Classification of inductive limits of continuous trace C*-algebras

A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely, the class of simple C*-algebras which are inductive limits of continuous trace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0,1]. In particular, a classification of simple stably AI algebras is obtained. Also, the range of the invariant is calculated. We start by approximating the building blocks appearing in a given inductive limit decomposition by certain special building blocks. The special building blocks are continuous trace C*-algebras with finite dimensional irreducible representations and such that the dimension of the representations, as a function on the interval, is a finite (lower semicontinuous) step function. It is then proved that these C*-algebras have finite presentations and stable relations. The advantage of having inductive limits of special subhomogeneous algebras is that we can prove the existence of certain gaps for the induced maps between the affine function spaces. These gaps are necessary to prove the Existence Theorem. Also the Uniqueness theorem is proved for these special building blocks.