Almost invariant vectors for a unitary representation

Let $G$ be a topological group and let $(\pi ,H)$ be a strongly continuous unitary representation of $G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) on a Hilbert space $H$ (Hilbert space). For a subset $Q\subseteq G$ and a real number $\epsilon >0$, a vector $\xi \in H$ is $(Q,\epsilon )$-invariant if $\mathrm{\parallel }\pi (x)\xi -\xi \mathrm{\parallel }<\epsilon \mathrm{\parallel }\xi \mathrm{\parallel }\text{for every }x\in Q\mathrm{.}$ The condition is vacuous when $Q=\mathrm{\varnothing }$; the definition of almost invariant vectors below still tests the compact singleton containing the identity. For $Q=\{e\}$, every unit vector is $(Q,\epsilon )$-invariant because $\pi (e)=I$. The representation $\pi$ has almost invariant vectors if 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 $\epsilon >0$, it has a $(Q,\epsilon )$-invariant unit vector. The representation has a nonzero invariant vector if there is $\xi \mathrm{\ne }0$ such that $\pi (g)\xi =\xi$ for every $g\in G$. In particular, the zero representation has no almost invariant vectors because it has no unit vectors. For a nonzero $\xi$, the $(Q,\epsilon )$-condition is unchanged by multiplying $\xi$ by a nonzero scalar, so it may be normalized to a unit vector.

For any topological group, almost invariant vectors imply $1_G\prec \pi$ in the finite-sum coefficient sense of Weak containment of unitary representations. Indeed, every coefficient of the trivial representation is a nonnegative constant $c$. If $c>0$, choose a unit vector $\xi$ that is $(Q,\delta )$-invariant, where $\delta =\eta \mathrm{/}c$ for a requested approximation tolerance $\eta >0$, and use $\sqrt{c}\xi$ as a vector for $\pi$. For every $x\in Q$, Cauchy–Schwarz gives $\mid c-⟨\pi (x)(\sqrt{c}\xi ),\sqrt{c}\xi ⟩\mid =c\mid ⟨\xi -\pi (x)\xi ,\xi ⟩\mid \le c\mathrm{\parallel }\pi (x)\xi -\xi \mathrm{\parallel }<\eta \mathrm{.}$ This diagonal coefficient is continuous and of positive type by Matrix coefficient of a unitary representation and Diagonal unitary coefficients have positive type, so it is an allowed one-term approximant; the zero coefficient ($c=0$) is represented by the zero vector. No choice is used, since a witness is selected separately for each given compact set and tolerance.

When $G$ is locally compact Hausdorff (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and AC is assumed (The Axiom of Choice), Weak containment of the trivial representation and almost invariant vectors proves the converse as well: $1_G\prec \pi$ is equivalent to almost invariant vectors. Under AC, the converse fails for general topological groups; that published lemma gives a counterexample in its final remarks, using Tychonoff and recursive compact-stage neighbourhood selections. The LCH equivalence and that counterexample are the two assertions invoked here under AC; the definition and the forward implication above are choice-free.