# C*-semigroup bundles and c*-algebras whose irreducible representations are all finite-dimensional

We investigate the structure of C*-algebras with a finite bound on the dimensions of their irreducible representations, sometimes called 'subhomogenous' In the first chapter we develop the theory of C*-semigroup bundles. These are C*-bundles over semigroups together with a 'structure map' which links the semigroup structure of the base space to the bundle. Under suitable conditions we prove the existence of 'enough' bounded sections, which are 'compatible' with the C*-semigroup bundle structure. Then we establish a complete duality between a certain class of C*-semigroup bundles and subhomogenous C*-algebras, namely the algebra of compatible sections of such a C*-semigroup bundle is subhomogenous and conversely, every subhomogenous C*-algebra is isomorphic to the algebra of compatible sections of such a C*-semigroup bundle. In this way we are able to even represent C*-algebras with non-Hausdorff spectrum as sections in bundles The second chapter is devoted to developing methods for the computation of the functor (PI)H(,R)('1), which classifies certain C*- bundles with varying finite dimensional fibres. (PI)H(,R)('1) is the C*- bundle analog of Cech-cohomology for bundles with one fibre type. The difficulty here is, that homotopy classes of cocycles of bundle imbeddings have to be computed, while only homotopies that satisfy a corresponding cocycle condition can be considered. We define a functor MH(,R)('1) which describes the multiplicities of the imbeddings of the fibres into the bundle and assignment of multiplicity matrices to cocycles yields a natural transformation: (PI)H(,R)('1) (--->) MH(,R)('1) Chapter three finally gives some applications. We calculate (PI)H(,R)('1) for C*-bundles over a two disk for an assignment of different finite dimensional fibres. The result is stated in terms of MH(,R)('1) and quotients of homotopy groups of bundle imbeddings. It provides a new way to describe the group C*-algebra of an interesting group called p4gm, which has been computed by I. Raeburn, and furthermore, our description yields complete invariants--in fact these are given by MH(,R)('1) A last example involving bundles over a three ball with 3 different fibres shows the fact that MH(,R)('1) does not always provide complete invariants and at the same time illustrates the limits of our methods