G. Lassner

Topological algebras of operators

Various methods of topologization of an algebra of unbounded operators are studied. For an algebra of unbounded operators with involution the uniform topology is defined and investigated. The notions of an O∗-algebra and AO∗-algebra are defined, which are generalizations of the notions of a C∗-algebra and B∗-algebra. Examples illustrate how the developed idea works.

Normal states on algebras of unbounded operators

Abstract The problem of the normality of positive functionals on algebras of unbounded operators is investigated. We formulate conditions under which the normal states are uniformly continuous and we prove the normality of any uniformly continuous state on the maximal Op ∗-algebras on certain unitary spaces.

Completely positive mappings and unbounded observables

Abstract Completely positive mappings on algebras of unbounded observables are investigated. The main theorem of Stinespring on the structure of completely positive mappings is generalized to algebras of unbounded observables. It is shown that the consideration of unbounded observables leads to a natural physical topology on the state space, with respect to which the entropy is continuous.