\eta_2), \eta_3
  \eta_4\rangle\\ & \leq & \|\eta_1 \eta_2\|_{\ell^2\Gamma} \|\eta_3
  \eta_4\|_{\ell^2\Gamma} \leq 1.\end{aligned}$$ In particular, $\langle \eta_1^2,\eta_2^2\rangle = \|\eta_1 \eta_2\|_{\ell^2\Gamma}^2 \geq
(1-\delta)^2$, which clearly implies that $\|\eta_1^2 - \eta_2^2\|_{\ell^2\Gamma} =
o(1)$. The elementary inequality $\|a-b\|_{\ell^4\Gamma} \leq \sqrt{\|a^2 - b^2\|_{\ell^2\Gamma}}$ valid for all $a,b:\Gamma \to \R_+$ in turn implies $\|\eta_1 -
\eta_2\|_{\ell^4\Gamma} = o(1)$. Similarly, $\|\eta_3 - \eta_4\|_{\ell^4\Gamma}  =o(1)$. By the first line of the series of inequalities above, we get $\langle
\lambda_{s^{-1}} \eta_1^2, \eta_3^2 \rangle =1+o(1)$, which indeed implies $\|\lambda_s \eta_3^2 - \eta_1^2\|_{\ell^2\Gamma} = o(1)$.

Preliminaries {#Sect-Preliminaries}
=============

We collect the necessary results and fix the notation. In particular, we introduce $L^p$-Fourier multipliers for any locally compact group.

General notation
----------------

For a Hilbert space ${\mathcal{H}}$, ${\mathcal{S}}_p({\mathcal{H}})$ denotes the Schatten class, i.e. the non-commutative $L^p$-space associated with $B({\mathcal{H}})$. We use brackets $[\: ]$ to denote the closure of an operator and $\cdot$ for the strong product of (unbounded) operators. Recall that $a \cdot b$, when it exists, is the closure of the operator defined on $\{x \in {\mathrm{dom}}(b), b(x) \in {\mathrm{dom}}(a)\}$ by $x \mapsto a(b(x))$. For $\varphi$ a normal, semi-finite, faithful weight on a von Neumann algebra ${\mathcal{M}}$, we set ${\mathfrak{n}_\varphi}= \{ x \in {\mathcal{M}}\mid \varphi(x^\ast x ) < \infty \}$ 