# Journal of Operator Theory

Volume 88, Issue 1, Summer 2022  pp. 117-138.

Classification of AH algebras with finitely many ideals

Authors:  Kun Wang
Author institution: Texas A&M University, College Station, TX, 77843, U.S.A.

Summary:  The class of AH algebras with the ideal property and no dimension growth is classified by the invariant $\mathrm{inv}(\cdot)$. In this paper, we introduce a new invariant, $\mathrm{Inv}(\cdot)$, a refined version of $\mathrm{inv}(\cdot)$ and show that they are equivalent for AH algebras with the ideal property and no dimension growth. Then we give a sufficient condition under which the Hausdorffified algebraic ${\rm K}_1$ group could be recovered from the traditional Elliott invariant. As an application, the class of AH algebras with the ideal property, no dimension growth, and finitely many ideals can be classified by the extended Elliott invariant.

DOI: http://dx.doi.org/10.7900/jot.2020dec04.2331
Keywords: ${C}^*$-algebra, AH algebra, classification, ideal property, Hausdorffified algebraic $\text{K}_1$ group