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# A Complete Classification of AI Algebras with the Ideal Property

Published:2011-02-15
Printed: Apr 2011
• Kui Ji,
Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, China
• Chunlan Jiang,
Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, China
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## Abstract

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
 MSC Classifications: 46L35 - Classifications of $C^*$-algebras 19K14 - $K_0$ as an ordered group, traces 46L05 - General theory of $C^*$-algebras 46L08 - $C^*$-modules

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