Vol. 5, No. 1, 2020

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

Guihua Gong, Chunlan Jiang and Liangqing Li

Vol. 5 (2020), No. 1, 43–78
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 ${\left(\underset{¯}{K}\left(A\right);\underset{¯}{K}{\left(A\right)}^{+};\Sigma A\right)}_{\Lambda }$ (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 $\left[p\right]\in \Sigma A$), with certain compatibility conditions, is the complete invariant for a certain well behaved class of ${C}^{\ast }$-algebras with the ideal property (e.g., $\mathit{AH}$ 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 $\left[p\right]\in \Sigma A$), 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 $\mathit{AH}$ algebras (of no dimension growth) with the ideal property.

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$C^*$-algebra, $\mathit{AH}$ algebras, ideal property , Elliott invariant, Hausdorffified algebraic $K_1$-group