We examine the ranks of operators in semi-finite ${{C}^{*}}$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple ${{C}^{*}}$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray–von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first author, this shows that slow dimension growth coincides with $Z$-stability for approximately subhomogeneous 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.

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.

2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.