Positive linear maps on operator algebras

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.     

Positive maps of low dimensional matrix algebras

It is shown that any positive map from M₂ into M₃ is a sum of completely positive and completely copositive maps. This result does not hold for maps into M₄. A generalization of the Kadison inequality is suggested and proved for positive maps defined on M₂ or M₃.     

Frobenius theory for positive maps of von Neumann algebras

Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.     

A Lie algebraic setting for Miura maps related to an energy dependent linear problem

We study the problem of complete parametrization of the moduli space ofSU(2) Yang-Mills-Higgs monopoles in terms of a nonlinear integrable system. It is shown that the moduli space is homeomorphic to the solution space of a new generalization of finite nonperiodic Toda equation called the complex cyclic-Toda hierarchy.     

Some remarks on covariant completely positive linear maps on C1-algebras

A Stinespring type theorem, for covariant completely positive linear maps of C¹-algebras, is proved. As a corollary, a criterion for a representation of a locally compact group to be a subrepresentation of an induced one is given. The relations between covariant completely positive maps of a C¹-algebra, and the completely positive maps of the associated covariance algebra are discussed.     

A contribution to the linear magnetoresistance problem

It is shown that semi-classical transport theory provides an extremely accurate prediction of the limiting high field transverse magnetoresistivities of the compensated metals Cd, Pb and W. These results invalidate those theoretical models based on scattering effects which have been proposed to explain the linear term in the magnetoresistivities of K, Al and In. We argue that the only remaining viable models are those based on the effects of sample imperfections and inhomogeneities.     

Dual isomonodromic deformations and moment maps to loop algebras

The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to dual pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painlevé transcendentsP _V and _VI .Research supported in part by the Natural Sciences and Engineering Research Council of Canada.     

n(G)-nonextendibility of linear positive maps

Linear positive maps of C¹-algebras into the algebra of all bounded operators acting on a Hilbert space are considered. A special class of n(G)-nonextendible maps, which contains the class of nonextendible positive maps introduced by S.L. Woronowicz [9], is defined and studied. A map is n(G)-positive if it is n-positive or covariant with respect to the action of some group G of automorphisms of a C¹-algebra. n(G)-nonextendible maps are those which are nonextendible in this class in the sense of [9].Any n(G)-positive map can be obtained from an n(G)-nonextendible one by restriction.     

Completely positive maps on the CCR-algebra

Given any operator on the testfunction space, the general form of the induced completely positive map of the C ^*-algebra of the canonical commutation relations is characterized.Postal address: Instituut voor Theoretische Fysica, KUL, Celestijnenlaan 200 D, B-3030 Heverlee, Belgium.     

A remark on matrix product operator algebras,anyons and subfactors

Letters in Mathematical Physics - We show that the mathematical structures in a recent work of...     

Representations of Lie algebras and the problem of noncommutative integrability of linear differential equations

An algorithm is proposed for integrating linear partial differential equations with the help of a special set of noncommuting linear differential operators — an analogue of the method of noncommutative integration of finite-dimensional Hamiltonian systems. The algorithm allows one to construct a parametric family of solutions of an equation satisfying the requirement of completeness. The case is considered when the noncommutative set of operators form a Lie algebra. An essential element of the algorithm is the representation of this algebra by linear differential operators in the space of parameters. A connection is indicated of the given method with the method of separation of variables, and also with problems of the theory of representations of Lie algebras. Let us emphasize that on the whole the proposed algorithm differs from the method of separation of variables, in which sets of commuting symmetry operators are used.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 95–100, April, 1991.     

A linear code for the sawtooth and cat maps

The sawtooth maps are a one-parameter set of piecewise linear area preserving maps on the torus. For positive integer values of the parameter K they are automorphisms of the torus, known as the cat maps. We present a symbolic dynamics for these maps in which the symbols are integers. This code is related to a practical problem of the stabilisation of a system which is perturbed by impulses. The code is linear in the sense that an orbit and its code are linearly related, so it is not difficult to obtain a good approximation to one from the other in practice. A stationary stochastic process for generating the code is given explicitly. The theory uses Green function methods, which are also used to study ordered periodic orbits and cantori. The problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed.     

Completely positive quasi-free maps on the CAR algebra

We construct and study a new class of quasi-free completely positive maps on theC*-algebra of the canonical anti-commutation relations.     

A one-dimensionalN Fermion problem with factorizedS matrix

A one dimensionalN Fermion problem with attractive or repulsive function interaction is solved by Bethe's hypothesis. TheS matrix factorizes and is explicitly given.     

Density matrix approach to the Hartree-Fock problem

The problem of determining the best one-body density matrix which minimizes the total energy of a many fermion system is studied, with the idea of providing an alternative approach to the method of Roothaan for solving the Hartree-Fock equation.     

On trace representation of linear functionals on unbounded operator algebras

In this paper we consider the following problem: Given a *-algebraA of unbounded operators, under what conditions is every strongly positive linear functionalf onA a trace functional, i.e. of the formf(a)=Trtta,aA, wheret is an appropriate positive nuclear operator. Further, the linear functionalsf onA which can be represented asf(a)=Trta (f andt not necessarily positive) are characterized by their continuity in a certain topology. Some applications (canonical commutation relations on the Schwartz space, integrable representations of enveloping algebras) are discussed.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

On positive functionals on algebras of test functions for quantum fields

It is shown that the *-algebraR of test-functions for a quantum field is reduced, i.e. for eachbR,b0, there exists a positive continuous linear functionalW(a) onR withW(b)0.On leave of absence from Mathematisches Institut, Karl-Marx-Universität Leipzig.     

Iso-spectral deformations of general matrix and their reductions on Lie algebras

We study an iso-spectral deformation of the general matrix which is a natural generalization of the nonperiodic Toda lattice equation. This deformation is equivalent to the Cholesky flow, a continuous version of the Cholesky algorithm, introduced by Watkins. We prove the integrability of the deformation and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szegö. Using the formula, the solution to the LU matrix factorization can the constructed explicitly. Based on the root spaces for simple Lie algebras, we consider several reductions of the equation. This leads to generalized Toda equations related to other classical semi-simple Lie algebras which include the integrable systems studied by Bogoyavlensky and Kostant. We show these systems can be solved explicitly in a unified way. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.     

A diagrammatic approach to spin algebras

A reduction formula for the thermal average of the time-ordered product of spin operators for a magnetic moment in a field H is presented. The reduction formula and the spin algebra are used to compute the magnetization. The procedure provides the basis for a diagrammatic approach to interacting spin systems.