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

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Rozprawy Matematyczne tom/nr w serii: 290 wydano: 1989
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EN

We investigate the structure of C*-algebras with a finite bound on the dimensions of their irreducible representations, sometimes called "subhomogeneous".
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 lo the bundle. Under suitable conditions we prove the existence of "enough" bounded sections, which arc "compatible" with the C*-semigroup bundle structure. Then we establish a complete duality between a certain class of C*-semigroup bundles and subhomogeneous C*-algebras, namely the algebra of compatible sections of such a C*-semigroup bundle is subhomogeneous and conversely, every subhomogeneous 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 $ΠH¹_R$, which classifies certain C*-bundles with varying finite dimensional fibres. $ΠH¹_R$ is the C*-bundle analog of Čech-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 $MH¹_R$ which describes the multiplicities of the imbeddings of the fibres into the bundle and assignment of multiplicity matrices to cocycles yields a natural transformation: $ΠH¹_R → MH¹_R$.
Chapter three finally gives some applications. We calculate $ΠH¹_R$ for C'-bundles over a two disk Tor an assignment of different finite dimensional fibres. The result is stated in terms of $MH¹_R$ 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 $MH¹_R$.
A last example involving bundles over a three ball with 3 different fibres shows the fact that $MH¹_R$ does not always provide complete invariants and at the same time illustrates the limits of our methods.
EN

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
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 290
Liczba stron
87
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCXC
Daty
wydano
1989
Twórcy
autor
Bibliografia
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