The Dunford-Pettis property in Jb*-Triples

JB*-triples occur in the study of bounded symmetric domainsin several complex variables and in the study of contractiveprojections on C*-algebras. These spaces are equipped with aternary product {·,·,·}, the Jordan tripleproduct, and are essentially geometric objects in that the linearisometries between them are exactly the linear bijections preservingthe Jordan triple product (cf. [23]).     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Inner Derivations and Primal Ideals of C*-Algebras

Let A be a C*-algebra. For a A let D(a, A) denote the innerderivation induced by a, regarded as a bounded operator on A,and let d(a, Z(A)) denote the distance of a from Z(A), the centreof A. Let K(A) be the smallest number in [0, ] such that d(a,Z(A)) K(A)||D(a, A)|| for all a A. It is shown that if A isnon-commutative and has an identity then either K(A) = , or K(A) = 1 / 3, or K(A) 1. Necessaryand sufficient conditions for these three possibilities aregiven in terms of the primitive and primal ideals of A. If Ais a quotient of an AW*-algebra then K(A) . Helly's Theorem is used to show that if A is aweakly central C*-algebra then K(A) 1.     

Derivations on Real and Complex JB*-Triples

At the regional conference held at the University of California,Irvine, in 1985 [24], Harald Upmeier posed three basic questionsregarding derivations on JB*-triples: (1) Are derivations automatically bounded? (2) When are all bounded derivations inner? (3) Can bounded derivations be approximated by inner derivations? These three questions had all been answered in the binary cases.Question 1 was answered affirmatively by Sakai [17] for C*-algebrasand by Upmeier [23] for JB-algebras. Question 2 was answeredby Sakai [18] and Kadison [12] for von Neumann algebras andby Upmeier [23] for JW-algebras. Question 3 was answered byUpmeier [23] for JB-algebras, and it follows trivially fromthe Kadison–Sakai answer to question 2 in the case ofC*-algebras. In the ternary case, both question 1 and question 3 were answeredby Barton and Friedman in [3] for complex JB*-triples. In thispaper, we consider question 2 for real and complex JBW*-triplesand question 1 and question 3 for real JB*-triples. A real orcomplex JB*-triple is said to have the inner derivation propertyif every derivation on it is inner. By pure algebra, every finite-dimensionalJB*-triple has the inner derivation property. Our main results,Theorems 2, 3 and 4 and Corollaries 2 and 3 determine whichof the infinite-dimensional real or complex Cartan factors havethe inner derivation property.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

Collinear Systems and Normal Contractive Projections on JBW*-Triples

Given a family {x _k }_k∈K of elements x _k in the predual A _* of a JBW^*-triple A, such that the support tripotents e _k of x _k form a collinear system in the sense of [31], necessary and sufficient criteria for the existence of a contractive projection from A _*. onto the subspace are provided. Preparatory to these results, and interesting in itself, is a set of necessary and sufficient algebraic conditions upon a contractive projection P on A for its range PA to be a subtriple. The results also provide criteria for the range of a normal contractive projection on A to be a Hilbert space. Supported by the Irish Research Council for Science, Engineering and Technology, Grant No. R 9854.     

Faithful representations of left C*-modules

The existence of a faithful modular representation of a left module $ \mathfrak{X} $ \mathfrak{X} over a C*-algebra $ \mathfrak{A}_\# $ \mathfrak{A}_\# possessing sufficiently many traces is proved.     

New Kinds of Fuzzy Ideals in BCI-algebras*

Following the seminal work of Xi on the definition of fuzzy ideal in BCI-algebras, three new kinds of definitions of fuzzy ideal of BCI-algebras are proposed. First, by the use of the relations between fuzzy points and fuzzy sets, the definition of a (s,t]-fuzzy ideals of BCI-algebras is introduced. The acceptable nontrivial concepts obtained in this manner are the - fuzzy ideals and -fuzzy ideals. Second, based on the concept of falling shadow, a theoretical approach of fuzzy ideal is established and the fuzzy ideal based on t-norm is proposed. Finally, by the use of the implication operators of fuzzy logic, an R-fuzzy ideals is proposed and relations between the (s,t]-fuzzy ideals and R-fuzzy ideals are discussed. *Foundation Item:Supported by NNSF of China(60274016). Biography:Zhang Guang-ji(1947-),male,Professor     

On Minimal and Maximal Left Ideals in Ordered Semigroups*

The concept of maximal left ideals is introduced in ordered semigroups. Characterizatations of minimal left and maximal left ideals are given.     

拓扑分次C*-代数中的对角不变理想

We study diagonal invariant ideals of topologically graded C＊-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author＇s previous works on this topic.     

Diagonal Invariant Ideals of Topologically Graded C*-algebras

We study diagonal invariant ideals of topologically graded C~*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author's previous works on this topic.     

基于L*-格值逻辑上的BCH-代数中的直觉不分明化理想

在L*-格值逻辑的语义框架下,以L*-格值上的Lukasiewicz蕴涵算子为工具定义了L*-格值逻辑上的直觉不分明化BCH-代数的概念,将用集论所刻画的BCH-代数与理想的概念在L*-格值谓词演算下给予了新的刻画,讨论了直觉不分明化BCH-代数中的理想、闭理想及q理想的有关性质.     

Ideals of a C *-algebra generated by an operator algebra

In this paper, we consider ideals of a C ^*-algebra C^*(B){C^*(\mathcal{B})} generated by an operator algebra B{\mathcal{B}} . A closed ideal J í C^*(B){J\subseteq C^*(\mathcal{B})} is called a K-boundary ideal if the restriction of the quotient map on B{\mathcal{B}} has a completely bounded inverse with cb-norm equal to K ⁻¹. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from C^*(B){C^*(\mathcal{B})} onto B{\mathcal{B}} which is a projection on B{\mathcal{B}} is completely bounded. Moreover, we prove that Kadison’s similarity problem is decided on one particular C ^*-algebra which is a completion of the *-double of M₂(\mathbbC){M_2(\mathbb{C})} .     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

Primary Ideals and Valuation Ideals in Semigroups

Let S be a cancellation torsion-free additive semigroup with identity 0 and let S {0}. We consider the relation between the valuation ideals and primary ideals of S. We give a characterization that each primary ideal is a valuation ideal in S and also give a characterization that each valuation ideal is a primary ideal in S.AMS Subject Classification (1991): 20M14     

BCH-代数的拟结合理想和拟结合模糊理想

在BCH-代数中,引入了拟结合理想和模糊拟结合理想的概念.研究了BCH-代数中模糊拟结合理想的相关性质,给出了BCH-代数拟结合理想的构造.     

Soft Ideals and Arithmetic Mean Ideals

This article investigates the soft-interior (se) and the soft-cover (sc) of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study in [10] of the multiplicity of traces. Many classical ideals are ‘soft’, i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals in [6, 7] and arithmetic mean operations at infinity (am-∞) were studied in [10]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-∞ interior of an ideal. Both authors were partially supported by grants of the Charles Phelps Taft Research Center; the second named author was partially supported by NSF Grants DMS 95-03062 and DMS 97-06911.