HOMOMORPHISMS INTO SIMPLE Z-STABLE C*-AIGEBRAS

作者：Lin, Huaxin^*; Niu, Zhuang

来源：Journal of Operator Theory, 2014, 71(2): 517-569.

DOI：10.7900/jot.2012jul10.1975

• Classification of C*-algebras
• AH-algebras
• Z-stable C*-algebras
• homotopy lemma
• uniqueness theorems

Let A and B be unital separable simple amenable C*-algebras which satisfy the universal coefficient theorem. Suppose that A and B are Z-stable and and are of rationally tracial rank no more than one. We prove the following: Suppose that phi,psi : A -> B are unital *-monomorphisms. There exists a sequence of unitaries {un} subset of B such that n ->infinity lim un*phi(a)un = psi(a) for all a is an element of A, if and only if [phi] = [psi] in KL(A,B), phi = psi and phi = psi, where phi,psi : Aff(T(A)) -> Aff(T(B)) and phi,psi : U(A) / CII(A) -> U(B)/CU(B) are the induced maps (where T(A) and T(B) are the tracial state spaces of A and B, and CU(A) and CU(B) are the closures of the commutator subgroups of the unitary groups of A and B, respectively). We also show that this holds if A is a rationally AH-algebra which is not necessarily simple. Moreover, for any strictly positive unit-preserving kappa is an element of KL(A,B), any continuous affine map A : Aff(T(A)) -> Aff(T(B)) and any continuous group homomorphism gamma : U(A) / CU(A) -> U(B)/CU(B) which are compatible, we also show that there is a unital homomorphism phi : A -> B so that ([phi], phi,psi) = (kappa, lambda, gamma), at least in the case that K-1 (A) is a free group.

• 出版日期2014
• 单位华东师范大学