The Factorization Property Of C~*-algebra Homomorphisms

Let X be a connected finite CW complex and A be a unital simple separable C*-algebra with tracial rank zero. Suppose thatφ:C(X)→A is a unital monomorphism.For dim(X)≤3, we prove thatφcan approximately factor through finite direct sums of matrix algebras over spaces of forms:{pt}, [0,1], S1,S2, S3, TⅡ,k, or TⅢ,k. For general X (with higher dimension), we prove a sim-ilar factorization theorem, but we allow the factoring maps to be almost multiplicative completely positive linear maps.For dim.(X)≤3, we prove thatφcan approximately factor through circle algebras if and only if the H2-part of [φ]∈KK(C(X),A) induces an zero homomorphism between K-groups and the H3-part of [φ] is zero.Also, we prove thatφis an AF-homomorphism (see 5.1 for definition) if and only if [φ](tor(K0(C(X))))=0, [φ]{K1(C(X)))= 0 and [φ](Ki(C(X);Z/n))= 0 for all n≥2.Let A be a unital simple C*-algebra with TR(A)≤k, X as above, andφ:C(X)→A be a unital monomorphism. We prove that there is another unital homomorphismφ:C(X)→A such that the induced mapsφ,ψ: CR(X)→AffT(A) are close andψcan factor through interval algebras.We also study a class of AF-algebras in which the identity homomor-phism can approximately factor through finite dimensional C*-algebras. We give an example which is a RFD AF-algebra but not in this class.