Hausdorffifized algebraic $K_1$ group and invariants for $C^{\ast }$-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 by the projections inside the ideal, as a closed two sided ideal. $C^{\ast }$-algebras with the ideal property are generalization and unification of real rank zero $C^{\ast }$-algebras and unital simple $C^{\ast }$-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $In{v}^0(A)$ (see the introduction), 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 of Elliott invariant of $pAp$ (for each $[p]\in \Sigma A$) with a certain compatibility, is the complete invariant for certain well behaved class of $C^{\ast }$-algebras with the ideal property (e.g., $AH$ algebras with no dimension growth). In this paper, we will construct two non isomorphic $A\mathbb{T}$ algebras $A$ and $B$ with the ideal property such that $In{v}^0(A)\cong In{v}^0(B)$. The invariant to differentiate the two algebras is the Hausdorffifized 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 complete invariant for $AH$ algebras (of no dimension growth) with the ideal property.

arXiv admin note: text overlap with arXiv:1607.07581