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Positive Forms on Nuclear *-Algebras and Their Integral Representations

Published online by Cambridge University Press:  20 November 2018

Alain Bélanger  and
Erik G. F. Thomas
Alain Bélanger
Affiliation:
Department of Mathematics York University North York, Ont. M3J 1P3
Erik G. F. Thomas
Affiliation:
Department of Mathematics University of Groningen, P.O. Box 800, 9700 AV Groningen, TheNetherlands
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Abstract.

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The main result of this paper establishes the existence and uniqueness of integral representations of KMS functionals on nuclear *- algebras. Our first result is about representations of *-algebras by means of operators having a common dense domain in a Hilbert space. We show, under certain regularity conditions, that (Powers) self-adjoint representations of a nuclear *-algebra, which admit a direct integral decomposition, disintegrate into representations which are almost all self-adjoint. We then define and study the class of self-derivative algebras. All algebras with an identity are in this class and many other examples are given. We show that if is a self-derivative algebra with an equicontinuous approximate identity, the cone of all positive forms on is isomorphic to the cone of all positive invariant kernels on These in turn correspond bijectively to the invariant Hilbert subspaces of the dual space This shows that if is a nuclear -space, the positive cone of has bounded order intervals, which implies that each positive form on has an integral representation in terms of the extreme generators of the cone. Given a continuous exponentially bounded one-parameter group of *-automorphisms of we can define the subcone of all invariant positive forms satisfying the KMS condition. Central functionals can be viewed as KMS functionals with respect to a trivial group action. Assuming that is a self-derivative algebra with an equicontinuous approximate identity, we show that the face generated by a self-adjoint KMS functional is a lattice. If is moreover a nuclear *-algebra the previous results together imply that each self-adjoint KMS functional has a unique integral representation by means of extreme KMS functionals almost all of which are self-adjoint.

Keywords

46 K 1046 A 1252 A 0746 H 0581 E 0582 A 15Nuclear spacereproducing kernels of Hilbert subspacestopological *-algebraself-adjoint representationcentral and KMS statesintegral representation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 3 , 01 June 1990 , pp. 410 - 469
Copyright
Copyright © Canadian Mathematical Society 1990

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