Hausdorffified algebraic K1-groups and invariants for C*-algebras with the ideal property

A C*-algebra A is said to have the ideal property if each closed two-sided ideal of A is generated as a closed two-sided ideal by the projections inside the ideal. C*-algebras with the ideal property are a generalization and unification of real rank zero C* -algebras and unital simple C* -algebras. It was long expected that an invariant that we call Inv(0). (A), consisting of the scaled ordered total K-group (K (A); K (A)(+); Sigma A)(Lambda) (used in the real rank zero case), along with the tracial state spaces T(pAp) for each cut-down algebra pAp, as part of the Elliott invariant of pAp (for each [p] is an element of Sigma A, with certain compatibility conditions, is the complete invariant for a certain well behaved class of C* -algebras with the ideal property (e.g., AH algebras with no dimension growth). In this paper, we construct two nonisomorphic AT algebras A and B with the ideal property such that Inv(0) (A)similar or equal to*Inv(0)(B), disproving this conjecture. The invariant to distinguish the two algebras is the collection of Hausdorffified algebraic K-1-groups U(pAp) / DU (pAp) (for each [p] is an element of Sigma A), along with certain compatibility conditions. We will prove in a separate article that, after adding this new ingredient, the invariant becomes the complete invariant for AH algebras (of no dimension growth) with the ideal property.