Simple nuclear C∗-algebras of tracial topological rank one

We give a classification theorem for unital separable nuclear C∗-algebras with tracial rank no more than one. Let A and B be two unital separable simple nuclear C∗-algebras with TR(A),TR(B) 1 which satisfy the universal coefficient theorem. We show that A∼= B if and only if there is an order and unit preserving isomorphism γ = (γ0,γ1,γ2) : K0(A),K0(A)+, [1A],K1(A), T (A) ∼= K0(B),K0(B)+, [1B],K1(B), T (B) , where γ−1 2 (τ )(x) = τ(γ0(x)) for each x ∈ K0(A) and τ ∈ T (B). © 2007 Elsevier Inc. All rights reserved