Topologies on Positive Type Functions on Groupoids, Weak Containment of Continuous Unitary Representations, and Property (T)

Abstract

We introduce two new topologies on the space of normalized positive type functions on a groupoid with Haar system, the weak* topology and fiberwise compact convergence topology. We demonstrate these topologies are equivalent when the groupoid is second countable and locally compact, thereby extending Raikov's theorem. Using the fiberwise compact convergence topology we introduce a notion of weak contaiment of continuous unitary representations of a groupoid. We characterize weak containment of the trivial representation with almost invariant sections which extends the idea of almost invariant vectors in group representations. This weak containment naturally suggests a definition of property (T) for groupoids which extends Kazhdan's group property (T). We briefly explore this property and attempt to generalize some of the classic results from the theory of groups.