Embedding Groupoid C^*-Algebras and Certain Bimodules in Groupoid C^*-Algebras

Abstract

The following thesis is divided into three main chapters. In Chapter 3, for an étale groupoid $G$ equipped with a fxed Fell line bundle L, and an open subgroupoid $H$ of $G$ containing $G^{(0)}$, we show that $$C_{0}(G^{(0)})\subseteq C_{r}^{*}(H,L_{H})\subseteq C_{r}^{*}(G,L).$$

This is an extension of what Brown, Fuller, Pitts, and Reznikoff proved in the Hausdorff setting, [2, Lemma 2.19].

Moreover, if $G$ is topologically free, we define the essential groupoid C^*-algebra $C_{ess}^{*}(G,L)$ to be the quotient of $C_{r}^*(G,L)$ by the largest ideal of $C_{r}^{*}(G,L)$ that has trivial intersection with $C_{0}(G^{(0)})$. We show that

$$C_{0}(G^{(0)})\subseteq C_{ess}^{*}(H,L_{H})\subseteq C_{ess}^{*}(G,L).$$

In Chapter 4, we gave an extension of a theorem due to Kumora. In particular, we study the closed $C_{0}(G^{(0)})$-bimodules for an étale groupoid $G$ equipped with a fxed Fell line bundle L. If $N_{G}$ is the $*$-subsemigroup of the set of all normalizer of $C_{0}(G^{(0)})$ within $C_{r}^{*}(G,L)$ which contains elements of bisection support, and $M$ a closed linear subspace of $C_{r}^{*}(G,L)$ we show that $M$ is generated by elements from $N_{G}$ if and only if there exist an open set $U\subseteq G$ such that $$M=\overline{\mathscr{C}_{c}(U,L}.$$

In Chapter 5, we study the example of a non-Hausdorff, étale, and effective groupoid given by Ruy Exel. Exel presented an example of a normalizer $u\in C_{r}^{*}(G)$ and showed that $u$ is not supported in a bisection and hence not in $N_{G}$. If $u$ is the normalizer that is not supported in a bisection and $M\subseteq C_{r}^{*}(G)$ is the closed $C_{0}(G^{(0)})$-bimodule generated by $u$, we show that there is not an open set $U\subseteq G$ for which $$M=\overline{\mathscr{C}_{c}(U)}.$$ Hence, in a non-Hausdorff setting, even if the groupoid $G$ is effective, our result is not extended to cover the set of all normalizers, and we have to restrict our choice to $N_{G}$.