125 research outputs found
Affine orbifolds and rational conformal field theory extensions of W_{1+infinity}
Chiral orbifold models are defined as gauge field theories with a finite
gauge group . We start with a conformal current algebra A associated
with a connected compact Lie group G and a negative definite integral invariant
bilinear form on its Lie algebra. Any finite group of inner
automorphisms or A (in particular, any finite subgroup of G) gives rise to a
gauge theory with a chiral subalgebra of local
observables invariant under . A set of positive energy
modules is constructed whose characters span, under some assumptions on
, a finite dimensional unitary representation of . We compute
their asymptotic dimensions (thus singling out the nontrivial orbifold modules)
and find explicit formulae for the modular transformations and hence, for the
fusion rules.
As an application we construct a family of rational conformal field theory
(RCFT) extensions of that appear to provide a bridge between two
approaches to the quantum Hall effect.Comment: 64 pages, amste
Sets of multiplicity and closable multipliers on group algebras
We undertake a detailed study of the sets of multiplicity in a second
countable locally compact group and their operator versions. We establish a
symbolic calculus for normal completely bounded maps from the space
of bounded linear operators on into the von
Neumann algebra of and use it to show that a closed subset
is a set of multiplicity if and only if the set is a set of operator multiplicity.
Analogous results are established for -sets and -sets. We show that
the property of being a set of multiplicity is preserved under various
operations, including taking direct products, and establish an Inverse Image
Theorem for such sets. We characterise the sets of finite width that are also
sets of operator multiplicity, and show that every compact operator supported
on a set of finite width can be approximated by sums of rank one operators
supported on the same set. We show that, if satisfies a mild approximation
condition, pointwise multiplication by a given measurable function defines a closable multiplier on the reduced C*-algebra
of if and only if Schur multiplication by the function , given by , is a closable
operator when viewed as a densely defined linear map on the space of compact
operators on . Similar results are obtained for multipliers on .Comment: 51 page
Local Operator Multipliers and Positivity
We establish an unbounded version of Stinespring's Theorem and a lifting
result for Stinespring representations of completely positive modular maps
defined on the space of all compact operators. We apply these results to study
positivity for Schur multipliers. We characterise positive local Schur
multipliers, and provide a description of positive local Schur multipliers of
Toeplitz type. We introduce local operator multipliers as a non-commutative
analogue of local Schur multipliers, and obtain a characterisation that extends
earlier results concerning operator multipliers and local Schur multipliers. We
provide a description of the positive local operator multipliers in terms of
approximation by elements of canonical positive cones.Comment: 31 page
Complexity and capacity bounds for quantum channels
We generalise some well-known graph parameters to operator systems by
considering their underlying quantum channels. In particular, we introduce the
quantum complexity as the dimension of the smallest co-domain Hilbert space a
quantum channel requires to realise a given operator system as its
non-commutative confusability graph. We describe quantum complexity as a
generalised minimum semidefinite rank and, in the case of a graph operator
system, as a quantum intersection number. The quantum complexity and a closely
related quantum version of orthogonal rank turn out to be upper bounds for the
Shannon zero-error capacity of a quantum channel, and we construct examples for
which these bounds beat the best previously known general upper bound for the
capacity of quantum channels, given by the quantum Lov\'asz theta number
Positive Herz-Schur multipliers and approximation properties of crossed products
For a -algebra and a set we give a Stinespring-type
characterisation of the completely positive Schur -multipliers on
. We then relate them to completely positive Herz-Schur
multipliers on -algebraic crossed products of the form
, with a discrete group, whose various versions were
considered earlier by Anantharaman-Delaroche, B\'edos and Conti, and Dong and
Ruan. The latter maps are shown to implement approximation properties, such as
nuclearity or the Haagerup property, for .Comment: 21 pages, v2 corrects a few minor typos. The paper will appear in the
Mathematical Proceedings of the Cambridge Philosophical Societ
Schur multipliers of Cartan pairs
We define the Schur multipliers of a separable von Neumann algebra M with
Cartan masa A, generalising the classical Schur multipliers of . We
characterise these as the normal A-bimodule maps on M. If M contains a direct
summand isomorphic to the hyperfinite II_1 factor, then we show that the Schur
multipliers arising from the extended Haagerup tensor product are strictly contained in the algebra of all Schur multipliers
Quantum no-signalling correlations and non-local games
We introduce and examine three subclasses of the family of quantum
no-signalling (QNS) correlations introduced by Duan and Winter: quantum
commuting, quantum and local. We formalise the notion of a universal TRO of a
block operator isometry, define an operator system, universal for stochastic
operator matrices, and realise it as a quotient of a matrix algebra. We
describe the classes of QNS correlations in terms of states on the tensor
products of two copies of the universal operator system, and specialise the
correlation classes and their representations to classical-to-quantum
correlations. We study various quantum versions of synchronous no-signalling
correlations and show that they possess invariance properties for suitable sets
of states. We introduce quantum non-local games as a generalisation of
non-local games. We define the operation of quantum game composition and show
that the perfect strategies belonging to a certain class are closed under
channel composition. We specialise to the case of graph colourings, where we
exhibit quantum versions of the orthogonal rank of a graph as the optimal
output dimension for which perfect classical-to-quantum strategies of the graph
colouring game exist, as well as to non-commutative graph homomorphisms, where
we identify quantum versions of non-commutative graph homomorphisms introduced
by Stahlke.Comment: 72 page
Homomorphisms of quantum hypergraphs
We introduce quantum homomorphisms between quantum hypergraphs through the
existence of perfect strategies for quantum non-local games, canonically
associated with the quantum hypergraphs. We show that the relation of
homomorphism of a given type satisfies natural analogues of the properties of a
pre-order. We show that quantum hypergraph homomorphisms of local type are
closely related, and in some cases identical, to the TRO equivalence of finite
dimensionally acting operator spaces, canonically associated with the
hypergraphs
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