Original programme exposition in Codex (OpenAI), September 2026; proof restoration and supporting proofs by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Spot-checked by GPT-6 Astra in a separate session.

Narrow spectral bands give nearly character orbits

A vector whose frequencies lie in a sufficiently narrow band around $p$ behaves, for a fixed compact set of group elements, almost like a $p$-eigenvector in norm. The width can be chosen uniformly for every vector in that spectral band. A norm-controlled Fourier cutoff and singleton synthesis make this precise even when individual orbits are only weak-star continuous. This is Takesaki II, Lemma XI.1.11 at arbitrary LCA and specified dual-Banach generality.

Programme proof written in Codex (OpenAI), September 2026; restoration and proof expansion, 5 October 2026. New expression is dedicated under CC0 to the extent of rights held. Human review is not asserted.

The positive Fourier convention and both eligible Banach settings are AF0. The exact earlier scalar inputs are LF0–1 for the complete dual-Haar correlation identity and norm-controlled local plateaus, LF4 for plateaus around compact sets, and SS1 for singleton synthesis. BS1–3 supply the full integrated action, filter bounds and cutoff laws in both Banach settings.

A flat-top cutoff with norm below two

Use the arbitrary locally compact Hausdorff abelian group $G$, its dual $H=\widehat G$, and the Fourier algebra $A(H)=\mathcal F L^1(G)$ of AF0. For any $p\in H$ and open neighborhood $O\ni p$, there is $h\in A_c(H)$ such that

$h=1\text{ near }p,\qquad \operatorname{supp}h\subset O,\qquad \|h\|_A<2. \tag{N1}$

Here is an explicit construction. Translate to $p=0$. Choose a relatively compact open identity neighborhood $N$ so small that $N-N\subset O$. Let $V\subset N$ be compact with positive Haar measure, and by outer regularity choose an open $W$ with $V\subset W$, compact closure inside $N$, and $m(W)<4m(V)$. Such $V,W$ can be chosen by first putting a positive-measure compact set inside a smaller open neighborhood and then shrinking its outer neighborhood. Define

$h(q)=\frac{1}{m(V)}\int_H 1_W(r)1_V(r-q)\,dr =\frac{m(W\cap(V+q))}{m(V)}. \tag{N2}$

The $L^2$ correlation is continuous and supported in the compact set $\overline W-V\subset N-N$. Since $V$ is compact and $W$ open, $V+q\subset W$ for every $q$ in some identity neighborhood, so $h=1$ there. With the chosen dual Haar normalization, put $u=\mathcal F^{-1}(1_W)$ and $v=\mathcal F^{-1}(1_V)$. The correlation in (N2) has inverse Fourier kernel $u\overline v/m(V)$. Plancherel and Cauchy–Schwarz give the required $L^1$ bound. Thus it lies in $L^1(G)$, and

$\|h\|_A \le\frac{\|1_W\|_2\|1_V\|_2}{m(V)} =\sqrt{\frac{m(W)}{m(V)}}<2. \tag{N3}$

Frequency translation moves this cutoff to $p$ without changing its $A$-norm. The construction uses compact Haar sets and nets of neighborhoods, not second countability or a smooth structure on $H$.

A vanishing filter has a small local product

The complete singleton-synthesis proof SS1, transported by the reflection in AF0, and (N1) give the following local multiplication lemma. If $g\in A(H)$ and $g(p)=0$, then for every $\eta>0$ there is an $h\in A_c(H)$, equal to $1$ near $p$, with

$\|h\|_A<2,\qquad \|hg\|_A<\eta. \tag{N4}$

To prove it, use $j(\{p\})=I(\{p\})$ to choose $g_0\in A_c(H)$ whose support misses $p$ and with $\|g-g_0\|_A<\eta/2$. Choose a neighborhood $O\ni p$ disjoint from $\operatorname{supp}g_0$, and construct $h$ from (N1) supported in $O$. Then $hg_0=0$, so $\|hg\|_A\le\|h\|_A\|g-g_0\|_A<\eta$. This proof identifies exactly where singleton synthesis is used; mere pointwise continuity of $g$ would not control the Fourier-algebra norm of $hg$.

The filtered orbit identity

Let $X=X_*^*$ and $\alpha$ satisfy the uniformly bounded normal action hypotheses of AF0 setting (D), with $C_\alpha=\sup_t\|\alpha_t\|$. If $X=\{0\}$, the conclusion is immediate; otherwise $C_\alpha\ge1$, so the divisions below are defined. Fix $p\in H$, a compact set $K\subset G$, and $\varepsilon>0$. Choose a compact neighborhood $V$ of $p$ and $f\in A_c(H)$ equal to $1$ on a neighborhood of $V$. For $t\in G$, set

$f_t(q)=(t,q)f(q),\qquad g_t(q)=f_t(q)-(t,p)f(q). \tag{N5}$

The actual integrated covariance BS1, in AF0’s positive convention, gives $\alpha_t\alpha_f=\alpha_{f_t}$, and $g_t(p)=0$. If $U$ is a relatively compact open neighborhood of $p$ with $\overline U\subset\operatorname{int}V$, and $x\in X_\alpha(U)$, then $\operatorname{Sp}_\alpha(x)\subset\overline U$ is compact. The proved compact-spectral cutoff BS3 gives $\alpha_f x=x$. Hence

$\alpha_t x-(t,p)x=\alpha_{g_t}x. \tag{N6}$

The map $t\mapsto g_t$ is norm continuous in $A(H)$: $t\mapsto f_t$ is norm continuous by translation continuity of the inverse $L^1$ kernel in L24 Lemma 3.1, and $t\mapsto(t,p)$ is continuous. This norm continuity is stronger than orbit weak-star continuity and is the input for the compact-uniform step.

One band for a whole compact time set

For each $t\in K$, apply (N4) to $g_t$ with $\eta=\varepsilon/(2C_\alpha)$. Obtain $h_t\in A_c(H)$, equal to $1$ on a neighborhood $U_t$ of $p$, with $\|h_tg_t\|_A<\eta$. By norm continuity of $s\mapsto g_s$, choose a neighborhood $W_t$ of $t$ such that

$\|h_tg_s\|_A<\varepsilon/C_\alpha \qquad(s\in W_t). \tag{N7}$

Finitely many $W_{t_1},\ldots,W_{t_m}$ cover $K$. Choose a relatively compact open neighborhood $U$ of $p$ whose closure lies inside $\operatorname{int}V$ and inside every neighborhood on which the finitely many $h_{t_j}$ equal $1$. For $x\in X_\alpha(U)$, the compact-spectral cutoff lemma gives $\alpha_{h_{t_j}}x=x$. If $s\in K\cap W_{t_j}$, use (N6), the module law and filter bound (AF2):

$\begin{aligned} \|\alpha_sx-(s,p)x\| &=\|\alpha_{g_s}\alpha_{h_{t_j}}x\|\\ &=\|\alpha_{h_{t_j}g_s}x\|\\ &\le C_\alpha\|h_{t_j}g_s\|_A\|x\| \le\varepsilon\|x\|. \end{aligned}\tag{N8}$

Thus one neighborhood $U$ works for every $s\in K$ and every vector $x\in X_\alpha(U)$. The last inequality is strict for $x\ne0$; the non-strict statement in (N8) also includes $x=0$. The estimate is meaningful even when $X_\alpha(U)=\{0\}$; lesson 88 will relate nonzero narrow-band vectors to membership of $p$ in the action spectrum.

The same estimate also holds in AF0 setting (B), on an arbitrary Banach space with norm-continuous orbits. Every step above uses the bounded integrated homomorphism, its norm bound, translation continuity and the vector-spectrum cutoff law. BS1–3 prove those statements in (B); the calculation has no additional weak-star limit. In fact it works for every vector with spectrum in $\overline U$, so it applies both to $X_\alpha(U)$ as defined in AF0 and to BS4’s filtered-span space $X_\alpha^0(U)\subset X_\alpha(\overline U)$.

Problem. On $X=\mathbb C^2$, let $\alpha_t=\operatorname{diag}(e^{it\lambda_1},e^{it\lambda_2})$ with $\lambda_1\ne\lambda_2$ and choose $p=\lambda_1$. How can $U$ be chosen so that (N8) holds with zero error?

Solution. Take $U$ containing $\lambda_1$ but not $\lambda_2$. Then $X_\alpha(U)$ is the first coordinate line, and $\alpha_t x=e^{it\lambda_1}x=(t,p)x$ there for every $t\in\mathbb R$. The general lemma replaces this exact finite-frequency separation by a uniform estimate for possibly continuous spectra. $\square$

The mathematical source is M. Takesaki, Theory of Operator Algebras II, Lemma XI.1.11, printed pages 321–322 (edition record). The norm-controlled correlation and complete singleton proof have their exact earlier locators above. The argument keeps the compact-uniform estimate at arbitrary LCA generality and preserves its finite-frequency example; it does not use the separate full spectral-transfer theorem.