Localizing The

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C∗-algebra D; we also explain how the known classification theorems for nuclear C∗-algebras fit into this concept. As a new result in this direction, we show that the class of separable, unital, simple C∗-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfy the Elliott conjecture localized at the Jiang–Su algebra Z. We also obtain a uniqueness theorem in terms of Rørdam’s KL-groups. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces; it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear, Z-stable and stably finite C∗-algebras of real rank zero as well as the (projectionless) Jiang–Su algebra itself.