Almost-invariant vectors

A unitary representation $\pi$ of a discrete group $G$ has almost-invariant vectors if for every finite $F\subseteq G$ and every $\varepsilon>0$ , there is a unit vector $v$ satisfying

\max_{g\in F}\|\pi(g)v-v\|<\varepsilon.

For a strongly continuous representation of a locally compact group, replace finite $F$ by compact $Q$ and maximum by supremum.

The quantifiers matter

The vector can depend on both $F$ and $\varepsilon$ . An invariant vector supplies all these approximations, but approximating vectors need not converge to a nonzero invariant vector. A spectral gap rules out such approximations on the specified representation space.

References

B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property (T), §1.1.