An \(AH\) algebra is a nuclear \(C^*\)-algebra, represented as an inductive limit of a sequence \(\left(A_n \right)\) of matrix algebras, as follows: \({A_n=\bigoplus_{i=1}^{t_n}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}}\), where, for the numbers \([n,i]\), \(X_{n,i}\) are compact metric spaces and \(P_{n,i}\) are projections in the matrix algebras \(M_{[n,i]}(C(X_{n,i}))\) of complex-valued continuous functions on \(X_{n,i}\). An \(AH\) algebra is said to have the ideal property if each closed two-sided ideal in the algebra is generated by the projections inside the ideal, as a closed two-sided ideal.
In this paper, the authors consider the problem of classifying \(AH\) algebras with the ideal property. They prove that, under the condition of very slow dimension growth, and assuming that \(K_{*}(A)\) is torsion free, then \(A\) is an approximate circle algebra (also called an \(A \mathbb T \) algebra). That is, \(A\) is the inductive limit of a sequence, \(B_n\), of finite direct sums of matrix algebras over the algebra of continuous functions on \(S^1\): \(B_n =\bigoplus_{i=1}^{s_n}M_{[n,i]}(C(S^1))\) (i.e., all the spaces \(X_{n,i}\) can be replaced by \(S^1\)). This is the reduction theorem, which is the main result of this paper. In a subsequent paper of the same authors [“A reduction theorem for AH-algebras with the ideal property”, preprint], the theorem is proved for a general setting, where the assumption of \(K_*(A)\) to be torsion free is relaxed and hence additional spaces besides \(S^1\) are involved.
The present article starts with a rich introduction of basic definitions and lemmas needed throughout the paper. In Section 2, several lemmas are developed in order to prove the other important theorem in this article, the decomposition theorem, which is proved here without assuming that \(K_*(A)\) is torsion free. This theorem is then used, in Section 3, to prove the reduction theorem. The paper ends with a large references section related to the subject of classifying \(AH\) algebras. Such work started with G.A.Elliott’s work during the 1990s in his famous program to classify nuclear \(C^*\)-algebras by their topological invariants including \(K\)-theory (for an overview, see [G. A. Elliott, in: Proceedings of the international congress of mathematicians, ICM ’94, August 3–11, 1994, Zürich, Switzerland. Vol. II. Basel: Birkhäuser, 922–932 (1995; Zbl 0946.46050)]). Intensive work has been done meanwhile towards classifying simple inductive limit \(C^*\)-algebras and real rank zero \(AH\) algebra. The \(AH\) algebras with the ideal property are of most interest, since they include both these classes. The results in this paper certainly are an important step in this direction, since they both extend and unify the results obtained in the literature for \(AH\) algebras of real rank zero and of simple \(AH\) algebras.