The Furstenberg boundary of groupoids

Let $\mathcal{G}$ be a locally compact groupoid. We show that there is a one-to-one correspondence between $\mathcal{G}$-spaces and the groupoid dynamical systems whose underling $C_0({\mathcal{G}}^{(0)})$-algebra is commutative. We study minimality and (strong) proximality for $\mathcal{G}$-actions and show that each locally compact groupoid $\mathcal{G}$ has a universal minimal (strongly) proximal $\mathcal{G}$-space (called the Furstenberg boundary).