive algebra. For the structural analysis of positive mappings the concept of a representation of an involutive algebra is needed. This is introduced and in the case of the involutive algebra \(\mathcal{B}(\mathcal{H})\) of bounded linear operators on a Hilbert space \(\mathcal{H}\) the general form of its representations is determined (Naimark’s theorem). The structure of positive linear functionals on an involutive algebra with unit is presented in the Gelfand-Naimark-Segal (GNS)-construction. Positive linear functionals f on an involutive algebra \(\mathcal{A}\) with unit I such that \(f(I)=1\) are called states. On a weakly closed subalgebra \(\mathcal{A}\) of \(\mathcal{B}(\mathcal{H})\) special states are of the form \(f(A)={\rm{Tr}}(AW)\) where W is a density matrix on \(\mathcal{H}\). These states are characterized in terms of an additional continuity condition (normality, complete additivity), and are called normal states. The Stinespring factorization theorem gives the general form of a completely positive map between C\(^*\)-algebras. When this result is combined with Naimark’s theorem of representations of \(\mathcal{B}(\mathcal{H})\) it allows to determine the general form of completely positive mappings on \(\mathcal{B}(\mathcal{H})\) in more detail.