Simple real rank zero algebras with locally Hausdorff spectrum

Abstract

Abstract. Let A be a unital, simple, separable C * -algebra with real rank zero, stable rank one, and weakly unperforated ordered K 0 group. Suppose, also, that A can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then A is tracially approximately finite dimensional (i.e., A has tracial rank zero).Hence, A is an AH-algebra with bounded dimension growth and is determined by K-theoretic invariants.The above result also gives the first proof for the locally AH case.

ON THE CLASSIFICATION OF SIMPLE ${{\mathcal Z}}$-STABLE $C^{*}$-ALGEBRAS WITH REAL RANK ZERO AND FINITE DECOMPOSITION RANK

“…We may assume that the elements of F are positive and normalized and that 1 A ∈ F . Choose strictly positive numbers α, δ and η such that (14) α + η + δ < µ and 2α + 4 δ + α…”

“…Similar results were obtained in [2] and [11], under the additional assumption that the algebras have only countably many extremal tracial states. In the (locally) AH real rank zero case, the trace space conditions are not necessary, as was shown in [12] and [14]. In the present article we show that the condition of [24] on the tracial state space can be removed if the algebras in question are Z-stable -in other words, we verify the Elliott conjecture for the class of separable simple unital Z-stable C * -algebras with real rank zero and finite decomposition rank (satisfying the UCT).…”

Z-Stable ASH Algebras

“…3.11 Based on Theorem 1 of [19], Nate Brown has shown us a proof of the following theorem, which will also follow from the (more general) results in [36]. The arguments of [19] and [36] are quite different; the latter is built upon the methods developed in [15], [34] and [35].…”

Commutative ${C}^*$-subalgebras of simple stably finite ${C}^*$-algebras with real rank zero

“…However, moving beyond these conditions of having small tracial simplex is difficult. This is closely related to certain recent classification problems, for simple C * -algebras, where a major difficulty lies in replacing the small tracial simplex condition by arbitrary tracial simplexes (see, for example [5], [27], [35], [51] and [52]).…”