Lifting KK-elements, asymptotic unitary equivalence and classification of simple [formula omitted]-algebras

Abstract

Let A and C be two unital simple C ∗ -algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by K K e ( C , A ) + + the set of those κ in K K ( C , A ) for which κ ( K 0 ( C ) + ∖ { 0 } ) ⊂ K 0 ( A ) + ∖ { 0 } and κ ( [ 1 C ] ) = [ 1 A ] . Suppose that κ ∈ K K e ( 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 f ( 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 f ( 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 ( K 1 ( C ) , Aff ( A ) ) . We show that there exists a unital monomorphism ϕ : C → A such that [ ϕ ] = [ h ] in K K ( C , A ) , τ ○ ϕ = τ ○ h for all tracial states τ and the associated rotation map can be given by γ. Denote by K K T ( C , A ) + + the set of compatible pairs ( κ , λ ) , where κ ∈ K L e ( C , A ) + + and λ is a continuous affine map from T ( A ) to T f ( C ) . Together with a result on asymptotic unitary equivalence in [H. Lin, Asymptotic unitary equivalence and asymptotically inner automorphisms, arXiv:math/0703610, 2007], this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from C to A to ( K K T ( C , A ) + + , Hom ( K 1 ( C ) , Aff ( T ( A ) ) ) / R 0 ) , where R 0 is a subgroup related to vanishing rotation maps. As an application, combining these results with a result of W. Winter [W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, arXiv:0708.0283v3, 2007], we show that two unital amenable simple Z -stable C ∗ -algebras are isomorphic if they have the same Elliott invariant and the tensor products of these C ∗ -algebras with any UHF-algebra have tracial rank zero. In particular, if A and B are two unital separable simple Z -stable C ∗ -algebras with unique tracial states which are inductive limits of C ∗ -algebras of type I, then they are isomorphic if and only if they have isomorphic Elliott invariants.