Geometric Property (T) and Kazhdan Projections

We characterise Geometric Property (T) by the existence of a certain projection in the maximal uniform Roe algebra (Formula presented.), extending the notion of Kazhdan projection for groups to the realm of metric spaces. We also describe this projection in terms of the decomposition of the metric space into coarsely connected components. © 2024 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.