Communications in Mathematical Physics

, Volume 210, Issue 2, pp 467–495 | Cite as

Locality in GNS Representations of Deformation Quantization

  • Stefan Waldmann


In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over ℂ[[λ]] and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish “thermal” from “pure” representations. The computation of the local commutant is xemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann's double commutant theorem is proved in the particular situation of a GNS representation with respect to a KMS functional and for the Schrödinger representation on cotangent bundles. Finally we prove a formal version of the Tomita–Takesaki theorem.


Local Operator Formal Version Cotangent Bundle Deformation Quantization Thermal Representation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Stefan Waldmann
    • 1
  1. 1.Département de Mathématique, Campus Plaine, C. P. 213, Boulevard du Triomphe, 1050 Bruxelles,¶Belgique. E-mail:

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