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A $C^*$-algebra $A$ is said to have the ideal property if each closedtwo-sided ideal of $A$ is generated by the projections inside the ideal, as aclosed two sided ideal. $C^*$-algebras with the ideal property aregeneralization and unification of real rank zero $C^*$-algebras and unitalsimple $C^*$-algebras. It is long to be expected that an invariant (see [Stev]and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $Inv^0(A)$ (see theintroduction), consisting of scaled ordered total $K$-group $(\underline{K}(A),\underline{K}(A)^{+},\Sigma A)_{\Lambda}$ (used in the real rank zero case),the tracial state space $T(pAp)$ of cutting down algebra $pAp$ as part ofElliott invariant of $pAp$ (for each $[p]\in\Sigma A$) with a certaincompatibility, is the complete invariant for certain well behaved class of$C^*$-algebras with the ideal property (e.g., $AH$ algebras with no dimensiongrowth). In this paper, we will construct two non isomorphic $A\mathbb{T}$algebras $A$ and $B$ with the ideal property such that $Inv^0(A)\congInv^0(B)$. The invariant to differentiate the two algebras is theHausdorffifized algebraic $K_1$-groups $U(pAp)/\overline{DU(pAp)}$ (for each$[p]\in\Sigma A$) with a certain compatibility condition. It will be proved in[GJL] that, adding this new ingredients, the invariant will become the completeinvariant for $AH$ algebras (of no dimension growth) with the ideal property.

Hausdorffifized algebraic $K_1$ group and invariants for $C^*$-algebras with the ideal property