245 research outputs found
Diagonals of normal operators with finite spectrum
Let X be a finite set of complex numbers and let A be a normal operator with
spectrum X that acts on a separable Hilbert space H. Relative to a fixed
orthonormal basis e_1,e_2, ... for H, A gives rise to a matrix whose diagonal
is a sequence d=(d_1,d_2,...) with the property that each of its terms d_n
belongs to the convex hull of X. Not all sequences with that property can arise
as the diagonal of a normal operator with spectrum X.
The case where X is a set of real numbers has received a great deal of
attention over the years, and is reasonably well (though incompletely)
understood. In this paper we take up the case in which X is the set of vertices
of a convex polygon in the complex plane. The critical sequences d turn out to
be those that accumulate rapidly in X in the sense that
We show that there is an abelian group -- a quotient of by a
countable subgroup with concrete arithmetic properties -- and a surjective
mapping of such sequences with the following
property: If s(d) is not 0, then d is not the diagonal of any such operator A.
We also show that while this is the only obstruction when X contains two
points, there are other (as yet unknown) obstructions when X contains more than
two points
A Simple Proof of the Fundamental Theorem about Arveson Systems
With every Eo-semigroup (acting on the algebra of of bounded operators on a
separable infinite-dimensional Hilbert space) there is an associated Arveson
system. One of the most important results about Arveson systems is that every
Arveson system is the one associated with an Eo-semigroup. In these notes we
give a new proof of this result that is considerably simpler than the existing
ones and allows for a generalization to product systems of Hilbert module (to
be published elsewhere).Comment: Publication data added, acknowledgements and a note after acceptance
added, corrects a number of inconveniences that have been produced in the
published version during the publication proces
The noncommutative Kubo Formula: Applications to Transport in Disordered Topological Insulators with and without Magnetic Fields
The non-commutative theory of charge transport in mesoscopic aperiodic
systems under magnetic fields, developed by Bellissard, Shulz-Baldes and
collaborators in the 90's, is complemented with a practical numerical
implementation. The scheme, which is developed within a -algebraic
framework, enable efficient evaluations of the non-commutative Kubo formula,
with errors that vanish exponentially fast in the thermodynamic limit.
Applications to a model of a 2-dimensional Quantum spin-Hall insulator are
given. The conductivity tensor is mapped as function of Fermi level, disorder
strength and temperature and the phase diagram in the plane of Fermi level and
disorder strength is quantitatively derived from the transport simulations.
Simulations at finite magnetic field strength are also presented.Comment: 10 figure
The Index of (White) Noises and their Product Systems
(See detailed abstract in the article.) We single out the correct class of
spatial product systems (and the spatial endomorphism semigroups with which the
product systems are associated) that allows the most far reaching analogy in
their classifiaction when compared with Arveson systems. The main differences
are that mere existence of a unit is not it sufficient: The unit must be
CENTRAL. And the tensor product under which the index is additive is not
available for product systems of Hilbert modules. It must be replaced by a new
product that even for Arveson systems need not coincide with the tensor
product
Algebraic approach to quantum field theory on non-globally-hyperbolic spacetimes
The mathematical formalism for linear quantum field theory on curved
spacetime depends in an essential way on the assumption of global
hyperbolicity. Physically, what lie at the foundation of any formalism for
quantization in curved spacetime are the canonical commutation relations,
imposed on the field operators evaluated at a global Cauchy surface. In the
algebraic formulation of linear quantum field theory, the canonical commutation
relations are restated in terms of a well-defined symplectic structure on the
space of smooth solutions, and the local field algebra is constructed as the
Weyl algebra associated to this symplectic vector space. When spacetime is not
globally hyperbolic, e.g. when it contains naked singularities or closed
timelike curves, a global Cauchy surface does not exist, and there is no
obvious way to formulate the canonical commutation relations, hence no obvious
way to construct the field algebra. In a paper submitted elsewhere, we report
on a generalization of the algebraic framework for quantum field theory to
arbitrary topological spaces which do not necessarily have a spacetime metric
defined on them at the outset. Taking this generalization as a starting point,
in this paper we give a prescription for constructing the field algebra of a
(massless or massive) Klein-Gordon field on an arbitrary background spacetime.
When spacetime is globally hyperbolic, the theory defined by our construction
coincides with the ordinary Klein-Gordon field theory on aComment: 21 pages, UCSBTH-92-4
The existence problem for dynamics of dissipative systems in quantum probability
Motivated by existence problems for dissipative systems arising naturally in
lattice models from quantum statistical mechanics, we consider the following
-algebraic setting: A given hermitian dissipative mapping is
densely defined in a unital -algebra . The identity
element in is also in the domain of . Completely
dissipative maps are defined by the requirement that the induced maps,
, are dissipative on the by complex
matrices over for all . We establish the existence of different
types of maximal extensions of completely dissipative maps. If the enveloping
von Neumann algebra of is injective, we show the existence of an
extension of which is the infinitesimal generator of a quantum
dynamical semigroup of completely positive maps in the von Neumann algebra. If
is a given well-behaved *-derivation, then we show that each of the
maps and is completely dissipative.Comment: 24 pages, LaTeX/REVTeX v. 4.0, submitted to J. Math. Phys.; PACS 02.,
02.10.Hh, 02.30.Tb, 03.65.-w, 05.30.-
Operator theory and function theory in Drury-Arveson space and its quotients
The Drury-Arveson space , also known as symmetric Fock space or the
-shift space, is a Hilbert function space that has a natural -tuple of
operators acting on it, which gives it the structure of a Hilbert module. This
survey aims to introduce the Drury-Arveson space, to give a panoramic view of
the main operator theoretic and function theoretic aspects of this space, and
to describe the universal role that it plays in multivariable operator theory
and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page
Strongly Incompatible Quantum Devices
The fact that there are quantum observables without a simultaneous
measurement is one of the fundamental characteristics of quantum mechanics. In
this work we expand the concept of joint measurability to all kinds of possible
measurement devices, and we call this relation compatibility. Two devices are
incompatible if they cannot be implemented as parts of a single measurement
setup. We introduce also a more stringent notion of incompatibility, strong
incompatibility. Both incompatibility and strong incompatibility are rigorously
characterized and their difference is demonstrated by examples.Comment: 27 pages (AMSart), 6 figure
Tensor products of subspace lattices and rank one density
We show that, if is a subspace lattice with the property that the rank
one subspace of its operator algebra is weak* dense, is a commutative
subspace lattice and is the lattice of all projections on a separable
infinite dimensional Hilbert space, then the lattice is
reflexive. If is moreover an atomic Boolean subspace lattice while is
any subspace lattice, we provide a concrete lattice theoretic description of
in terms of projection valued functions defined on the set of
atoms of . As a consequence, we show that the Lattice Tensor Product Formula
holds for \Alg M and any other reflexive operator algebra and give several
further corollaries of these results.Comment: 15 page
CQG algebras: a direct algebraic approach to compact quantum groups
The purely algebraic notion of CQG algebra (algebra of functions on a compact
quantum group) is defined. In a straightforward algebraic manner, the
Peter-Weyl theorem for CQG algebras and the existence of a unique positive
definite Haar functional on any CQG algebra are established. It is shown that a
CQG algebra can be naturally completed to a -algebra. The relations
between our approach and several other approaches to compact quantum groups are
discussed.Comment: 14 pp., Plain TeX, accepted by Lett. Math. Phy
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