Locality in GNS representations of deformation quantization. (English) Zbl 0976.81019

The author applies the formal version of the GNS construction in the framework of the deformation quantization, namely the states are constructed as a sort of positive functional on the deformed algebra on a manifold \(M\) with a star product. Then the locality of the star product implies that states have a well defined support on \(M\). Many properties of the states can be naturally expressed in terms of their supports, e.g. two subspaces are orthogonal if and only if their supports are disjoint. The main results of the paper are: a physically reasonable distinction between thermal and pure states in terms of the commutant with local operators, an analogue of the von Neumann’s double commutant theorem, and a formal version of the Tomita-Takesaki theorem.


81S10 Geometry and quantization, symplectic methods
46N50 Applications of functional analysis in quantum physics
47N50 Applications of operator theory in the physical sciences
53D55 Deformation quantization, star products
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