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Compatibility and Schur Complements of Operators on Hilbert C~*-Module

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Chinese Annals of Mathematics,Series B 2011年第1期32卷 69-88页

作者： xiaochun fang Jing YU Department of Mathematics Tongji University Shanghai 200092 China Department of Mathematics Qingdao University Qingdao 266071 Shandong China

Let E be a Hilbert C＊-module,and Y be an orthogonally complemented closed submodule of *** authors generalize the definitions of Y-complementability and Y-compatibility for general（adjointable） operators from Hilbert space to Hilbert C＊-module,and discuss the relationship between each *** equivalent statements about Y-complementability and Y-compatibility,and several representations of Schur complements of Y-complementable operators（especially,of Y-compatible operators and of positive Y-compatible operators） on a Hilbert C＊-module are *** addition,the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of Y-complementable operators and Y＊-complementable operators on a Hilbert C＊-module.

关键词： Hilbert C*-module Compatibility Complementability Schur complement Quotient property

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

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Chinese Annals of Mathematics,Series B 2009年第2期30卷 179-186页

作者： Qingzhai FAN xiaochun fang School of Mathematical Sciences Fudan University Shanghai 200433 China Department of Mathematics Tongji University Shanghai 200092 China

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.

关键词： C*-algebra Stable rank one Real rank zero

Certain classes of C＊-algebras preserved by tracial approximation

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Frontiers of Mathematics in China 2012年第3期7卷 449-458页

作者： Qiugzhai FAN xiaochun fang Institute of Mathematics Shanghai Maritime UniversityShanghai 201306China Institute of Mathematics Tongji UniversityShanghai 200092China

We show that the following classes of C＊-algebras in the classes t are inherited by simple unital C＊-algebras in the classes TAft ：（1） simple unital purely infinite C＊-algebras, （2） unital isometrically rich ... 详细信息

关键词： C*-algebras isometrically rich C*-algebras Riesz interpolationC*-algebras

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

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Chinese Annals of Mathematics,Series B 2016年第3期37卷 367-372页

作者： Changjing LI xiaochun fang School of Mathematical Sciences Shandong Normal University Jinan 250014 China. Department of Mathematics Tongji University Shanghai 200092 China.

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.

关键词： C^*-algebras Von Neumann algebras Haagerup property

The tracial topological rank of certain C*-algebras

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Science China Mathematics 2011年第11期54卷 2295-2307页

作者： fang xiaochun ZHAO YiLe Department of Mathematics Tongji University Shanghai 200092 China

Let 0 →I → A →A/I →0 be a short exact sequence of C^＊-algebras with A unital. Suppose that I has tracial topological rank no more than one and A/I belongs to a class of certain C^＊-algebras. We show that A has trazial topological rank no more than one if the extension is quasidiagonal, and A has the property （P1） if the extension is tracially quasidiagonal.

关键词： tracial topological rank quasidiagonal extension tracially quasidiagonal extension

Approximately isometric lifting in quasidiagonal extensions

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Science China Mathematics 2009年第3期52卷 457-467页

作者： fang xiaochun ZHAO YiLe Department of Mathematics Tongji UniversityShanghai 200092China

Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number ε, two positive elements (projections, partial isometries, unitary elements, respectively) $ \bar a,\bar b $ in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of $ \bar a $ , there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of $ \bar b $ such that ∥a?b∥ < $ \left\| {\bar a - \bar b} \right\| + \varepsilon $ . As an application, it is shown that for any positive numbers ε and $ \bar u $ in U(A/I) 0 , there exists u in U(A)0 which is a lifting of $ \bar u $ such that cel(u) < cel $ (\bar u) + \varepsilon $ .

关键词： commutativity lifting quasidiagonal extension 46L05

Some Properties of Tracially Quasidiagonal Extensions

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Chinese Annals of Mathematics,Series B 2019年第1期40卷 97-110页

作者： Yile ZHAO xiaochun fang Xiaoming XU Department of Mathematics Hangzhou Normal University Department of Mathematics Tongji University School of Science Shanghai Institute of Technology

Suppose that 0→ I→ A→ A/I→ 0 is a tracially quasidiagonal extension of C*-algebras. In this paper, the authors give two descriptions of the K_0, K_1 index maps which are induced by the above extension and show that for any ∈ > 0, any τ in the tracial state space of A/I and any projection p ∈ A/I(any unitary u ∈ A/I), there exists a projection p ∈ A(a unitary u ∈ A) such that |τ(p)-τ(π(p))| < ∈(|τ(u)-τ(π(u))| < ∈).

关键词： Tracially topological rank Quasidiagonal extension Tracially quasidiagonal extension

CROSSED PRODUCTS BY FINITE GROUP ACTIONS WITH CERTAIN TRACIAL ROKHLIN PROPERTY

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Acta Mathematica Scientia 2018年第3期38卷 829-842页

作者： Qingzhai Fan xiaochun fang Department of Mathematics Shanghai Maritime Universit Department of Mathematics Tongji Universit

We introduce a special tracial Rokhlin property for unital C~＊-algebras. Let A be a unital tracial rank zero C~＊-algebra（or tracial rank no more than one C~＊-algebra）. Suppose that α ： G → Aut（A） is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~＊-algebra. Then, the crossed product C~＊-algebra C~＊（G, A, α） has tracia rank zero（or has tracial rank no more than one）. In fact,we get a more general results.

关键词： C~*-algebras Rokhlin property crossed product C~*-algebra

Continuity of functors with respect to generalized inductive limits

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Frontiers of Mathematics in China 2019年第3期14卷 551-566页

作者： Jiajie HUA xiaochun fang Xiao-Ming XU School of Mathematics Physics and Information Engineering Jiaxing University Jiaxing 314001 China Department of Mathematics Tongji University Shanghai 200092 China School of Science Shanghai Institute of Technology Shanghai 201418 China

Let(Ai,φi,i+1) be a generalized indue Live system of a sequeiiee (Ai) of unital separable C^*-algebras,with A =limi→∞(Ai,φi,i+1). Set φj,i=φi-1,i^0…0φj+1,j+2^0 φj,j+1 for all i>j. We prove that if φj,i are order zero completely positive contractions for all j and i>j, And L:=inf{λ|λ∈σ(φj,i(1Aj)) for all j uud i>j}>0, where σ(φj,i(1Aj)) is the speetrum of φj,i(1Aj),than limi→∞(Cu(Ai),Cu((φi,i+1))=Cu(A), where Cu(A) is a stable version of the Cuntz semigroup of C^*-algebra A. Let (An,φm,n) be a generalized inductive syfitem of C^*-algahrafl, with the ipmkn order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of ,4n is no more t han some integer k for each n, then the decompostition rank (nuclear dimension) of A is also no more than k.

关键词： Completely positive map order zero generalized inductive limits classification of C^*-algebra