# Unbounded Operator Algebras and Representation Theory

Part of the Operator Theory: Advances and Applications book series (OT, volume 37)

Advertisement

Part of the Operator Theory: Advances and Applications book series (OT, volume 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.

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

- DOI https://doi.org/10.1007/978-3-0348-7469-4
- Copyright Information Springer Basel AG 1990
- Publisher Name Birkhäuser, Basel
- eBook Packages Springer Book Archive
- Print ISBN 978-3-0348-7471-7
- Online ISBN 978-3-0348-7469-4
- Series Print ISSN 0255-0156
- Series Online ISSN 2296-4878
- Buy this book on publisher's site