Lemma: Fourier localization and predual control

If $f(u)=\int g(t)e^{itu}dt$, with $f\in {C_c}^{\infty }(\mathbb{R})$, then

The integral is weak, defined against normal functionals.

With predual conventions

one has

Consequently uniformly bounded $w_n$ whose vectors are supported in bands shrinking to $s$ satisfy ${\Vert \phi w_n-e^s{w}_n\phi \Vert }_{M_{\ast }}\to 0$.

Proof. The inverse Fourier transform of $f$ is integrable. The weak integral has the asserted operator-norm bound. Applying it to $\xi$ and using the spectral theorem proves the vector identity.

For the second claim, put $c=e^{s/2}$ and $\zeta =\xi w-cw\xi$. Modular conjugation gives $J\zeta =w^{\ast }\xi -c\xi w^{\ast }$. For $\Vert y\Vert \le 1$, write

Replacing $w^{\ast }\xi$ by $c\xi w^{\ast }$ costs at most $\Vert \zeta \Vert$. Commutation of left and right actions gives $⟨y\xi ,\xi w^{\ast }⟩=⟨y(\xi w),\xi ⟩$. Replacing $\xi w$ by $cw\xi$ costs at most $c\Vert \zeta \Vert$. The resulting expression is $c^2\phi (yw)$, proving (eq:predualband).

On a band $[s-{\delta }_n,s+{\delta }_n]$, functional calculus bounds the last norm by

which tends to zero.