#### 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.

##### Keywords
$C^*$-algebra, $\mathit{AH}$ algebras, ideal property , Elliott invariant, Hausdorffified algebraic $K_1$-group
##### Mathematical Subject Classification 2010
Primary: 19K14, 19K35, 46L35, 46L80
##### Milestones
Received: 28 May 2018
Revised: 8 June 2019
Accepted: 24 June 2019
Published: 21 March 2020
##### Authors
 Guihua Gong College of Mathematics and Information Science Hebei Normal University Shijiazhuang, Hebei China Department of Mathematics University of Puerto Rico Rio Piedras, PR United States Chunlan Jiang College of Mathematics and Information Science Hebei Normal University Shijiazhuang, Hebei China Liangqing Li Department of Mathematics University of Puerto Rico Rio Piedras, PR United States