An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit limn→∞(An =⊕i=1Ani,ϕ nm), where each Ani is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], 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 K1(A) are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic K1-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of 𝒜ℋ𝒟 algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

中文翻译：

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简单 TAI 代数直接和的归纳极限

ATAI（或分别为 ATAF）代数，在 [C. 江，一类非简单 C*-代数的分类第一：简单 TAI C*-代数的有限直和的归纳极限，J. 白杨。肛门。 3(2011) 385–404] （或在 [XC Fang, The classification of certain non-simple C*-algebras of tracial rank zero,J. 功能。肛门。 256(2009) 3861–3891]，分别）是一个感应极限林n→∞(一种n =⊕一世=1一种n一世,φ n米), 其中每个一种n一世是具有 UCT 属性的简单可分核 TAI（或 TAF）C*-代数。在 [C. 江，一类非简单 C*-代数的分类第一：简单 TAI C*-代数的有限直和的归纳极限，J. 白杨。肛门。 3(2011) 385–404]，第二作者将所有 ATAI 代数分类为由有序总数组成的不变量ķ-在一个额外的限制下，削减代数的理论和种族状态空间，其中所有元素ķ1(一种)是扭转。在本文中，我们去掉了这个限制，并用 Hausdorffized 代数得到了所有 ATAI 代数的分类ķ1-group 作为 [C. 江，一类非简单 C*-代数的分类第一：简单 TAI C*-代数的有限直和的归纳极限，J. 白杨。肛门。 3(2011) 385–404]。该定理通过将类归约到分类定理来证明𝒜ℋ𝒟具有理想性质的代数，在 [G. 龚，C.江和L.李，具有理想性质的归纳极限C*-代数的分类，预印本（2016），arXiv：1607.07681]。我们的定理推广了 [XC Fang, The classification of certain non-simple C*-algebras of tracial rank 0,J. 功能。肛门。 256(2009) 3861–3891]，[C. 江，一类非简单 C*-代数的分类第一：简单 TAI C*-代数的有限直和的归纳极限，J. 白杨。肛门。 3(2011) 385–404]（见推论 4.3）。