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

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.

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

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.

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

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).

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