# 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](OA-FLOW-AF.md#af-0). 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](OA-FLOW-AF.md#af-0). 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](OA-FLOW-L24.md#oa-flow.grp.translations), 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](https://doi.org/10.1007/978-3-662-10451-4)). 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.