Abstract
We report on recent progress in the program to classify separable amenable C∗^*-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, andZ\mathcal {Z}-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.
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