Stable ASH Algebras

The Jiang–Su algebra $Z$ has come to prominence in the classification program for nuclear ${{C}^{*}}$ -algebras of late, due primarily to the fact that Elliott’s classification conjecture in its strongest form predicts that all simple, separable, and nuclear ${{C}^{*}}$ -algebras with unperforated $\text{K}$ -theory will absorb $Z$ tensorially, i.e., will be $Z$ -stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and $Z$ -stable ${{C}^{*}}$ -algebras. We prove that virtually all classes of nuclear ${{C}^{*}}$ -algebras for which the Elliott conjecture has been confirmed so far consist of $Z$ -stable ${{C}^{*}}$ -algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible ${{C}^{*}}$ -algebras are $Z$ -stable.