ON THE CLASSIFICATION OF C-ASTERISK-ALGEBRAS OF REAL RANK ZERO .2.

A classification is given of what may turn out to be all separable nuc lear simple C-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 C-algebras considered are those that can be expressed as the inductive limit of a sequence of finite direct sums of homogeneous C-algebras with spectrum 3-dimensional finite CW complexes. This cla ssification is also extended to include certain nonsimple algebras. Th e invariant used is the abelian group K- = K-0 + K-1, together with t he distinguished subset arising from partial unitaries in the algebra, the graded dimension range. With the semigroup generated by the grade d dimension range as positive cone, K- is an ordered group with the R iesz decomposition property which, in a suitable sense (allowing torsi on) is unperforated. In fact, K- is an arbitrary (countable) graded o rdered group with these two properties. (This extends the theorem of E ffros, Handelman, and Shen.)