A Complete Classification of AI Algebras with the Ideal Property

Ji, Kui ; Jiang, Chunlan

Canadian journal of mathematics, Tome 63 (2011) no. 2, pp. 381-412

Résumé

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.

Détail

DOI : 10.4153/CJM-2011-005-9

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