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 finite direct sums of building blocks of the form
$$\left\{ f\in C\left( \mathbb{T} \right)\otimes {{M}_{n}}:f\left( {{x}_{i}} \right)\in {{M}_{{{d}_{i}}}},i=1,2,...,N \right\},$$
where
${{x}_{1}},\,{{x}_{2.}},...,{{x}_{N\,}}\in \mathbb{T},{{d}_{1}},{{d}_{2}},.\,.\,.,{{d}_{N}}$
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.