Factoriality of groupoid von Neumann algebras

We give a characterisation of factoriality of the groupoid von Neumann algebra $L(\mathcal{G})$ associated to a discrete measured groupoid $(\mathcal{G},\mu )$. We introduce the notion of groupoids with `infinite conjugacy classes’ and show that this property together with ergodicity of the groupoid is equivalent to factoriality of $L(\mathcal{G})$.