On the inductive limit of direct sums of simple TAI algebras

Abstract

An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit $\underset{n\to \infty }{\lim }(A_n=\underset{i=1}{⨁}{A}_n^i,\ddot{I}{•}_{nm})$, 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 $\mathcal{A}\mathcal{H}\mathcal{D}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).

24 pages. arXiv admin note: text overlap with arXiv:1607.07581