A Complete Classification of AI Algebras with the Ideal Property
A Complete Classification of AI Algebras with the Ideal Property
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DOI：
10.4153/cjm-2011-005-9
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发表时间：
2011-04
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影响因子：
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通讯作者：
K. Ji;Chunlan Jiang
中科院分区：
文献类型：
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作者：
K. Ji;Chunlan Jiang
Abstract 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.
Abstract 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.