22,516 research outputs found
Partial orders on partial isometries
This paper studies three natural pre-orders of increasing generality on the
set of all completely non-unitary partial isometries with equal defect indices.
We show that the problem of determining when one partial isometry is less than
another with respect to these pre-orders is equivalent to the existence of a
bounded (or isometric) multiplier between two natural reproducing kernel
Hilbert spaces of analytic functions. For large classes of partial isometries
these spaces can be realized as the well-known model subspaces and
deBranges-Rovnyak spaces. This characterization is applied to investigate
properties of these pre-orders and the equivalence classes they generate.Comment: 30 pages. To appear in Journal of Operator Theor
A classification of 2D fermionic and bosonic topological orders
The string-net approach by Levin and Wen, and the local unitary
transformation approach by Chen, Gu, and Wen, provide ways to classify
topological orders with gappable edge in 2D bosonic systems. The two approaches
reveal that the mathematical framework for 2+1D bosonic topological order with
gappable edge is closely related to unitary fusion category theory. In this
paper, we generalize these systematic descriptions of topological orders to 2D
fermion systems. We find a classification of 2+1D fermionic topological orders
with gappable edge in terms of the following set of data , that satisfy a set of non-linear
algebraic equations. The exactly soluble Hamiltonians can be constructed from
the above data on any lattices to realize the corresponding topological orders.
When , our result recovers the previous classification of 2+1D
bosonic topological orders with gappable edge.Comment: 19 page 5 figures, RevTeX
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
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