Simple nuclear $C^{\ast }$-algebras of tracial topological rank one

We give a classification theorem for unital separable nuclear simple \CA s with tracial rank no more than one. Let $A$ and $B$ be two unital separable simple nuclear \CA s with $TR(A),TR(B)\le 1$ which satisfy the universal coefficient theorem. We show that $A\cong B$ if and only if $$(K_0(A),K_0(A{)}_{+},[1_A],K_1(A),T(A))\cong (K_0(B),K_0(B{)}_{+},[1_B],K_1(B),T(B))\mathrm{.}$$