2008/02/11 by Huaxin Lin, Lin, Huaxin, Zhuang Niu +1
Mathematics · #46L05 #46L80 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.0802.1484
openalex publication_date 2008/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A and C be two unital simple C*-algebas with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by KKe(C,A)++ the set of those κ for which κ(K0(C)+∖\0\)⊂ K0(A)+∖\0\ and κ([1C])=[1A]. Suppose that κ∈ KKe(C,A)++. We show that there is a unital monomorphism ϕ: C→ A such that [ϕ]=κ. Suppose that C is a unital AH-algebra and λ: T(A)→ T_\mathttf(C) is a continuous affine map for which τ(κ([p]))=λ(τ)(p) for all projections p in all matrix algebras of C and any τ∈ T(A), where T(A) is the simplex of tracial states of A and T_\mathttf(C) is the convex set of faithful tracial states of C. We prove that there is a unital monomorphism ϕ: C→ A such that ϕ induces both κ and λ. Suppose that h: C→ A is a unital monomorphism and γ∈ Hom(\Kone(C), \aff(A)). We show that there exists a unital monomorphism ϕ: C→ A such that [ϕ]=[h] in KK(C,A), τ∘ ϕ=τ∘ h for all tracial states τ and the associated rotation map can be given by γ. Applications to classification of simple C*-algebras are also given.