#### Date of Award

Spring 1-1-2018

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### First Advisor

Judith Packer

#### Second Advisor

Carla Farsi

#### Third Advisor

Martin Walter

#### Fourth Advisor

Robin Deeley

#### Fifth Advisor

Elizabeth Gillaspy

#### Abstract

We consider a ℤ-action σ on a directed graph -- in particular a rooted tree *T* -- inherited from the odometer action. This induces a ℤ-action by automorphisms on *C**(*T*). We show that the resulting crossed product *C**(*T*) ⋊_{σ}ℤ is strongly Morita equivalent to the Bunce-Deddens algebra. The Pimsner-Voiculescu sequence allows us to reconstruct the *K*-theory for the Bunce-Deddens algebra in a new way using graph methods. We then extend to a ℤ^{k}-action σ̃ on a *k*-graph when *k* = 2, show that *C**(*T*_{1} ✕ *T*_{2})⋊_{σ}ℤ^{2} is strongly Morita equivalent to a generalized Bunce-Deddens algebra of type Orfanos, and invoke the Künneth theorem to determine this new crossed product's *K*-theory. We end by generalizing the results for all *k*.

#### Recommended Citation

Davidoff, Nathan, "On the K-Theory of Generalized Bunce-Deddens Algebras" (2018). *Mathematics Graduate Theses & Dissertations*. 68.

https://scholar.colorado.edu/math_gradetds/68