Classification ofC*-Algebras of Real Rank Zero and UnsuspendedE-Equivalence Types

In this article, examples are given to prove that the graded scaled orderedK-group is not the complete invariant for aC*-algebra in the class of unital separable nuclearC*-algebras of real rank zero and stable rank one, even for aC*-algebra in the subclass which consists of those real rank zero, stable rank oneC*-algebras being expressed as inductive limits of ⊕kni=1 M[n, i](C(Xn, i)), whereXn, iare two-dimensional finite CW complexes and [n, i] are positive integers. (In the case of simple suchC*-algebras, it has been proved that the above invariant is the complete invariant by George Elliott and the author.) These examples prove that the classification conjecture of Elliott for the case of non simple real rank zeroC*-algebras should be revised—one needs extra invariants. The obstruction preventing two suchC*-algebras with the same graded scaled orderedK-group from being isomorphic is that they have different unsuspendedE-equivalence types (a refinement ofKK-equivalence type ofC*-algebras due to Connes and Higson). In this article, it is proved that for the above class of inductive limitC*-algebras, the obstruction of unsuspendedE-equivalence type is the only obstruction (i.e., if twoC*-algebras in the class are unsuspendedE-equivalence, then they are isomorphic). It is a surprise that in the case of simple suchC*-algebras, or even the case ofC*algebras with finitely many ideals, the obstruction will disappear (see Section 4).