Partially Commutative Moment Problems

Abstract

Partially commutative tensor algebras occur naturally in the algebraic formulation of WIGETYAN field theory. A state on an algebra of this type leads via GNS-construction to a partially commutative family of hermitean operators on HILBEET space. We d i~c u~e the question when these operators can be extended to self adjoint operators preserving the commutation properties and state a necessary and sufficient condition for the existence of such an extension in terms of a positivity property of the state.

Positivity of Wightman functionals and the existence of local nets

“…W are not algebras. There exist stronger positivity conditions [1,10,29] that are sufficient in all cases, but their precise relation to Theorem 3.1 has still to be worked out.…”

“…Extending Φ(f) to a self adjoint operator with the required commutation properties is a special case of a noncommutative moment problem. General formalisms for dealing with such problems have been developed in [15,23,1,29,10]. For the special case at hand a simple criterion due to Powers [22] applies after a suitable modification.…”

“…Remark. Using the formalism of [10] it can be shown that for centrally positive states not only the operator π ω (^4 0 )» but all the operators π ω (A) 9 AeSΆ h have self adjoint extensions in an enlarged Hubert space, such that bounded functions of π ω (A 0 ) commute with bounded functions of π ω (A) for all A e 2I Λ . We shall not need this strengthening of Proposition 2.1 here, however.…”

Integrable Representations of Involutive Algebras and Ore Localization

“…Results of this kind are not unknown in some important cases. For instance, in [4] Borchers and Yngvason work out the case of Borchers algebras, appearing naturally in quantum field theory.…”