C(X)A上的自同构群

本文研究了连续函数代数C（X）与某个C*-代数A的张量积C（X）A的自同构群．当A是有单位元且具有平凡中心的C*-代数时,本文完全刻划了C（X）A的自同构群．利用AF-代数的K-理论,本文还刻划了当X是全不连通的紧致Hausdorff空间时,C（X）与紧算子理想的张量积的自同构群．     

有限Hopf C~*-代数的表示与对偶

设A是有限维Hopf C-代数,H是Hilbert空间.如果存在A在L(H)上的作用γ,在此作用下,L(H)成为具有共轭性质的模代数且H上内积是A-不变的,则A存在惟一的C-表示(θ,H),L(H)的A-不变子空间恰好是θ(A)的换位子.     

C(X) A上的自同构群

本文研究了连续函数代数C(X)与某个C*-代数 A的张量积C(X) A的自同构群.当 A是有单位元且具有平凡中心的C*-代数时,本文完全刻划了C(X) A的自同构群.利用AF-代数的K-理论,本文还刻划了当X是全不连通的紧致Hausdorff空间时,C(X)与紧算子理想的张量积的自同构群.     

一类具有弱无孔性质的c*-代数

证明了一类C~*-代数的弱无孔性质可以遗传到通过此类C~*-代数迹逼近后得到的C~*-代数中.同时证明了具有弱无孔性质的C~*-代数经过具有迹Rokhlin性质的有限群作用后得到的交叉积C~*-代数也具有弱无孔性质。     

纯无限单的C*-代数通过某些C*-代数扩张的非稳定K-理论

设0→B■E■A→0是有单位元C~*-代数E的一个扩张,其中A是有单位元纯无限单的C~*-代数,B是E的闭理想.当B是E的本性理想并且同时是单的、可分的而且具有实秩零及性质(PC)时,证明了K_0(E)={[p]| p是E\B中的投影};当B是稳定C~*-代数时,证明了对任意紧的Hausdorff空间X,有■(C(X,E))/■_0(C(X,E))≌K_1(C(X,E)).     

一类Smale空间上的C~*-代数自同构的熵

Ian Putnam利用Smale空间上的渐近等价关系定义了广群C~*-代数及其典则自同构.本文在零维Smale空间的情形下,计算此类C~*-自同构的逼近熵,证明了相应C~*-动力系统关于CNT熵和逼近熵的"变分原理"成立.由此推演出此类Smale空间上的Bowen测度诱导的C~*-代数上的态是此典则自同构的唯一平衡态.     

Banach * 代数

Ｖ．Ｐｔáｋ［１］在有单位元的情形下详细研究了Ｂａｎａｃｈ代数．本文将推广他的结果到一般情形．特别，我们引入了可正开拓的正线性泛函，并对它进行了刻划．     

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.     

C~*代数上保持不定正交性的线性映射

设A和B是含单位元的C~*代数,s∈A和t∈B是可逆自伴元,对任意的x∈A及z∈B,定义x~+=s~(-1)x~*s,z~+=t~(-1)z~*t。假定A是实秩零的并且Φ:A→B是有界线性满射。证明了对任意的 都成立的充要条件是Φ(1)可逆,Φ(1)~+Φ(1)=Φ(1)Φ(1)~+∈Z(B)(B的中心),并且存在从A到B上的满+同态Ψ,使得对所有的x∈A都有Φ(x)=Φ(1)Ψ(x)成立。对于一般C~*代数上保正交性的线性映射Φ,在假定Φ(1)可逆的条件下,也得到类似的结果。     

纯无限单C*-代数的扩张代数的K-理论Ⅱ

给出了有单位元的纯无限单的C*-代数A通过K的扩张代数E的K-理论的一种刻画.证明了K0(E)同构于E中所有具有无限余投影的无限投影的Murry-yon Neumann等价类全体所成的交换群,它还同构于上述投影的同伦等价类或酉等价类全体所成的交换群.还证明了对扩张代数E中的任·满的正元a,存在元索z ∈E,使得x*ax=1,其中K为可分无限维Hilbert空间上紧算子全体所成的C*一代数.     

A separable Brown-Douglas-Fillmore theorem and weak stability

We give a separable Brown-Douglas-Fillmore theorem. Let be a separable amenable -algebra which satisfies the approximate UCT, be a unital separable amenable purely infinite simple -algebra and be two monomorphisms. We show that and are approximately unitarily equivalent if and only if We prove that, for any 0$"> and any finite subset , there exist 0$"> and a finite subset satisfying the following: for any amenable purely infinite simple -algebra and for any contractive positive linear map such that

for all there exists a homomorphism such that

provided, in addition, that are finitely generated. We also show that every separable amenable simple -algebra with finitely generated -theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple -algebras. As an application, related to perturbations in the rotation -algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number and any 0$"> there is 0$"> such that in any unital amenable purely infinite simple -algebra if

for a pair of unitaries, then there exists a pair of unitaries and in such that

     

Compact range property and operators on -algebras

We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.

     

Simple corona -algebras

Let be a non-unital and -unital simple -algebra. We show that if is simple, then is purely infinite. We also show that is simple if and only if has a continuous scale provided that is not isomorphic to the compact operators.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .

     

Regularity of operators on essential extensions of the compacts

A semiregular operator on a Hilbert -module, or equivalently, on the -algebra of `compact' operators on it, is a closable densely defined operator whose adjoint is also densely defined. It is shown that for operators on extensions of compacts by unital or abelian -algebras, semiregularity leads to regularity. Two examples coming from quantum groups are discussed.

     

Derivations implemented by local multipliers

A condition on a derivation of an arbitrary C*-algebra is presented entailing that it is implemented as an inner derivation by a local multiplier.

     

FS-property for -algebras

A -algebra is said to have the FS-property if the set of all self-adjoint elements in has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal -tensor products even in case of separable nuclear -algebras.

     

Tracially AF -algebras

Inspired by a paper of S. Popa and the classification theory of nuclear -algebras, we introduce a class of -algebras which we call tracially approximately finite dimensional (TAF). A TAF -algebra is not an AF-algebra in general, but a ``large' part of it can be approximated by finite dimensional subalgebras. We show that if a unital simple -algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated -group. All nuclear simple -algebras of real rank zero, stable rank one, with weakly unperforated -group classified so far by their -theoretical data are TAF. We provide examples of nonnuclear simple TAF -algebras. A sufficient condition for unital nuclear separable quasidiagonal -algebras to be TAF is also given. The main results include a characterization of simple rational AF-algebras. We show that a separable nuclear simple TAF -algebra satisfying the Universal Coefficient Theorem and having and is isomorphic to a simple AF-algebra with the same -theory.

     

A {\tf C}*-ALGEBRA APPROACH TO THE IRREDUCIBILITY OF COWEN-DOUGLAS OPERATORS

The authors consider the irreducibility of the Cowen-Douglas operator $T$. It is proved that $T$ is irreducible iff the unital $C^*$-algebra generated by some non-zero blocks in the decomposition of $T$ with respect to $\bigoplus^\infty_{n=0}\limits(\Ker T^{n+1}\ominus\Ker T^n)$ is $\text{M}_n(\Bbb C).$