Classification of simple $C^{\ast }$-algebras of tracial topological rank zero

Abstract

We give a classification theorem for unital separable simple nuclear $C^{\ast }$-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if $A$ and $B$ are two such $C^{\ast }$-algebras and $$(K_0(A),K_0(A{)}_{+},[1_A],K_1(A))$$ $$\cong (K_0(B),K_0(B{)}_{+},[1_B],K_1(B)),$$ then $A\cong B.$