# K-exact group C*-algebras

#### Abstract

In this thesis we investigate the relationship between coarse embeddability and K-exactness of countable discrete groups. The main result is that under an additional technical assumption, every group coarsely embeddable into a Hilbert space is K-exact. The fact that every coarsely embeddable group satisfies the coarse Baum-Connes conjecture [1, 2] combined with the main result in my thesis suggest the study of this fundamental conjecture for K-exact groups. Hence, K-exactness draws its significance from its relationship to coarse embeddability and hence to the Baum-Connes conjecture. In Chapter 1 we have surveys on K-theory, KK-theory, E-theory, and the Baum-Connes conjecture. In Chapter 2 we define and develop the notion of K-exact C*-algebras and K-exact groups which is the K-theoretic analogue of exactness. A group Γ is K-exact if the minimal tensor product by [special characters omitted]Γ preserves the K-theoretic six-term exact sequence regardless of whether it preserves short exact sequences of C*-algebras. A theorem on equivalent definitions of K-exactness with several examples are presented. We also lay the foundations for further exploration of K-exactness and of related properties (e.g. generalizing K-exactness to KK-theory and E-theory for crossed product algebras); and the relation of K-exactness to geometric and other properties of groups. Chapter 3 is devoted to the relationship between K-exact and coarsely embeddable groups. In this part we prove our main result stated above.

#### Degree

Ph.D.

#### Advisors

Kaminker, Purdue University.

#### Subject Area

Mathematics

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