Type of Document Dissertation Author Spakula, Jan URN etd-07102008-220512 Title K-theory of uniform Roe algebras Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Guoliang Yu Committee Chair Bruce Hughes Committee Member Dietmar Bisch Committee Member Gennadi Kasparov Committee Member Thomas Kephart Committee Member Keywords

- coarse geometry
- C*-algebras
- K-theory
Date of Defense 2008-05-14 Availability unrestricted AbstractWe construct a uniform version of the analytic K-homology theory and prove its basic properties such as a Mayer-Vietoris sequence. We show that uniform K-homology is isomorphic to a directlimit of K-theories of certain C*-algebras.

Furthermore, we construct an index map (or uniform

assembly map) from uniform K-homology into the K-theory of uniform Roe C*-algebras. In an analogy to the coarse Baum--Connes conjecture,

this can be viewed as an attempt to provide an algorithm for computing K-theory of uniform Roe

algebras. Furthermore, as an application of uniform K-homology, we prove a criterion for amenability.

In contrast, we show that uniform Roe C*-algebras of a large class of expanders are not even K-exact. Consequently, their K-theory is in principle not computable by means of exact sequences.

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