| This paper concerns the KK-theory of the class C of Elliott-Thomsen algebras,as a generalization of bifunctor of K-homology and K-theory,KK-theory was first intro-duced by Russian mathematician Kasparlov in 1980,it was created on the founda of Hilbert modular.This theory was influenced by Atiyah-Singer index theory and BDF theorem,it influenced the development of operator algebra and nuclear C*-algebra,which is very important,also there a lot of mathematicians who developed some other relevant bifunctors,such as Connes and Higson’s E-theory and so on.Here,the class we concern is the Elliott-Thomsen class,which was first given by Elliott and Thomsen in 1990s.which has much more properties and a more and more mathematicians pay attention to it.As its spectrum is very special and the pull back construction,mathe-maticians also call it subhomogenous graph algebra or one dimensional CW complex.We concern the KK-theory of Elliott-Thomsen algebras carefully,and construct a rel-evant order,and then,we get a direct existence theorem and counterexample without dynamic behaviour.This shous the obstruction for KK-lifting.At last,combining with the important condition of real rank 0,we give a existence theorem with dynamic behaviour,which is effected by the decomposition theorem,and finish an important work of classification.The paper is organized as follows:In Section 1,we list some preliminaries con-cerning Elliott-Thomsen algebras and mod-p K-theory with the Dadarlat-Loring order.定义 0.7 For a C*-algebra A,the total K-theory of A is defined by with K*(A;Zp)=K*(A)for p = 0,K*(A;Zp)= 0 for p = 1,and K*(A;Zp)=KK(Ip,A(?)C(S1))for p>2.In Section 2,we formulate a description(as a quotient group)of the KK-group of two Elliott-Thomsen algebras.定理 0.8 Let A,B EC.Then we have a natural isomorphism of groups KK(A,B)≌C(A,B)/M(A,B).In Section 3,we give a useful sufficient condition for KK-lifting for Elliott-Thomsen algebras:引理 0.9 Let A,B ∈C be minimal.Let the diagram A E C(A,B),be given,such that λ is positive,then,AO is positive.If for any i ∈{1,2,…,p},j’∈{1,2,…,l’},andthen there is a homomorphism from A to Mτ(B)for some integer r inducing the dia,gram λ.This gives a complete criterion for a certain subclass,denoted by CO.(The suffi-ciency will be useful,when we check the Dadarlat-Loring order.)定理0.10 Let A,B∈C be minimal,with A∈Co.A KK-element α∈KK(A,B)can be lifted to a homomorphism if,and only if,χ(α)∈C(A,B)/M(A,B)is positive,where χ is the isomorphism defined following Theorem 2.13.In Section 4,we prove that for the subclass CO of Elliott-Thomsen algebras,which in fact includes the generalized dimension drop interval algebras,a KK-clement preserving the Dadarlat-Loring order can be lifted to a homomorphism定理 0.11 Let A∈Co and B∈C.Then a KK-element γ∈KK((A,B)can be lifted to a homomorphism if and only ifγ(K+(A))(?)K+(B).This is contrary to what was stated in Theorem 1.1 of[26].We show that,as was suggested by the work[26],there is a genuine difficulty present,and the lifting theorem does not hold for all of C.In the last Section,we give an existence theorem with property real rank 0 for a larger class than Co.定理 0.12 Let A=(?)(An,φn,m),B=(?)(Bn,φn,m)be two AD algebras with real rank 0.If we also have(K(A),K+(A),[1A])≌(K(B),K+(B),[1B]),then A and B are weakly shape equivalent. |
