2019/05/28 by Guihua Gong, Chunlan Jiang, Gong, Guihua +3
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #math.OA
paper · pdf · doi:10.48550/arxiv.1905.12453
arXiv admin note: text overlap with arXiv:1607.07581
arxiv created 2019/05/28 · arxiv updated 2019/05/30
A C^*-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^*-algebras with the ideal property are generalization and unification of real rank zero C^*-algebras and unital simple C^*-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it Inv0(A) (see the introduction), consisting of scaled ordered total K-group (\underlineK(A), \underlineK(A)+,ΣA)Λ (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]∈ΣA) with a certain compatibility, is the complete invariant for certain well behaved class of C^*-algebras with the ideal property (e.g., AH algebras with no dimension growth). In this paper, we will construct two non isomorphic A\mathbbT algebras A and B with the ideal property such that Inv0(A)≅ Inv0(B). The invariant to differentiate the two algebras is the Hausdorffifized algebraic K1-groups U(pAp)/DU(pAp) (for each [p]∈ΣA) 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.