On weakly amenable groupoids

In this work, we study groupoids and their approximation properties, generalizing both the definitions and some known results for the group case. More precisely, we introduce weak amenability for groupoids using the definition of the Fourier algebra given by Renault. We prove that weakly amenable groupoids are inner exact. We also generalize its algebraic counterpart, the CBAP. To do this we introduce the notion of a quasi Cartan pair $(B,A)$ and see that $(C_r^*(G),C_0(G^0))$ can be viewed as such. We then define what it means for a pair $(B,A)$ to have the CBAP. We introduce the Cowling-Haagerup constants associated to these approximation properties and prove that $\Lambda_{\text{cb}}(C_r^*(G),C_0(G^0)) \leq \Lambda_{\text{cb}}(G)$. We then study some classes of groupoids where we could achieve equality, that is, $\Lambda_{\text{cb}}(G) = \Lambda_{\text{cb}}(C_r^*(G),C_0(G^0))$. They are discrete groupoids and groupoids arising from partial actions of a discrete group $\Gamma$ on a locally compact Hausdorff space $X$.