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A Complete Classification of AI Algebras with the Ideal Property
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 2 / 01 April 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 381-412
- Print publication:
- 01 April 2011
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- Article
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- You have access
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Let
$A$ be an 
$\text{AI}$ algebra; that is, $A$ is the 
${{\text{C}}^{*}}$-algebra inductive limit of a sequence
$${{A}_{1}}\xrightarrow{{{\phi }_{1,2}}}{{A}_{2}}\xrightarrow{{{\phi }_{2,3}}}{{A}_{3}}\to \cdot \cdot \cdot \to {{A}_{n}}\to \cdot \cdot \cdot ,$$where
${{A}_{n}}=\oplus _{i=1}^{{{k}_{n}}}{{M}_{\left[ n,i \right]}}\left( C\left( X_{n}^{i} \right) \right),X_{n}^{i}$ are [0, 1], 
${{k}_{n}}$, and 
$\left[ n,\,i \right]$ 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 $\text{AI}$ algebras with the ideal property.
