We write arbitrary separable nuclear $\mathrm {C}^*$-algebras as limits of inductive systems of finite-dimensional $\mathrm {C}^*$-algebras with completely positive connecting maps. The characteristic feature of such ${\mathrm {CPC}^*}$-systems is that the maps become more and more orthogonality preserving. This condition makes it possible to equip the limit, a priori only an operator space, with a multiplication turning it into a $\mathrm {C}^*$-algebra. Our concept generalizes the NF systems of Blackadar and Kirchberg beyond the quasidiagonal case.

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.

We show that, if $A$ is a separable simple unital $C^{*}$-algebra that absorbs the Jiang–Su algebra ${{\mathcal Z}}$ tensorially and that has real rank zero and finite decomposition rank, then $A$ is tracially approximately finite-dimensional in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of $C^{*}$-algebras as above that, additionally, satisfy the universal coefficients theorem. In particular, such algebras are completely determined by their ordered $K$-theory. They are approximately homogeneous of topological dimension less than or equal to three, approximately subhomogeneous of topological dimension at most two and their decomposition rank also is no greater than two.

We analyze the decomposition rank (a notion of covering dimension for nuclear C*-algebras introduced by E. Kirchberg and the author) of subhomogeneous C*-algebras. In particular, we show that a subhomogeneous C*-algebra has decomposition rank $n$ if and only if it is recursive subhomogeneous of topological dimension $n$, and that $n$ is determined by the primitive ideal space.

As an application, we use recent results of Q. Lin and N. C. Phillips to show the following. Let $A$ be the crossed product C*-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then the decomposition rank of $A$ is dominated by the covering dimension of the underlying manifold.