Unbounded Operator Algebras and Representation Theory, 1990 Operator Theory: Advances and Applications Series, Vol. 37

*-algebras of unbounded operators in Hilbert space, or more generally algebraic systems of unbounded operators, occur in a natural way in unitary representation theory of Lie groups and in the Wightman formulation of quantum field theory. In representation theory they appear as the images of the associated representations of the Lie algebras or of the enveloping algebras on the Garding domain and in quantum field theory they occur as the vector space of field operators or the *-algebra generated by them. Some of the basic tools for the general theory were first introduced and used in these fields. For instance, the notion of the weak (bounded) commutant which plays a fundamental role in thegeneraltheory had already appeared in quantum field theory early in the six­ ties. Nevertheless, a systematic study of unbounded operator algebras began only at the beginning of the seventies. It was initiated by (in alphabetic order) BORCHERS, LASSNER, POWERS, UHLMANN and VASILIEV. J1'rom the very beginning, and still today, represen­ tation theory of Lie groups and Lie algebras and quantum field theory have been primary sources of motivation and also of examples. However, the general theory of unbounded operator algebras has also had points of contact with several other disciplines. In particu­ lar, the theory of locally convex spaces, the theory of von Neumann algebras, distri­ bution theory, single operator theory, the momcnt problem and its non-commutative generalizations and noncommutative probability theory, all have interacted with our subject.

1. Preliminaries.- I. O*-Algebras and Topologies.- 2. O-Families and Their Graph Topologies.- 3. Spaces of Linear Mappings Associated with O-Families and Their Topologization.- 4. Topologies for O-Famiiies with Metrizable Graph Topologies.- 5. Ultraweakly Continuous Linear Functionals and Duality Theory.- 6. The Generalized Calkin Algebra and the*-Algebra 𝓛+(𝓓).- 7. Commutants.- II:*-Representations.- 8. Basics of*-Representations.- 9. Self-Adjoint Representations of Commutative*-Algebras.- 10. Integrable Representations of Enveloping Algebras.- 11. n-Positivity and Complete Positivity of*-Representations.- 12. Integral Decompositions of*-Representations and States.- Symbol Index.