The following is the definition of weak containment: Let $\pi$ and $\rho$ be two unitary representations of the group $G$. Then, we say $\pi$ is weakly contained in $\rho$ denoted as $\pi \prec \rho$ if for every positive definite function $\phi$ associated to $\pi$, compact subset $K$ of $G$, and $\varepsilon >0$, there exist $\eta_{1},..., \eta_{n}$ in the Hilbert space $\mathcal{H}^{\rho}$ such that $$ \left|\phi(x)- \sum_{i=1}^{n} \langle \rho(x)\eta_{i}, \eta_{i}\rangle\right| < \varepsilon,~~\text{for every}~~x\in K.$$
Two representation $\pi$ and $\rho$ are weakly equivalent if $\pi \prec \rho$ and $\rho \prec \pi$.
Suppose $\pi$ and $\rho$ are two unitary representations of locally compact groups $G$ and $H,$ respectively. Then the outer tensor product of representations $(\pi \times \rho) _{|_{G \times \{e\}}}$ where $e$ is the identity of $H$ is weakly equivalent to $\pi$. How to show this? I could show that $\pi \prec (\pi \times \rho) _{|_{G \times \{e\}}}$ but couldn't show the other way.
This argument is directly given in the proof of Corollary $5.4(i)$ of
Bekka, Mohammed E. B., Amenable unitary representations of locally compact groups, Invent. Math. 100, No. 2, 383-401 (1990). ZBL0702.22010.
