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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 AI algebras, K-group, tracial state, ideal property, classification
MSC Classifications: 46L35, 19K14, 46L05, 46L08 show english descriptions Classifications of $C^*$-algebras
$K_0$ as an ordered group, traces
General theory of $C^*$-algebras
$C^*$-modules
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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