Classifying ${}^{\ast }$-homomorphisms I: Unital simple nuclear $C^{\ast }$-algebras

Abstract

We classify the unital embeddings of a unital separable nuclear $C^{\ast }$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^{\ast }$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^{\ast }$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.

Section 5.5 removed; new addendum 1.5 added; various other small changes, including removal of 164 footnotes. 128 pages. Acta Math., to appear