On the classification of $\mathrm{C}^\ast$-algebras of real rank zero, II


A classification is given of what may turn out to be all separable nuclear simple $\mathrm{C}^\ast$-algebras of real rank zero and stable rank one. (These terms refer to density of the invertible elements in the sets of self-adjoint elements and all elements, respectively, after adjunction of a unit.)

The $\mathrm{C}^\ast$-algebras considered are those that can be expressed as the inductive limit of a sequence of finite direct sums of homogeneous $\mathrm{C}^\ast$-algebras with spectrum $3$-dimensional finite CW complexes.

This classification is also extended to include certain nonsimple algebras.

The invariant used is the abelian group $\mathrm{K}_\ast = \mathrm{K}_0 \oplus \mathrm{K}_1$, together with the distinguished subset arising from partial unitaries in the algebra, the graded dimension range. With the semigroup generated by the graded dimension range as positive cone, $\mathrm{K}_\ast$ is an ordered group with the Riesz decomposition property which, in a suitable sense (allowing torsion) is unperforated. In fact, $\mathrm{K}_\ast$ is an arbitrary (countable) graded ordered group with these two properties. (This extends the theorem of Effros, Handelman, and Shen.)




George A. Elliott

Guihua Gong