Abstract
We discuss some properties of a non-commutative generalization of the classical moment problem (them-problem) previously introduced. It is shown that there is a connexion between the determination of the problem and the self-adjointness properties in the corresponding Hilbert space. This generalizes the well-known connexion between the determination of the measure in the classical moment problem and the self-adjointness properties of the polynomials as operators in the correspondingL 2-space. The dependence of them-problem on the choice ofC*-semi-norms and on the action of *-homomorphisms is also investigated. As an application, it is shown that if a quantum field (in a very general sense) is essentially self-adjoint then them-problem for the Wightman functional is determined on the quasi-localizableC*-algebra and that the corresponding representation of the localizable algebra generates the bounded observables of the field. It is pointed out that (ultraviolet and spatially) cut-off fields fall in this class and, therefore, are in one to one correspondance with states on the quasi-localizableC*-algebra.
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Communicated by H. Araki
Laboratoire associé au Centre National de la Recherche Scientifique.
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Dubois-Violette, M. A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. II. Commun.Math. Phys. 54, 151–172 (1977). https://doi.org/10.1007/BF01614135
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DOI: https://doi.org/10.1007/BF01614135