Classification of certain simple $C^*$-algebras with torsion in $K_1$

We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of sequences of finite direct sums of building blocks of the form $$\{f\in C(\T)\otimes M_n: f(x_i)\in M_{d_i}, i=1,2,...,N\},$$ where $x_1,x_2,...,x_N\in\T$, $d_1,d_2,...,d_N$ are integers dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$. Furthermore we prove existence and uniqueness theorems for *-homomorphisms between such algebras and we identify the range of the invariant.