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On the classification of \(C^*\)-algebras of real rank zero. II. (English) Zbl 0867.46041

Summary: A classification is given of what may turn out to be all separable nuclear simple \(C^*\)-algebras of real rank zero and stable rank one. (These terms refer to density of the invertible elements in the sets of selfadjoint 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 classification is also extended to include certain nonsimple algebras.
The invariant used is the Abelian group \(K_*= K_0\oplus K_1\), 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, \(K_*\) is an ordered group with the Riesz decomposition property which, in a suitable sense (allowing torsion) is unperforated. In fact, \(K_*\) is an arbitrary (countable) graded ordered group with these two properties. (This extends the theorem of Effros, Handelman, and Shen.)
[For part I see the first author, J. Reine Angew. Math. 443, 179-219 (1993; Zbl 0809.46067)].

MSC:

46L35 Classifications of \(C^*\)-algebras
46M40 Inductive and projective limits in functional analysis
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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