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.     

Completely Positive Definite Maps Over Topological^＊—algebras

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Norm Attainability of Elementary Operators and Derivations

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Elementary Operators on Standard Operator Algebras

We present a new simple proof of the structural result for elementary operators on standard operator algebras. Using the idea of this short proof we can characterize surjective maps between standard operator algebras having a certain multiplicativity-like property appearing in the abstract definition of elementary operators of length one. In particular, we show that such maps are automatically additive.     

Elementary Operators on Standard Operator Algebras

We present a new simple proof of the structural result for elementary operators on standard operator algebras. Using the idea of this short proof we can characterize surjective maps between standard operator algebras having a certain multiplicativity-like property appearing in the abstract definition of elementary operators of length one. In particular, we show that such maps are automatically additive.     

The Lower Socle and Finite Rank Elementary Operators

Abstract

We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a _k , b _k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ? a ₁ xb ₁ + ? + a _n xb _n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.     

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     

Extremal Unital Completely Positive Maps and Their Symmetries

We consider the set \(P_1({\mathcal A},{\mathcal M})\) (respectively \(CP_1({\mathcal A},{\mathcal M})\) of unital positive (completely) maps from a \(C^*\) algebra \({\mathcal A}\) to a von-Neumann sub-algebra \({\mathcal M}\) of \({\mathcal B}({\mathcal H})\), the algebra of bounded linear operators on a Hilbert space \({\mathcal H}\). We study the extreme points of the convex set \(P_1({\mathcal A},{\mathcal M})\) (\(CP_1({\mathcal A},{\mathcal M})\)) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from \(\hat{{\mathcal A}}\) to \({\mathcal M}\), where \({\mathcal A}^{**}\) is the universal enveloping von-Neumann algebra over \({\mathcal A}\). If \({\mathcal A}={\mathcal M}\) then a (completely) positive map \(\tau \) admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if \({\mathcal M}\) is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on \({\mathcal M}\) to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role.     

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.     

用线性弱正算子逼近

用线性正算子的逼近理论飞速发展,但正性是一个较强的限制,孙永生,王仁宏等研究过减弱正性限制,作者研究用线性弱正算子逼近,推广Korovkin定理和Grundmann定理等等.     

保持算子乘积投影的线性映射

设B(H)是复Hilbert空间H上的有界线性算子全体且dim H≥2.本文证明了B(H)上的线性满射φ保持两个算子乘积非零投影性的充分必要条件是存在B(H)中的酉算子U以及复常数λ满足λ~2=1,使得φ(X)=λU~*XU,(?)X∈B(H).同时也得到了线性映射保持两个算子Jordan三乘积非零投影的充分必要条件.     

Fuzzy带算子环和Fuzzy带算理想

提出Fuzzy带算子环、Fuzzy带算理想和Fuzzy带算商环的定义,讨论它们的一些初等性质。     

Matrix Summability and Positive Linear Operators

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C_ρ1 into a weighted space B_ρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.     

Hyperbolic Sequences of Linear Operators and Evolution Maps

For a nonautonomous dynamics defined by a sequence of linear operators on a Banach space, we give a characterization of its nonuniform hyperbolicity in terms of the hyperbolicity of an associated evolution map. We also obtain a generalization of this characterization to the class of nonuniformly hyperbolic cocycles over a locally compact topological space with values on the set of bounded linear operators.     

共变完全多正线性映射的共变投射表示

研究了C*-代数中的共变完全多正线性映射,证明了共变完全多正线性映射可以诱导Hilbert C*-模上的共变投射表示,并且给出了共变完全多正线性映射的KS- GNS(Kasparov,Stinespring,Gel’fand,Naimark,Segal)构造.     

局部C线性算子群

本文定义了一种局部C群,讨论了局部C群与局部C半群的关系及一些基本性质,并建立了局部C群的生成定理．     

以仿正规算子为参数的初等算子的性质

若对x∈H,‖Tx‖~2≤‖T~2x‖‖x‖,则称T是仿正规算子.d_(AB)表示δ_(AB)或△_(AB),其中δ_(AB)和△_(AB)分别表示Banach空间B(H)上的广义导算子和初等算子,其定义为δ_(AB)X=AX-XB,△_(AB)X=AXB-X,X∈B(H).若A和B~*是仿正规算子,则可证d_(AB)是polaroid算子,f∈H(σ(d_(AB))),f(d_(AB))满足广义Weyl定理,f(d_(AB)~*)满足广义a-Weyl定理,其中H(σ(d_(AB)))表示在σ(d_(AB))的某邻域上解析的函数全体.     

Linear Operators Generated by a Countable Number of Quasi-differential Expressions

The theory of linear ordinary quasi-differential operators has been considered in Lebesgue locally integrable spaces on a single interval of the real line. Such spaces are not Banach spaces but can be considered as complete, locally convex, linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete, locally convex, linear topological space but now with the topology derived as a strict inductive limit. This article extends the previous single interval results to the case when a finite or countable number of intervals of the real line is considered. Conjugate and preconjugate linear quasi-differential operators are defined and relationships between these operators are developed.     

Complete Surjections and Complete Epimorphisms over Completely Distributive Lattices

Generators and lattice properties of the poset of complete homomorphisic images of a completely distributive lattice are exploited via the localic methods. Some intrinsic and extrinsic conditions about this poset to be a completely distributive lattice are given. It is shown that the category of completely distributive lattices is co-well-powered,and complete epimorphisms on completely distributive lattice are not necessary to be surjections. Finally, some conditions about complete epimorphisms to be surjections are given.