Hausdorffified algebraic $K_1$-groups and invariants for $C^*$-algebras with the ideal property

Abstract

A $C^{\ast }$-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. $C^{\ast }$-algebras with the ideal property are a generalization and unification of real rank zero $C^{\ast }$-algebras and unital simple $C^{\ast }$-algebras. It was long expected that an invariant that we call ${Inv}^0\left(A\right)$, consisting of the scaled ordered total $K$-group ${\left(\underset{¯}{K}\left(A\right);\underset{¯}{K}{\left(A\right)}^{+};\mathrm{\Sigma }A\right)}_{\mathrm{\Lambda }}$ (used in the real rank zero case), along with the tracial state spaces $T\left(pAp\right)$ for each cut-down algebra $pAp$, as part of the Elliott invariant of $pAp$ (for each $\left[p\right]\in \mathrm{\Sigma }A$), with certain compatibility conditions, is the complete invariant for a certain well behaved class of $C^{\ast }$-algebras with the ideal property (e.g., $AH$ algebras with no dimension growth). In this paper, we construct two nonisomorphic $A\mathbb{T}$ algebras $A$ and $B$ with the ideal property such that ${Inv}^0\left(A\right)\cong {Inv}^0\left(B\right)$ , disproving this conjecture. The invariant to distinguish the two algebras is the collection of Hausdorffified algebraic $K_1$-groups $U\left(pAp\right)∕\overline{DU\left(pAp\right)}$ (for each $\left[p\right]\in \mathrm{\Sigma }A$ ), 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.