Many of the known manifestations of the Elliott conjecture may be interpreted as classification up to D-stability, where D is a fixed strongly self-absorbing C*-algebra, such as the Cuntz algebra O∞, O2 or the Jiang-Su algebra Z. We formalize this point of view by introducing the concept of ’localizing the Elliott conjecture at a strongly self-absorbing C*-algebra D’. We explain how existing classification results fit into this framework.

Our main theorem is a new classification result up to Z-stability: Let A be the class of separable, unital, simple C*-algebras with locally finite decomposition rank, such that projections separate traces, and satsifying the UCT as well as a mild K-theory condition. Using results of H. Lin, we show that A satisfies the Elliott conjecture localized at Z.

This in particular confirms the Elliott conjecture for separable, unital, simple, monotracial and Z-stable ASH algebras with finitely generated K-theory, a class known to contain the UHF algebras, the irrational rotation algebras and the (projectionless) JiangSu algebra itself.

Our result does not depend on an inductive limit structure in any way; in the monotracial case it does not depend on the existence or nonexistence of projections.

Our proof also reveals a strategy of how to possibly remove the trace space condition entirely.