Unbounded Operator Algebras and Representation Theory

1. Front Matter

   Pages 1-12

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2. Preliminaries

   1. Preliminaries

      • Konrad Schmüdgen

      Pages 13-32

3. O*-Algebras and Topologies

   1. Front Matter

      Pages 33-34

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   2. O-Families and Their Graph Topologies

      • Konrad Schmüdgen

      Pages 35-63

   3. Spaces of Linear Mappings Associated with O-Families and Their Topologization

      • Konrad Schmüdgen

      Pages 64-100

   4. Topologies for O-Families with Metrizable Graph Topologies

      • Konrad Schmüdgen

      Pages 101-122

   5. Ultraweakly Continuous Linear Functionals and Duality Theory

      • Konrad Schmüdgen

      Pages 123-154

   6. The Generalized Calkin Algebra and the *-Algebra \( {\mathcal{L}^{^ + }}\left( \mathcal{D} \right)\)

      • Konrad Schmüdgen

      Pages 155-174

   7. Commutants

      • Konrad Schmüdgen

      Pages 175-198

4. *-Representations

   1. Front Matter

      Pages 199-200

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   2. Basics of *-Representations

      • Konrad Schmüdgen

      Pages 201-235

   3. Self-Adjoint Representations of Commutative *-Algebras

      • Konrad Schmüdgen

      Pages 236-259

   4. Integrable Representations of Enveloping Algebras

      • Konrad Schmüdgen

      Pages 260-299

   5. n-Positivity and Complete Positivity of *-Representations

      • Konrad Schmüdgen

      Pages 300-329

   6. Integral Decompositions of *-Representations and States

      • Konrad Schmüdgen

      Pages 330-361

5. Back Matter

   Pages 362-380

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*-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.

• Hilbert space
• algebra
• field
• lie algebra
• lie group
• operator algebra
• operator theory
• representation theory

• Sektion Mathematik, Karl-Marx Universität, Leipzig, DDR

  Konrad Schmüdgen