CONTENTS
0. Introduction........................................................................................................................................................................5
I. C*-semigroup bundles and C*-algebras whose irreducible representations are all finite dimensional.................................7
1. C*-semigroup bundles and their morphisms......................................................................................................................7
2. The universal C*-semigroup bundle of a C*-algebra........................................................................................................10
3. Abelian and associative C*-semigroup bundles and the extension of compatible sections..............................................13
4. Existence and "uniqueness" of representation semigroups and C*-semigroup bundles..................................................23
5. Duality between certain C*-semigroup bundles and certain C*-algebras.........................................................................29
6. The core of a representation semigroup..........................................................................................................................35
II. The calculation of $ΠH¹_R$ for certain C*-bundles..........................................................................................................39
1. The functor $ΠH¹_R$.......................................................................................................................................................39
2. C*-bundle embeddings, multiplicity bundles and $MH¹_R$..............................................................................................44
3. Finite order C*-bundles....................................................................................................................................................52
4. Third order C*-bundles with finite dimensional fibres over cones over pairs of compact Riemannian manifolds..............58
5. A remark on the continuity of the map $f: X → S_A$ of I.5.3.3.........................................................................................66
III. Applications and open problems......................................................................................................................................67
1. Applications and final remarks.........................................................................................................................................67
2. Applications.....................................................................................................................................................................76
3. Open problems and final remarks....................................................................................................................................81
Appendix. A simple proof of Dupre's classification Theorem II.1.1 for a restricted class of bundles.....................................82
References..........................................................................................................................................................................87