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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."
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