343 research outputs found
Locality in GNS Representations of Deformation Quantization
In the framework of deformation quantization we apply the formal GNS
construction to find representations of the deformed algebras in pre-Hilbert
spaces over 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 exemplified 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 Schr\"odinger representation on cotangent bundles. Finally we prove
a formal version of the Tomita-Takesaki theorem.Comment: LaTeX2e, 29 page
The H-Covariant Strong Picard Groupoid
The notion of H-covariant strong Morita equivalence is introduced for
*-algebras over C = R(i) with an ordered ring R which are equipped with a
*-action of a Hopf *-algebra H. This defines a corresponding H-covariant strong
Picard groupoid which encodes the entire Morita theory. Dropping the positivity
conditions one obtains H-covariant *-Morita equivalence with its H-covariant
*-Picard groupoid. We discuss various groupoid morphisms between the
corresponding notions of the Picard groupoids. Moreover, we realize several
Morita invariants in this context as arising from actions of the H-covariant
strong Picard groupoid. Crossed products and their Morita theory are
investigated using a groupoid morphism from the H-covariant strong Picard
groupoid into the strong Picard groupoid of the crossed products.Comment: LaTeX 2e, 50 pages. Revised version with additional examples and
references. To appear in JPA
- …