Simple nuclear C∗C∗-algebras of tracial topological rank one

We give a classification theorem for unital separable nuclear C∗C∗-algebras with tracial rank no more than one. Let A and B   be two unital separable simple nuclear C∗C∗-algebras with TR(A),TR(B)⩽1TR(A),TR(B)⩽1 which satisfy the universal coefficient theorem. We show that A≅BA≅B if and only if there is an order and unit preserving isomorphismγ=(γ0,γ1,γ2):(K0(A),K0(A)+,[1A],K1(A),T(A))≅(K0(B),K0(B)+,[1B],K1(B),T(B)), where γ2−1(τ)(x)=τ(γ0(x)) for each x∈K0(A)x∈K0(A) and τ∈T(B)τ∈T(B).