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

Thomas Müller

GDML_Books, (1989), p.

Résumé

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.

CONTENTS0. Introduction........................................................................................................................................................................5I. C*-semigroup bundles and C*-algebras whose irreducible representations are all finite dimensional.................................71. C*-semigroup bundles and their morphisms......................................................................................................................72. The universal C*-semigroup bundle of a C*-algebra........................................................................................................103. Abelian and associative C*-semigroup bundles and the extension of compatible sections..............................................134. Existence and “uniqueness” of representation semigroups and C*-semigroup bundles..................................................235. Duality between certain C*-semigroup bundles and certain C*-algebras.........................................................................296. The core of a representation semigroup..........................................................................................................................35II. The calculation of $ΠH¹_R$ for certain C*-bundles..........................................................................................................391. The functor $ΠH¹_R$.......................................................................................................................................................392. C*-bundle embeddings, multiplicity bundles and $MH¹_R$..............................................................................................443. Finite order C*-bundles....................................................................................................................................................524. Third order C*-bundles with finite dimensional fibres over cones over pairs of compact Riemannian manifolds..............585. A remark on the continuity of the map $f: X → S_A$ of I.5.3.3.........................................................................................66III. Applications and open problems......................................................................................................................................671. Applications and final remarks.........................................................................................................................................672. Applications.....................................................................................................................................................................763. Open problems and final remarks....................................................................................................................................81Appendix. A simple proof of Dupre’s classification Theorem II.1.1 for a restricted class of bundles.....................................82References..........................................................................................................................................................................87

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