Z-stability and infinite tensor powers of C∗-algebras

We prove that under a mild hypothesis, the infinite tensor power of a unital separable C∗-algebra absorbs the Jiang–Su algebra Z tensorially. Combining this result with a recent theorem of Winter, we complete Elliott’s classification program for strongly self-absorbing ASH algebras. We also give a succinct universal property for Z in an ambient category so large that there are no unital separable C∗-algebras without characters that are known to lie outside it. This category contains the vast majority of our stock-in-trade separable amenable C∗-algebras, and is closed under passage to quotients and separable superalgebras. In particular, the category is closed under the formation of unital direct limits, unital tensor products, and crossed products by countable discrete groups. Finally, we take a significant step toward the confirmation of Elliott’s classification conjecture for the C∗-algebras of minimal diffeomorphisms. © 2008 Elsevier Inc. All rights reserved. MSC: primary 46L35; secondary 46L80