Let X be a compact space and Y a closed subset of X. For M(,k), the complex k x k-matrices, consider the C*-algebra of continuous functions f : X (--->) M(,k) with the property that f(x) is a diagonal matrix for all x (ELEM) Y. We shall study the K-theory of this C*-algebra and some closely related C*-algebras for various spaces X and Y. The tools used in this study are a Mayer-Vietoris Sequence and a Puppe Sequence for K-theory of C*-algebras, both of which reduce to the respective sequence in K-theory of locally compact spaces if the involved C*-algebras are commutative First we set up K-theory of unital C*-algebras, following the approach of Karoubi. We define relative K-groups K(,(alpha))((phi)) for unital C*-morphisms (phi) and prove two excision theorems, which will allow us to define K-theory of non-unital C*-algebras. Moreover, we show that the K-functors do not distinguish between homotopic C*-morphisms. This will enable us to define K(,n) of a C*-algebra for all n (ELEM) and to establish a long exact sequence in K-theory associated to a short exact sequence of C*-algebras. We also define a cup product in K-theory of C*-algebras, which will be a (,2)-graded bilinear map K(,*)(A) x K(,*)(B) (--->) K(,*)(A(' )(CRTIMES)(' )B), give some of its basic properties, and use it to define module structures on the K-groups Finally we prove a non-commutative splitting principle which generalizes the well known splitting principle for vector bundles over compact spaces / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_27659 |
| Date | January 1982 |
| Contributors | Hilgert, Joachim (Author) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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