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Search: All articles in the CJM digital archive with keyword ideal property

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1. CJM 2017 (vol 69 pp. 1385)

Pasnicu, Cornel; Phillips, N. Christopher
The Weak Ideal Property and Topological Dimension Zero
Following 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: $\bullet$ The weak ideal property implies topological dimension zero. $\bullet$ For a separable C*-algebra~$A$, topological dimension zero is equivalent to ${\operatorname{RR}} ({\mathcal{O}}_2 \otimes A) = 0$, to $D \otimes A$ 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 ${\mathcal{O}}_{\infty} \otimes B$ contains a nonzero projection. $\bullet$ Extending the known result for ${\mathbb{Z}}_2$, 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. $\bullet$ If $A$ and $B$ are separable, one of them is exact, $A$ has the ideal property, and $B$ has the weak ideal property, then $A \otimes_{\mathrm{min}} B$ has the weak ideal property. $\bullet$ If $X$ is a totally disconnected locally compact Hausdorff space and $A$ is a $C_0 (X)$-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). $\bullet$ 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. $\bullet$ 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 hope for.

Keywords:ideal property, weak ideal property, topological dimension zero, $C_0 (X)$-algebra, purely infinite C*-algebra
Category:46L05

2. CJM 2011 (vol 63 pp. 381)

Ji, Kui ; Jiang, Chunlan
A Complete Classification of AI Algebras with the Ideal Property
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit of a sequence $$ A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots, $$ where $A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$, $X^{i}_n$ are $[0,1]$, $k_n$, 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.

Keywords:AI algebras, K-group, tracial state, ideal property, classification
Categories:46L35, 19K14, 46L05, 46L08

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