Simple generalized inductive limits of C*-algebras

We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C~*-algebra. We also show that if a unital C~*-algebra can be approximately embedded into some tensorially self absorbing C~*-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_2),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.     

On Quasi-Chebyshevity Subsets of Unital Banach Algebras

In this paper,first,we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev.Examples of those algebras are given including the algebras of continuous functions on compact sets.We also see some results in C*-algebras and Hilbert C*-modules.Next,by considering some conditions,we study Chebyshev of subalgebras in C*-algebras.     

迹稳定秩一的C~*-代数

本文引入了一类迹稳定秩一的C*-代数,证明了迹稳定秩一的C*-代数与AF-代数的张量积是迹稳定秩一的,得到了一个可分的单的有单位元的迹稳定秩一的,并且具有SP性质的C*-代数是稳定秩一的.同时,还讨论了迹稳定秩一的C*-代数的K-群的某些性质.     

Stable Rank One and Real Rank Zero for Crossed Products by Finite Group Actions with the Tracial Rokhlin Property

The authors prove that the crossed product of an infinite dimensional simple separable unital C＊-algebra with stable rank one by an action of a finite group with the tracial Rokhlin property has again stable rank one. It is also proved that the crossed product of an infinite dimensional simple separable unital C＊-algebra with real rank zero by an action of a finite group with the tracial Rokhlin property has again real rank zero.     

On the classification of certain real rank zero C*-algebras

In this paper,we give a classification of real rank zero C^*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.     

The(**)-Haagerup Property for C*-Algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the(**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki's question(2013), and then obtain several results of(**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the(**)-Haagerup property. Some heredity results concerning the(**)-Haagerup property are also proved.     

Examples of different minimal diffeomorphisms giving the same C*-algebras

We give examples of minimal diffeomorphisms of compact connected manifolds which are not topologically orbit equivalent, but whose transformation group C*-algebras are isomorphic. The examples show that the following properties of a minimal diffeomorphism are not invariants of the transformation group C*-algebra: having topologically quasidiscrete spectrum; the action on singular cohomology (when the manifolds are diffeomorphic); the homotopy type of the manifold (when the manifolds have the same dimension); and the dimension of the manifold. These examples also give examples of nonconjugate isomorphic Cartan subalgebras, and of nonisomorphic Cartan subalgebras, of simple separable nuclear unital C*-algebras with tracial rank zero and satisfying the Universal Coefficient Theorem. The research was partially supported by the NSF grants DMS 9706850, DMS 0070776, DMS 0302401, and by the MSRI.     

Certain classes of <Emphasis Type="Italic">C</Emphasis>*-algebras preserved by tracial approximation

We show that the following classes of C*-algebras in the classes Ω are inherited by simple unital C*-algebras in the classes TAΩ: (1) simple unital purely infinite C*-algebras, (2) unital isometrically rich C*-algebras, (3) unital Riesz interpolation C*-algebras.     

Uniqueness of the trace and simplicity

It is proved that for certain large classes of unital C*-algebras, the existence of a unique, faithful tracial state implies simplicity. An example is given to show that this implication does not hold for all unital C*-algebras.

     

Purely infinite corona algebras of simple C*-algebras

In this paper, we study the problem of when the corona algebra of a non-unital C*-algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition.     

PrO-C*-代数的顺从性和核性

研究了Pro—C^*-代数的顺从性和核性．主要证明了(1)顺从Pro—C^*代数的闭理想是顺从的；(2)核Pro—C^*代数类对归纳极限封闭；(3)交换σ-C^*-代数和核C^*-代数都是核，σ-C^*-代数并且核σ-C^*-代数类对于商运算、张量积运算和可数逆向极限封闭．进一步得到核，σ-C^*-代数的扩张保持核性的条件。     

On tracial approximation

Let C be a class of unital C*-algebras. The class TAC of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in C is studied (Lin considered the case that C consists of finite-dimensional C*-algebras or the tensor products of such with C([0,1])). In particular, the question is considered whether, for any simple separable A∈TAC, there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in C, such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.)     

AF Embeddability of Crossed Products of AT Algebras by the Integers and Its Application

We will show that the crossed products of unital simple real rank zero AT algebras by the integers are AF embeddable. This is a generalization of Brown's AF embedding theorem. As an application, we will prove the AF embeddability of crossed product algebras arising from certain minimal dynamical systems induced by two commuting homeomorphisms.     

On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.     

Matricial quantum Gromov-Hausdorff distance

We develop a matricial version of Rieffel's Gromov-Hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital C∗-algebras. Our approach yields a metric space of “isometric” unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting, as for example in Rieffel's approximation of the sphere by matrix algebras using Berezin quantization. Within the metric subspace of metrized unital C∗-algebras we establish the convergence of sequences which are Cauchy with respect to a larger Leibniz distance, and we also prove an analogue of the precompactness theorems of Gromov and Rieffel.     

实厄米Banach*-代数

对任意实厄米的B anach*-代数进行系统研究,即对不需任何附加条件限制的代数进行研究,讨论方法与传统的假定代数具有单位元和*-运算是连续的两个重要假定条件下的方法不同.     

Approximately central matrix units and the structure of noncommutative tori

The property of approximate divisibility forC*-algebras is introduced and studied. Simple approximately divisibleC*-algebras are shown to have nice nonstableK-theory properties. Non-rational noncommutative tori are shown to be approximately divisible. It follows that every simple noncommutative torus (in particular, every irrational rotation algebra) has stable rank one and real rank zero.     

C*-Algebras with the Approximate Positive Factorization Property

We say that a unital -algebra has the approximate positive factorization property (APFP) if every element of is a norm limit of products of positive elements of . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial , and stable rank 1. In the unital case, the group separates the tracial states. The APFP passes to matrix algebras, and if is an ideal in such that and have the APFP, then so does . We also give some new examples of -algebras with the APFP, including type factors and infinite-dimensional simple unital direct limits of homogeneous -algebras with slow dimension growth, real rank zero, and trivial group. Simple direct limits of homogeneous -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a -algebra that may be of interest in the future.

     

Ordered *-Algebras with an Order Unit

We develop a theory of ordered *-algebras with an order unit. These are complex algebras equipped with a conjugate-linear vector space involution whose hermitian elements form an ordered real algebra with an order unit but not necessarily with a multiplicative identity. Our main result is a representation theorem for ordered *-algebras with an order unit. This can be viewed as the complex and the non-unital version of the representation theorem of Stone and Kadison for ordered real algebras containing an order unit which is a multiplicative identity. A key role in our approach is played by certain weighted function algebras.