A classification of inductive limit $C^{*}$-algebras with ideal property

Let $A$ be an $AH$ algebra. Suppose that $A$ has the ideal property: each closed two sided ideal of $A$ is generated by the projections inside the ideal, as closed two sided ideal. In this article, we will classify all $AH$ algebras with ideal property of no dimension growth. This result generalizes and unifies the classification of $AH$ algebras of real rank zero in [EG] and [DG] and the classification of simple $AH$ algebras in [G5] and [EGL1]. By this paper, we have completed one of two so considered most important possible generalizations of [EGL1](see the introduction of [EGL1]). The invariants for the classification including scale ordered total $K-$group, for each $[p]\in\Sigma A$, $T(pAp)$---the tracial state space of cut down algebra $pAp$ with certain compatibility, and a new ingredient, the invariant $U(pAp)/\overline{DU(pAp)}$, with certain compatibility condition. We will also present a counterexample if this new ingredient is missed in the invariant. The discovery of this new invariant is an analogy to that of order structure on total K-theory when one advanced from the classification of simple real rank zero $C^*$-algebras to that of non simple real rank zero $C^*$-algebras in [G2], [Ei], [DL] and [DG] (see introduction below). We will also prove that the so called $ATAI$ algebras in [Jiang1] (and $ATAF$ algebra in [Fa])---the inductive limits of direct sums of simple $TAI$ (and $TAF$) algebras with $UCT$---, are in our class. Here the concept of simple $TAI$ (and $TAF$ ) algebras was introduced by Lin to axiomatize the decomposition theorem of [G5] (and of [EG2] respectively) and is partially inspired by Popa's work [Popa].