Geometric property (T) and Kazhdan projections (Q6644172)

The open property (T) dedicated to David Kazhdan, for locally compact groups, is a negation of amenability. Indeed, if G is a locally compact group which has Property (T) and is amenable, then it is bound to be compact. This property has an analogue called Geometric Property (T) in the realm of metric spaces.\N\NA bounded geometry, monogenic metric space \(X\) with at most countably many coarse components is said to have Property (T) if for every controlled generating set \(E\subseteq X\), there exists \(c>0\) such that, for every representation \(\pi: C_u[X]\rightarrow B(H)\) and for each \(\xi \in( {H^{\pi}})^{\bot},\) there is a partial translation \(v\) with support in \(E\) satisfying\[\|(\pi(v)-\pi(vv^*))\xi\|\geq c\|\xi\|,\] where \(C_u[X]\) stands for the \(*\)-algebra of finite propagation operators.\N\NThe main result of the present paper is to give a characterization for this property. The author proves that Property (T) is equivalent to the existence of a projection \(P\in C^*_{u, \max}(X)\) such that, for every representation \(\pi: C_u[X] \rightarrow B(H),\) \(\pi(P)\) is the orthogonal projection onto the subspace of invariant vectors \(H^{\pi},\) where \(C^*_{u, \max}(X)\) is the maximal uniform Roe algebra of \(X\).