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.     

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     

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     

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 .     

标准算子代数上完全保对合性的可加映射

刻画了无限维实或复Banach空间上的标准算子代数间完全保对合性的可加映射,证明了这样的映射是同构的常数倍或(复情形下)共轭同构的常数倍.     

Nonlocal Symmetries for Bilinear Equations and Their Applications

In this paper, nonlocal symmetries for the bilinear KP and bilinear BKP equations are re-studied. Two arbitrary parameters are introduced in these nonlocal symmetries by considering gauge invariance of the bilinear KP and bilinear BKP equations under the transformation     . By expanding these nonlocal symmetries in power series of each of two parameters, we have derived two types of bilinear NKP hierarchies and two types of bilinear NBKP hierarchies. An impressive observation is that bilinear positive and negative KP (NKP) and BKP hierarchies may be derived from the same nonlocal symmetries for the KP and BKP equations. Besides, as two concrete examples, we have derived bilinear Bäcklund transformations for   t ₋₂  -flow of the NKP hierarchy and   t ₋₁  -flow of the NBKP hierarchy. All these results have made it clear that more nice integrable properties would be found for these obtained NKP hierarchies and NBKP hierarchies. Because KP and BKP hierarchies have played an essential role in soliton theory, we believe that the bilinear NKP and NBKP hierarchies will have their right place in this field.