Theorem: Bounded modular-band witnesses

Assume (eq:scalarcentralizer). Fix $T\in \mathcal{B}(H)$, $s\in \mathbb{R}$, and the mean above. Let ${\delta }_n>0$ tend to zero, and let unit vectors $h_n$ have $D$-support in $[s-{\delta }_n,s+{\delta }_n]$. If

then a subsequence has $v_j\in M$ and constants $C_{\ast }<\infty ,\eta >0$ such that

This is the bounded-recovery statement of OpenAI29002 (Theorem 4.1); the proof below follows its supplied construction.

Proof. Choose $f_0\in {C_c}^{\infty }(\mathbb{R})$ with $0\le f_0\le 1$, $f_0=1$ on $[-1,1]$, and support in $[-2,2]$. Write $f_0(u)=\int g_0(t)e^{itu}dt$. For $\rho >0$, set

Then ${\Vert g_{\rho }\Vert }_1=C_1$, and

Pass to a subsequence on which the averages in (eq:positiveaverage) are at least $4\gamma$ for some $\gamma >0$. Density of $M\xi$, filtering by $f_{{\delta }_n}(D)$, and normalization give $y_n\in M$ satisfying

and, by sufficiently close approximation,

Here the averaged-seminorm difference is bounded by $\Vert T\Vert \Vert y_n\xi -h_n\Vert$. Discard finitely many terms so that $d_n\le 1$. The modular identity yields

There is no uniform bound on $\Vert y_n\Vert$.

Fix $n$, write $y=y_n$, and define

The functional $\psi$ is a normal state. By Lemma lem:ergodic,

Although $B_k$ varies with $k$,

Thus, with this $n$ fixed, all sufficiently large $k$ satisfy

and hence

The positive ultrafilter average ensures that the set of $k$ with

belongs to $\mathcal{U}$. It meets the cofinite set on which (eq:continuousmoments) holds.

For one such $k$, approximate the three continuous scalar integrands

on the compact interval or square by one common finite partition. Strong-star continuity on bounded sets justifies these approximations. This gives $z_i={\sigma }_{t_i}(y)$ and weights $p_i\ge 0$, ${\sum }_i{p}_i=1$, with

such that

The times and weights may depend on $n$; $C,\gamma$ do not.

Let ${ϵ}_i$ be independent uniform circle phases and put $z={\sum }_i\sqrt{p_i}{ϵ}_i{z}_i$. Its vector remains in the band of radius $d_n$, and

Expanding the fourth moment, the two surviving phase pairings give

The subtracted terms are positive, so $\mathbb{E}\phi (|z{|}^4)\le C$. This is the circle-phase fourth-moment device underlying the Haagerup–Musat construction HaagerupMusat.

For polar decomposition $z=u|z|$, truncate at a fixed $K>0$:

Then

Set $v={\mathcal{F}}_{d_n}(z^{[K]})$. Every realization satisfies

Since $f_{d_n}(D)$ fixes $z\xi$,

Therefore

Choose once and for all $K\ge 2\Vert T\Vert \sqrt{C/\gamma }$. The resulting average is at least $\sqrt{\gamma }/2$.

The phase torus is compact. The truncation is norm continuous, because

Filtering is bounded linear, so ${\Vert Tv\xi \Vert }^2$ attains a maximum at least its expectation. Choose one maximizing realization for each $n$. It gives $C_{\ast }=C_{1}K$, $\eta =\sqrt{\gamma }/2$, and band radius $2d_n=4{\delta }_n$.