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

Abstract

Let A be an AH algebra A = lim n→∞ (A n = tn i=1P n,i M [n,i] (C(X n,i ))P n,i , φ n,m ), where X n,i are compact metric spaces, t n and [n, i] are positive integers, and P n,i ∈ M [n,i] (C(X n,i )) are projections. 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-that is Sup n,i dim(X n,i ) < +∞. 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 (K(A), K(A) + , ΣA)(as already used in real rank zero case in [DG]), for each [p] ∈ ΣA, T (pAp)-the tracial state space of cut down algebra pAp with certain compatibility, (which is used by [Stev] and [Ji-Jiang] for AI algebras with ideal property), and a new ingredient, the invariant U(pAp)/DU(pAp), with certain compatibility condition, where DU(pAp) is the closure of commutator subgroup DU(pAp) of the unitary group U(pAp) of the cut down algebra pAp. 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 AT AI algebras in [Jiang1] (and AT AF algebra in [Fa])-the inductive limits of direct sums of simple T AI (and T AF ) algebras with UCT -, are in our class. Here the concept of simple T AI (and T AF ) 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].

$C^*$ exponential length of commutators unitaries in $AH$ algebras

“…I k is called Elliott dimension drop interval algebra. As in [9], we denote by HD the class of algebras of direct sums of building blocks of forms M l (I k ) and P M n (C(X))P , with X being one of the spaces {pt}, [0, 1], S 1 and T II,k , and with P ∈ M n (C(X)) being a projection. We will call a C *algebra an AHD algebra, if it is an inductive limit of algebras in HD.…”

Rationally AF algebras and KMS states of Z-absorbing C*-algebras

“…In the infinite (O ∞ -stable) case an ideal-related KK-equivalence based isomorphism theorem was outlined in [47]; a different proof of this was given in [33]. In the finite (non-simple) case, after earlier results, a K-theoretical classification of inductive limits of finite direct sums of matrix algebras over commutative C * -algebras (AH, or approximately homogeneous, algebras) with no dimension growth in the spectra and with the ideal property (every closed two-sided ideal generated by projections) was given in [36]. The next step in this direction would be to classify inductive limits of sequences of C * -subalgebras of matrix algebras over commutative C * -algebras (ASH, or approximately subhomogeneous, algebras), with no dimension growth and with the ideal property.…”

“…By [35], every C * -algebra which is both RAF and in the AH class considered in [36] (see above) is AF. The present class, thus, projects directly into the unknown territory of the non-simple ASH class.…”