Kazhdan's property (T)

Let $G$ be a topological group. Then $G$ has Kazhdan's property (T) if every strongly continuous unitary representation of $G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) on a Hilbert space $H$ (Hilbert space) that has almost invariant vectors (Almost invariant vectors for a unitary representation) has a nonzero $G$-invariant vector. Equivalently, whenever such a representation $(\pi ,H)$ has, for every compact $Q\subseteq G$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) and every $\epsilon >0$, a unit vector $\xi$ satisfying $\mathrm{\parallel }\pi (x)\xi -\xi \mathrm{\parallel }<\epsilon \text{for every }x\in Q\mathrm{.}$ Then $\pi$ has a nonzero vector fixed by every $\pi (g)$, $g\in G$.

• The two formulations are exactly the definition of “almost invariant vectors” and the definition of “nonzero invariant vector” from Almost invariant vectors for a unitary representation. The inequality is pointwise for each $x\in Q$; no supremum over $Q$ is introduced, so the empty compact set causes no undefined supremum.
• The zero representation on $H=\{0\}$ has no unit vectors and therefore has no almost invariant vectors. The property-(T) implication is vacuous for that representation.
• No local compactness, Hausdorffness, countability, or choice assumption is part of this definition.