# On the inductive limit of direct sums of simple TAI algebras

## TL;DR

This paper extends the classification of ATAI algebras by removing a torsion restriction on K-theory, using an invariant that includes the Hausdorffized algebraic K1-group, and connects to classification of AHD algebras.

## Contribution

It generalizes previous classification results of ATAI algebras by incorporating the Hausdorffized algebraic K1-group, removing the torsion restriction on K1.

## Key findings

- Classification of all ATAI algebras with the new invariant.
- Reduction to classification of AHD algebras with ideal property.
- Generalization of previous main theorems in the field.

## Abstract

An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},\phi_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).