The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

C*-代数的迹迹秩

引入C*-代数迹迹秩的概念,讨论它的基本性质.另外,迹迹秩为零和迹拓扑秩为零的C*-代数等价,同时讨论这类代数的拟对角扩张性质.设0→I→ A→A/I→0是拟对角扩张的短正合列,证明如果TTR(I)≤k且TTR(A/I)=0,则TTR(A)≤k.     

C~*-代数的迹迹秩

引入C~*-代数迹迹秩的概念,讨论它的基本性质.另外,迹迹秩为零和迹拓扑秩为零的C~*-代数等价,同时讨论这类代数的拟对角扩张性质.设O→I→A→A/I→O是拟对角扩张的短正合列,证明如果TTR(I)≤k且TTR(A/I)=0,则TTR(A)≤k.     

迹稳定秩1的C*-代数的稳定有限性与弱迹稳定秩1

主要给出了迹稳定秩1的C*-代数的稳定有限性,证明了如果A是有单位元迹稳定秩1的C*-代数,则A是稳定有限的,引入了弱迹稳定秩1的定义,并且证明了如果有单位元的C*-代数A是迹稳定秩1的,则A是弱迹稳定秩1的.对于单的具有SP性质的有单位元的C*-代数A,如果A是弱迹稳定秩1的,则A是迹稳定秩1的.同时给出了迹稳定秩1的C*-代数的一个等价条件,证明了一个有单位元的可分的C*-代数A是迹稳定秩1的,等价于A=(t4)limn→∞(An,Pn),其中tsr(AN)=1.     

迹稳定秩1的C~*-代数的稳定有限性与弱迹稳定秩1

主要给出了迹稳定秩1的C~*-代数的稳定有限性,证明了如果A是有单位元迹稳定秩1的C~*-代数,则A是稳定有限的,引入了弱迹稳定秩1的定义,并且证明了如果有单位元的C~*-代数A是迹稳定秩1的,则A是弱迹稳定秩1的．对于单的具有SP性质的有单位元的C~*-代数A,如果A是弱迹稳定秩1的,则A是迹稳定秩1的．同时给出了迹稳定秩1的C~*-代数的一个等价条件,证明了一个有单位元的可分的C~*-代数A是迹稳定秩1的,等价于A=(t_4)limn→∞(A_n,p_n),其中tsr(A_n)=1．     

单的迹稳定秩一C*-代数的消去性质

本文证明了一个单的有单位元的迹稳定秩一的C*-代数具有消去律,利用此结果证明了单的有单位元的迹稳定秩一的C*-代数是稳定秩一的.最后讨论了迹稳定秩一的C*-代数的K0群的性质.     

Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces

Let X be a connected finite CW complex and d _X : K ₀(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K ₀(A) = ℚ⊕ kerd _x and K ₁(A) = K ₁(C(X)), then A is isomorphic to an inductive limit of M _{n !}(C(X)). Received April 19, 2001, Accepted April 27, 2001.     

Topological and Nonstandard Extensions

We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the Stone-ech compactification X of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a natural topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.     

Topological and Nonstandard Extensions

We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the Stone-ech compactification X of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a natural topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.     

The Rank of Abelian Varieties over Infinite Galois Extensions

G. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112-128) asked if every Abelian variety A defined over a number field k with dim A>0 has infinite rank over the maximal Abelian extension k^ab of k. We verify this for the Jacobians of cyclic covers of P¹, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).     

Separation Properties of Simultaneous Topological Extensions

Let X be a set and s i( x ) be a (possibly zero) filter in X_i X for i I and x X . A topology pi on X is said to be a simultaneous extension compatible with...     

Generalized Principal Extensions in Topological Algebras

The ??generalized principal extension?? of a subset of a topological algebra is, by definition, a subset of another topological algebra. Hence, the generalized spectrum of an element, ??local spectrum??, of a topological algebra is thus a subset of a topological algebra, not necessarily of ${{\mathbb C}}$ . We also give a criterion for the continuity of the (generalized) Newburg map (Theorem 3.1).     

A Hilbert $$C^*$$ -Module with Extremal Properties

Functional Analysis and Its Applications - We construct an example of a Hilbert $$C^*$$ -module which shows that Troitsky’s theorem on the geometric essence of $$ {\mathcal A} $$ -compact...     

Boolean Topological Distributive Lattices and Canonical Extensions

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. The second author was supported by Slovak grants VEGA 1/3026/06 and APVV-51-009605.     

Central Extensions of Steinberg Lie Superalgebras of Small Rank

It was shown by Mikhalev and Pinchuk (2000 Mikhalev , A. V. , Pinchuk , I. A. ( 2000 ). Universal central extensions of the matrix Lie superalgebras sl(m,n,A) . Int. Conf. in H.K.U., AMS , 111 – 125 . [Google Scholar]) that the second homology group H ₂(𝔰𝔱(m,n,R)) of the Steinberg Lie superalgebra 𝔰𝔱(m,n,R) is trivial for m + n ≥ 5. In this article, we will work out H ₂(𝔰𝔱(m,n,R)) explicitly for m + n = 3, 4.     

Hilbert $$C^*$$-modules as a subcategory of operator systems and injectivity

In this paper, we study a category whose objects are Hilbert \(C^*\)-modules and whose morphisms are completely semi-\(\phi \)-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-\(\phi \)-maps on Hilbert \(C^*\)-modules, leading to an analog of the Arveson’s extension theorem for completely semi-\(\phi \)-maps (in contrast with \(\phi \)-maps). This theorem together with previous results suggest that the completely semi-\(\phi \)-maps are proper generalizations of the completely positive maps.     

K0-group Transmissibilities of the Tracial Limit of C^＊-algebras

We introduce the tracial limit A = (t4) limn→n∞ (An,pn) and show that if K0(An) has ordered relation, K0(A) has ordered relation naturally. In the case that A is simple and K0(An) is weakly unperforated for every n, K0(A) is weakly unperforated too. Furthermore, the Riesz interpolation property of K0(An) can be transmitted to K0(A).     

Compact Operators and Uniform Structures in Hilbert $$C^*$$ -Modules

Functional Analysis and Its Applications - Quite recently a criterion for the $$\mathcal{A}$$ -compactness of an ajointable operator $$F\colon {\mathcal M} \to\mathcal{N}$$ between Hilbert $$C^*$$...