Three Commuting,Unital, Completely Positive Maps that have no Minimal Dilation

In this note we prove that there exist at least two examples of three commuting, unital, completely positive maps that have no dilation on a type I factor, and no minimal dilation on any von Neumann algebra.     

Decomposition of Completely Positive Maps

The paper is concerned with completely positive maps on the algebra of unbounded operatore L+(D) and on its completion L(D, D⁺). A decomposition theorem for continuous positive functionals is proved in [Tim. Loef.), and [Scholz 91] contains a generalization to maps into operator algebra on finite dimensional Hilbert spaces H₀. The aim of the present paper is to construct an analogous decomposition without the assumption that H₀ is finite dimensional. Moreover, the Kraus - theorem [Kraus] is proved for normal completely positive mappings on L(D, D⁺). The paper is organized as follows. Section 1 contains the necessary definitions and notations. In Section 2 we prove the decomposition theorem. Section 3 deal with the structure of the normal completely positive mappings.     

Completely Positive Definite Maps Over Topological^＊—algebras

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The Purification of Completely Positive Maps

We generalize the concept of purification of states to completelypositive maps and prove that a CP map has a purification ifand only if it is orthogonality preserving.     

Elementary Operators and Completely Positive Linear Maps

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C*-Convex Sets and Completely Positive Maps

We prove a geometric version of an operator valued Hahn–Banach theorem and use it to study sets K that are A-convex over a unital C*-algebra A in the sense that \({\sum_{j=1}^{n} a_{j}^{*}y_{j}a_{j}\in K}\) whenever \({y_{j}\in K}\) and \({a_{j}\in A}\) with \({\sum_{j=1}^{n}a_{j}^{*}a_{j}=1}\). We show how weak* compact such sets can be realized as concrete sets of unital completely positive maps. An application to C*-extreme points is also presented.     

The Curvature and Index of Completely Positive Maps

We study conjugacy invariants for completely positive maps thatare inspired by the concept of curvature introduced for commutingd-tuples of contractions by Arveson. 2000 Mathematics SubjectClassification 46L53, 46L55, 46L57, 46L87, 47L55, 46L07, 46L08     

Completely Positive Definite Maps on σ-C*-algebras

Recently,there has been increased interest[1-8]in topological*-algebras that are inverse     

Strict Completely Positive Maps between Locally C*-Algebras and Representations on Hilbert Modules

The structure of the continuous strict completely positive linearmaps between locally C^*-algebras is described.     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

伪轨跟踪与完全正熵

本文主要给出了紧致度量空间上的连续满射f具有完全正熵的一个必要条件及当f具有伪轨跟踪性质时,f有完全正熵的一些等价条件．     

Tight Frames and Their Symmetries

The aim of this paper is to investigate the symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.     

Characterising Completely Positive Elementary Operators

Let S be an elementary operator on a C*-algebra with minimallength l(S)1. Then S is completely positive if and only if Sis n-positive for some natural number n(l(S)–1)/2. 1991Mathematics Subject Classification 47B47.     

Hypergeometric Type Functions and Their Symmetries

The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F ₁ equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed.     

Positive Solutions and Extremal Points for Differential Equations

Extremal point properties are examined for the boundary value problem x⁽ⁿ⁾ + ? ^n-2 _i=0 A_i(t)x⁽ⁱ⁾ = 0, x⁽ⁱ⁾(0) = x^(n-2)(T) = 0, 0?i?n-2, where the A_i's, a characterization for the first extremal point is given in terms of the existence of a solution which is positive with respect to a cone in a Banach space. Also, an existence theorem is obtained for a related nonlinear problem     

关于完全正矩阵的几点注记(英文)

本文给出了一个 n×n非负、对称、弱对角占优矩阵 A为完全正的一个充分条件 .我们还给出了较好的算法 ,用以获得关于矩阵 A(当 A为完全正时 )的分解指数的一个上界 .     

关于圈的完全正矩阵实现

本文给出了一个关联图为圈的非负、半正定矩阵A为完全正的一个充要条件.我们还证明了这样的矩阵A（当A为完全正时）的分解指数即为A的阶数.     

Factorization Index for Completely Positive Graphs

In this paper, we prove that the set of all factorization indices of a completely positive graph has no gaps. In other words, we give an affirmative answer to a question raised by N. Kogan and A. Berman [8] in the case of completely positive graphs. Received December 9, 1997, Accepted May 16, 2002