A Survey on Classification of C∗-Algebras with the Ideal Property

2020 ◽  
pp. 267-287
Author(s):  
Guihua Gong ◽  
Chunlan Jiang ◽  
Kun Wang
Keyword(s):  
C Algebras ◽  
Ideal Property ◽  
The Ideal

scholarly journals Inductive limit of direct sums of simple TAI algebras

2019 ◽  
pp. 1-26
Author(s):  
Bo Cui ◽  
Chunlan Jiang ◽  
Liangqing Li
Keyword(s):  
Inductive Limit ◽  
Direct Sums ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Ideal Property ◽  
Funct Anal

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

scholarly journals The Weak Ideal Property and Topological Dimension Zero

2017 ◽  
Vol 69 (6) ◽  
pp. 1385-1421 ◽  
Author(s):  
Cornel Pasnicu ◽  
N. Christopher Phillips
Keyword(s):  
Hausdorff Space ◽  
Crossed Products ◽  
Topological Dimension ◽  
C Algebra ◽  
Dimension Zero ◽  
C Algebras ◽  
Ideal Property ◽  
Locally Compact Hausdorff Space ◽  
Totally Disconnected ◽  
The Ideal

AbstractFollowing up on previous work, we prove a number of results for C* -algebras with the weak ideal property or topological dimension zero, and some results for C* -algebras with related properties. Some of the more important results include the following:The weak ideal property implies topological dimension zero.For a separable C* -algebra A, topological dimension zero is equivalent to , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C* -algebras B such that contains a nonzero projection.Extending the known result for , the classes of C* -algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups.If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ B has the weak ideal property.If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C* -algebras, including all separable locally AH algebras.The weak ideal property does not imply the ideal property for separable Z-stable C* -algebras.We give other related results, as well as counterexamples to several other statements one might conjecture.

scholarly journals Tensor Products of C*-Algebras with the Ideal Property

2000 ◽  
Vol 177 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Cornel Pasnicu ◽  
Mikael Rørdam
Keyword(s):  
Tensor Products ◽  
C Algebras ◽  
Ideal Property ◽  
The Ideal

On invariants of C*-algebras with the ideal property

2018 ◽  
Vol 12 (3) ◽  
pp. 1195-1221 ◽  
Author(s):  
Kun Wang
Keyword(s):  
C Algebras ◽  
Ideal Property ◽  
The Ideal

Hausdorffified algebraic K1-groups and invariants for C∗-algebras with the ideal property

2020 ◽  
Vol 5 (1) ◽  
pp. 43-78 ◽  
Author(s):  
Guihua Gong ◽  
Chunlan Jiang ◽  
Liangqing Li
Keyword(s):  
C Algebras ◽  
Ideal Property ◽  
The Ideal

A Complete Classification of AI Algebras with the Ideal Property

2011 ◽  
Vol 63 (2) ◽  
pp. 381-412 ◽  
Author(s):  
Kui Ji ◽  
Chunlan Jiang
Keyword(s):  
Inductive Limit ◽  
Complete Classification ◽  
C Algebra ◽  
Ideal Property ◽  
Positive Integers ◽  
The Ideal

Abstract Let A be an AI algebra; that is, A is the C*-algebra inductive limit of a sequencewhere are [0, 1], kn, and [n, i] are positive integers. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property.

On classification of non-unital amenable simple C*-algebras, II

2020 ◽  
Vol 158 ◽  
pp. 103865
Author(s):  
Guihua Gong ◽  
Huaxin Lin
Keyword(s):  
C Algebras

Managerial identity development across the age-spectrum from an ideal self and values perspective

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Udayan Dhar
Keyword(s):  
Identity Development ◽  
Professional Identity ◽  
Personal Values ◽  
Ideal Self ◽  
Professional Identity Development ◽  
Content Type ◽  
Management Professionals ◽  
Inductive Thematic Analysis ◽  
The Ideal

PurposeThe purpose of this study is to investigate professional identity development among management professionals through the lens of the ideal self and personal values.Design/methodology/approachDetailed career vision essays based on the ideal self and personal values of 48 participants ranging in age from 22 to 54 were analyzed using an inductive thematic analysis. A theory-based classification of their personal values, collected through a survey, was also conducted as a supplemental analysis.FindingsThe visions of older management professionals were less career-oriented, more holistic, involved in a greater multiplicity of career roles, had more clarity and placed higher emphasis on work–life balance and on developing others. The older participants also reported having fewer self-enhancement values.Originality/valueThe findings demonstrate the relevance of the ideal self as a lens to study identity development and advance our understanding of professional identity development in the context of modern careers.

scholarly journals Applying Biomimetic Principles to Thermoelectric Cooling Devices for Water Collection

2017 ◽  
Vol 7 (3) ◽  
pp. 27
Author(s):  
Kyle B Davidson ◽  
Bahram Asiabanpour ◽  
Zaid Almusaied
Keyword(s):  
Large Scale ◽  
Cost Effective ◽  
Thermoelectric Cooling ◽  
Device Structure ◽  
Cooling Systems ◽  
Cooling Devices ◽  
Water Collection ◽  
Maximum Water ◽  
The Ideal

The shortage of freshwater resources in the world has developed the need for sustainable, cost-effective technologies that can produce freshwater on a large scale. Current solutions often have extensive manufacturing requirements, or involve the use of large quantities of energy or toxic chemicals. Atmospheric water generating solutions that minimize the depletion of natural resources can be achieved by incorporating biomimetics, a classification of design inspired by nature. This research seeks to optimize thermoelectric cooling systems for use in water harvesting applications by analyzing the different factors that affect surface temperature and water condensation in TEC devices. Further experiments will be directed towards developing a robust, repeatable system, as well as an accurate measurement system. Surface modifications, device structure and orientation, and power generation will also be studied to better understand the ideal conditions for maximum water collection in thermoelectric cooling systems.

scholarly journals Reduction to Dimension Two of the Local Spectrum for an AH Algebra with the Ideal Property

2017 ◽  
Vol 60 (4) ◽  
pp. 791-806 ◽  
Author(s):  
Chunlan Jiang
Keyword(s):  
Inductive Limit ◽  
Local Spectrum ◽  
Direct Sums ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Ideal Property ◽  
Uniformly Bounded ◽  
The Ideal

AbstractA C*-algebra Ahas the ideal property if any ideal I of Ais generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit C*-algebra A of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that A can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion H2 groups.

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