| The problem people we considered is to classify C*algebras by their K-theoretical data.The first result of this kind was the classification of inductive limits of sequence of finite direct sums of matrix algebras(called AF algebras).This result was extended in the following work.It replaced the matrix algebras by matrix algebras over the unit circle and restricted the limit of C*algebras to have real rank zero.The current results is that a classification was given to all separable nuclear simple C*algebras of real rank zero.In this paper,we consider the real rank zero C*-algebra which can be written as an inductive limit of the Elliott-Thomsen building blocks:and obtain some results about the classification.This paper is organized as follows.In section 1,we will introduce some notations collect some known results.In Section 2,we use the generalized pairing lemma to get some comparable results about the spectra.In Section 3,we prove a decomposition result for the connective homomorphisms,The decomposition theory plays an important role in the classification theorem(see[18],[12]),which says thatφm,n can be approximately decomposed as a sum of two parts,φ1 and φ2;One part having a very small support projection,and the other part factoring through a finite dimensional algebra,this technique will be used in the classification theorem.Theorem 1Let A=A(F1,F2,ψ0,ψ1),B=B(F1’,F2’,ψ0’,ψ1’)∈C,assume that F2’ has only one block.Let G(?)A be a finite set,for any positive integers J,L,there existsη>0 such that if a unital homomorphismφ:A→ B satisfies φ(H(η))(?)1/6{f∈B |∫has finite spectrum},then there exists a projection q ∈ B and a unital homomor-phism ψ:A→(1—q)B(1-q)with finite dimensional image such that(1)L · rank(φ(e)-ψ(e))<rank ψ(e)for any projection e∈A,(2)||qφ(g)-φ(g)q||<4/J for any g∈G,(3)||φ(g)-qφ(g)q⊕ψ(g)||<4/J for any g∈G.Theorem 2Let A,B∈D with K1(A),K1(B)torsion free,let G(?)A be a finite set,then for any ε>0,positive integer L,there exists η>0 such that if an injective ho-momorphism φ:A →B satisfies the condition in 5.13 and φ(H(η))(?)1/6 {f∈B|f has finite spectrum},then there exists a projection q ∈B and a unital homo-morphism ψ:A→(1—q)B(1—q)with finite dimensional image such that(1)L·[φ(e)—ψ(e)]<[ψ(e)]for any projection e ∈A,(2)||φ(g)-qφ(g)q⊕ψ(g)||<4ε for any g∈G.In Section 4,we will give a weak variation result.Theorem 3(?)be a real rank zero inductive limits of direct sum of basic building blocks.Let F(?)Am be a finite subset,then we have(?)In Section 5 and 6,we will prove the uniqueness theorem for the real rank zero C*-algebra which can be expressed as the inductive limits of a subclass of the Elliott-Thomsen building blocks.Theorem 4(?)be real rank zero inductive limits of direct sums of basic building blocks,For any ε:>0 and finite set F(?)Am with ω(F)<ε,there are integers m2>m1 ≥ m satisfy that if two homomorphisms φ,ψ:Am1→Am2 with the property that KK(φ)＝ KK(ψ),then there is a unitary U∈Am2 Am2 such thatTheorem 5(?)and(?)are real rank zero inductive limits of direct sums of basic building blocks,if(An,φn,m)and(Bn,φn,m)are weakly shape equivalent,then A ≌ B. |
