2010/03/09 by Huaxin Lin, Lin, Huaxin, Zhuang Niu +1 · 1 citation
Mathematics · #46L05 #46L80 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1003.1760
openalex publication_date 2010/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A and B be unital separable simple amenable \CA s which satisfy the Universal Coefficient Theorem. Suppose that A and B are \mathcal Z-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that ϕ, ψ: A→ B are unital monomorphisms. There exists a sequence of unitaries \un\⊂ B such that limn→∞ un^*ϕ(a) un=ψ(a)\tforal a∈ A, if and only if [ϕ]=[ψ] in KL(A,B), ϕ\sharp=ψ\sharp\andeqnϕ^‡=ψ^‡, where ϕ\sharp, ψ\sharp: \aff(T(A))→ \aff(T(B)) and ϕ^‡, ψ^‡: U(A)/CU(A)→ U(B)/CU(B) are the induced maps and where T(A) and T(B) are tracial state spaces of A and B, and CU(A) and CU(B) are closure of commutator subgroups of unitary groups of A and B, respectively. We also show that this holds for some AH-algebras A. Moreover, if κ∈ KL(A,B) preserves the order and the identity, λ: \aff(\tr(A))→ \aff(\tr(B)) is a continuous affine map and γ: U(A)/CU(A)→ U(B)/CU(B) is a \hm which are compatible, we also show that there is a unital \hm ϕ: A→ B so that ([ϕ],ϕ\sharp,ϕ^‡)=(κ, λ, γ), at least in the case that K1(A) is a free group,