Weak containment of unitary representations

Let $G$ be a topological group and let $\pi$ and $\rho$ be strongly continuous unitary representations on Hilbert spaces $H_{\pi }$ and $H_{\rho }$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners). Write $\pi \prec \rho$ and say that $\pi$ is weakly contained in $\rho$ if every continuous function of positive type associated to $\pi$ can be approximated, uniformly on every compact subset of $G$, by finite sums of functions of positive type associated to $\rho$: for every $\xi \in H_{\pi }$, every compact $Q\subseteq G$ and every $ϵ>0$ there exist finitely many ${\eta }_1,\dots ,{\eta }_n\in H_{\rho }$ with ${\sup }_{g\in Q}\mid ⟨\pi (g)\xi ,\xi ⟩-{\sum }_{i=1}^n⟨\rho (g){\eta }_i,{\eta }_i⟩\mid <ϵ\mathrm{.}$ Write $\pi \sim \rho$ when both $\pi \prec \rho$ and $\rho \prec \pi$.

• Coefficient form. The vector $\xi$ of the definition is arbitrary, so the functions tested are exactly the diagonal matrix coefficients $c_{\xi ,\xi }(g)=⟨\pi (g)\xi ,\xi ⟩$ of $\pi$ (Matrix coefficient of a unitary representation); each is continuous and of positive type (Diagonal unitary coefficients have positive type), and so is each of the approximating functions (Continuous positive-type functions and normalization). Containment of a representation in another, when defined by subrepresentations, plainly implies weak containment; no multiplicity or dimension hypotheses are imposed, and the zero representation is allowed on either side.
• Reflexivity and invariance of the relation. Taking $n=1$ and ${\eta }_1=\xi$ shows $\pi \prec \pi$. If $U:H_{\pi }\to H_{{\pi }^{\mathrm{\prime }}}$ is a unitary intertwiner and $V:H_{\rho }\to H_{{\rho }^{\mathrm{\prime }}}$ is one, then $V$ carries every diagonal coefficient of $\rho$ to a diagonal coefficient of ${\rho }^{\mathrm{\prime }}$, so $\pi \prec \rho$ implies ${\pi }^{\mathrm{\prime }}\prec {\rho }^{\mathrm{\prime }}$: the relation is well defined on unitary equivalence classes.
• Transitivity. If $\pi \prec \rho$ and $\rho \prec \sigma$, then $\pi \prec \sigma$. Indeed, fix $\xi \in H_{\pi }$, compact $Q$ and $ϵ>0$. Since $\pi \prec \rho$, choose ${\eta }_1,\dots ,{\eta }_n\in H_{\rho }$ with ${\sup }_Q\mathrm{\mid }{c}_{\xi ,\xi }-{\sum }_j{c}_{{\eta }_j,{\eta }_j}\mathrm{\mid }<ϵ\mathrm{/}2$. Applying $\rho \prec \sigma$ to each of the finitely many vectors ${\eta }_j$ on the same compact $Q$ with tolerance $ϵ\mathrm{/}(2n)$ produces, for each $j$, finitely many vectors ${\zeta }_{j,k}\in H_{\sigma }$ with ${\sup }_Q\mathrm{\mid }{c}_{{\eta }_j,{\eta }_j}-{\sum }_k{c}_{{\zeta }_{j,k},{\zeta }_{j,k}}\mathrm{\mid }<ϵ\mathrm{/}(2n)$; summing the $n$ inequalities gives a finite family of vectors of $H_{\sigma }$ whose coefficient sum differs from $c_{\xi ,\xi }$ on $Q$ by less than $ϵ$.