19,976 research outputs found
Superselection Theory for Subsystems
An inclusion of observable nets satisfying duality induces an inclusion of
canonical field nets. Any Bose net intermediate between the observable net and
the field net and satisfying duality is the fixed-point net of the field net
under a compact group. This compact group is its canonical gauge group if the
occurrence of sectors with infinite statistics can be ruled out for the
observable net and its vacuum Hilbert space is separable.Comment: 28 pages, LaTe
Duality for convex monoids
Every C*-algebra gives rise to an effect module and a convex space of states,
which are connected via Kadison duality. We explore this duality in several
examples, where the C*-algebra is equipped with the structure of a
finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra
or group algebra of a finite group, the resulting state spaces form convex
monoids. We will prove that both these convex monoids can be obtained from the
other one by taking a coproduct of density matrices on the irreducible
representations. We will also show that the same holds for a tensor product of
a group and a function algebra.Comment: 13 page
2-C*-categories with non-simple units
We study the general structure of 2-C*-categories closed under conjugation,
projections and direct sums. We do not assume units to be simple, i.e. for i_A
the 1-unit corresponding to an object A, the space Hom(i_A, i_A) is a
commutative unital C*-algebra. We show that 2-arrows can be viewed as
continuous sections in Hilbert bundles and describe the behaviour of the fibres
with respect to the categorical structure. We give an example of a
2-C*-Category giving rise to bundles of finite Hopf-algebras in duality. We
make some remarks concerning Frobenius algebras and Q-systems in the general
context of tensor C*-categories with non-simple units.Comment: 47 page
Notions of Infinity in Quantum Physics
In this article we will review some notions of infiniteness that appear in
Hilbert space operators and operator algebras. These include proper
infiniteness, Murray von Neumann's classification into type I and type III
factors and the class of F{/o} lner C*-algebras that capture some aspects of
amenability. We will also mention how these notions reappear in the description
of certain mathematical aspects of quantum mechanics, quantum field theory and
the theory of superselection sectors. We also show that the algebra of the
canonical anti-commutation relations (CAR-algebra) is in the class of F{/o}
lner C*-algebras.Comment: 11 page
Gabor Duality Theory for Morita Equivalent -algebras
The duality principle for Gabor frames is one of the pillars of Gabor
analysis. We establish a far-reaching generalization to Morita equivalent
-algebras where the equivalence bimodule is a finitely generated
projective Hilbert -module. These Hilbert -modules are equipped with
some extra structure and are called Gabor bimodules. We formulate a duality
principle for standard module frames for Gabor bimodules which reduces to the
well-known Gabor duality principle for twisted group -algebras of a
lattice in phase space. We lift all these results to the matrix algebra level
and in the description of the module frames associated to a matrix Gabor
bimodule we introduce -matrix frames, which generalize superframes and
multi-window frames. Density theorems for -matrix frames are
established, which extend the ones for multi-window and super Gabor frames. Our
approach is based on the localization of a Hilbert -module with respect to
a trace.Comment: 36 page
Categorical formulation of quantum algebras
We describe how dagger-Frobenius monoids give the correct categorical
description of certain kinds of finite-dimensional 'quantum algebras'. We
develop the concept of an involution monoid, and use it to construct a
correspondence between finite-dimensional C*-algebras and certain types of
dagger-Frobenius monoids in the category of Hilbert spaces. Using this
technology, we recast the spectral theorems for commutative C*-algebras and for
normal operators into an explicitly categorical language, and we examine the
case that the results of measurements do not form finite sets, but rather
objects in a finite Boolean topos. We describe the relevance of these results
for topological quantum field theory.Comment: 34 pages, to appear in Communications in Mathematical Physic
Remarks on Morphisms of Spectral Geometries
Having in view the study of a version of Gel'fand-Neumark duality adapted to
the context of Alain Connes' spectral triples, in this very preliminary review,
we first present a description of the relevant categories of geometrical
spaces, namely compact Hausdorff smooth finite-dimensional orientable
Riemannian manifolds (or more generally Hermitian bundles of Clifford modules
over them); we give some tentative definitions of the relevant categories of
algebraic structures, namely "propagators" and "spectral correspondences" of
commutative Riemannian spectral triples; and we provide a construction of
functors that associate a naive morphism of spectral triples to every smooth
(totally geodesic) map. The full construction of spectrum functors
(reconstruction theorem for morphisms) and a proof of duality between the
previous "geometrical' and "algebraic" categories are postponed to subsequent
works, but we provide here some hints in this direction. We also show how the
previous categories of "propagators" of commutative C*-algebras embed in the
mildly non-commutative environments of categories of suitable Hilbert
C*-bimodules, factorizable over commutative C*-algebras, with composition given
by internal tensor product.Comment: 9 pages, AMS-LaTeX2e. Reformatted, heavily revised and corrected
version, only for arXiv, of a previous review paper published in East-West
Journal of Mathematics. The main results presented in this review are now
part of F.Jaffrennou PhD thesis "Morphisms of Spectral Geometries" (Mahidol
University, June 2014
Realization of minimal C*-dynamical systems in terms of Cuntz-Pimsner algebras
In the present paper we study tensor C*-categories with non-simple unit
realised as C*-dynamical systems (F,G,\beta) with a compact (non-Abelian) group
G and fixed point algebra A := F^G. We consider C*-dynamical systems with
minimal relative commutant of A in F, i.e. A' \cap F = Z, where Z is the center
of A which we assume to be nontrivial. We give first several constructions of
minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra
associated to a suitable Z-bimodule. These examples are labelled by the action
of a discrete Abelian group (which we call the chain group) on Z and by the
choice of a suitable class of finite dimensional representations of G. Second,
we present a construction of a minimal C*-dynamical system with nontrivial Z
that also encodes the representation category of G. In this case the C*-algebra
F is generated by a family of Cuntz-Pimsner algebras, where the product of the
elements in different algebras is twisted by the chain group action. We apply
these constructions to the group G = SU(N).Comment: 34 pages; References updated and typos corrected. To appear in
International Journal of Mathematic
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