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Compact Bott projections and spreading

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The class to be killed is the rank-zero Bott class of the torus spanned by the commuting pair \((z,x)\) in the Heisenberg group \(\mathsf H\). This lesson fixes a smooth compactly supported representative of it, rescales it into tiny rectangles of the torus, one for each height of the tower, and spreads each rescaled symbol over many translates by powers of \(y\). Conjugation by \(y\) shears the torus, and an irrational slope makes the sheared rectangles pairwise disjoint; the spread symbols then multiply like matrices while their support projections have tiny trace. These are the symbols that the weighted voltage incidences of the next lessons will approximate [OpenAI-BCI, Section 5.1, Lemma 5.1].

We use the normal form of The Heisenberg group and its laws: \(\mathsf H\) is generated by \(x,y\) with \(z=[x,y]\) central, and \(yxy^{-1}=xz^{-1}\). We use the identity trace of equivariant operators, Lemma 2.1 of Separated bands and equivariant traces, and from the course K-theory of C*-algebras that homotopic projections are unitarily equivalent, hence define the same class in \(K_0\) (Idempotents, projections and their equivalences, Corollary 3.3).

1. Torus coordinates

On \(\ell^2(\mathsf H)\) let \(\lambda(g)\delta_h=\delta_{gh}\) be the left regular representation. The subgroup \(\Lambda=\langle z,x\rangle\) is free abelian of rank two (Corollary 1.2 of the lesson on the Heisenberg group) and normal, since \(yzy^{-1}=z\) and \(yxy^{-1}=xz^{-1}\). Write \(\mathbb T^2=(\mathbb R/\mathbb Z)^2\) with coordinates \((\vartheta,\xi)\) and \(e(t)=e^{2\pi it}\).

Lemma 1.1 (torus coordinates). There is a unique way to let every bounded Borel function \(F\) on \(\mathbb T^2\) act as an operator \(F(\lambda(z),\lambda(x))\) on \(\ell^2(\mathsf H)\) such that:

  1. \(F\mapsto F(\lambda(z),\lambda(x))\) is a \(*\)-homomorphism of norm one, sending \(e(c\vartheta+n\xi)\) to \(\lambda(z^cx^n)\);
  2. uniformly bounded pointwise convergence of functions gives strong convergence of operators;
  3. the operators commute with the right translations \(\delta_h\mapsto\delta_{hk}\), and \(\langle F(\lambda(z),\lambda(x))\delta_1,\delta_1\rangle=\int_{\mathbb T^2}F\);
  4. \(\lambda(y)^jF(\lambda(z),\lambda(x))\lambda(y)^{-j}\) equals \(G(\lambda(z),\lambda(x))\) with \(G(\vartheta,\xi)=F(\vartheta,\xi-j\vartheta)\).

For continuous \(F\) the operator lies in the reduced group \(C^*\)-algebra \(C^*_r(\mathsf H)\), in the copy of \(C^*_r(\Lambda)\cong C(\mathbb T^2)\).

Proof. Decompose \(\ell^2(\mathsf H)=\bigoplus_{\Lambda h}\ell^2(\Lambda h)\) over the right cosets. On \(\ell^2(\Lambda h)\), the unitary \(\delta_{\gamma h}\mapsto\delta_\gamma\) onto \(\ell^2(\Lambda)\) intertwines \(\lambda(\gamma)\) for \(\gamma\in\Lambda\) with left translation, and the Fourier transform \(\ell^2(\Lambda)\to L^2(\mathbb T^2)\), \(\delta_{z^cx^n}\mapsto e(c\vartheta+n\xi)\), turns left translation by \(z^cx^n\) into multiplication by \(e(c\vartheta+n\xi)\). Let \(F\) act by multiplication on every summand. Properties (1) and (2) hold for multiplication operators (the second by dominated convergence), and uniqueness follows from (1), (2) and the density of trigonometric polynomials in the bounded pointwise sense. Right translation by \(k\) maps \(\ell^2(\Lambda h)\) to \(\ell^2(\Lambda hk)\) and is the identity in these coordinates, so it commutes with the multiplication operators. The vector \(\delta_1\) is the constant function \(1\) on the coset \(\Lambda\), which gives the trace formula in (3).

For (4), \(\lambda(y)\) maps \(\ell^2(\Lambda h)\) onto \(\ell^2(\Lambda yh)\) by \(\delta_{\gamma h}\mapsto\delta_{(y\gamma y^{-1})yh}\); in the coordinates of the two cosets it is the composition with the automorphism \(\sigma(\gamma)=y\gamma y^{-1}\), \(\sigma(z^cx^n)=z^{c-n}x^n\). Its Fourier transform is composition with the dual map \((\vartheta,\xi)\mapsto(\vartheta,\xi-\vartheta)\) of \(\mathbb T^2\). Hence conjugation by \(\lambda(y)\) turns multiplication by \(F\) into multiplication by \(F(\vartheta,\xi-\vartheta)\), and induction gives (4). For continuous \(F\), uniform approximation by trigonometric polynomials and (1) give norm limits of elements of the group algebra. \(\square\)

We write \(F\) for \(F(\lambda(z),\lambda(x))\); matrix-valued functions act on \(\ell^2(\mathsf H)\otimes\mathbb C^\ell\) entrywise. By (3), these operators commute with the right translations, so the identity trace \(\tau(A)=\sum_k\langle A(\delta_1\otimes e_k),\delta_1\otimes e_k\rangle\) of Separated bands and equivariant traces applies, and \(\tau(F)=\int_{\mathbb T^2}\operatorname{Tr}F\).

2. A compact Bott projection

Let \(h(t)=e^{-1/t}\) for \(t>0\) and \(h(t)=0\) for \(t\le0\), a smooth function, and \[ \begin{gathered} \zeta(r)=\frac{h(r-\frac14)}{h(r-\frac14)+h(1-r)},\\ \theta(r)=2r+(\pi-2r)\zeta(r)\qquad(r\ge0). \end{gathered} \] Then \(\zeta\) is smooth, \(\zeta=0\) on \([0,\frac14]\) and \(\zeta=1\) on \([1,\infty)\), so \(\theta(r)=2r\) near \(0\) and \(\theta(r)=\pi\) for \(r\ge1\). For \((t,u)\in\mathbb R^2\) with \(r=(t^2+u^2)^{1/2}>0\) put \[ \begin{gathered} p(t,u)=\frac12\,\times\\ \begin{pmatrix}1+\cos\theta&\bar w\\w&1-\cos\theta\end{pmatrix},\\ w=\frac{\sin\theta}r(t+iu), \end{gathered}\tag{2.1} \] with \(\theta=\theta(r)\), and \(p(0,0)=\operatorname{diag}(1,0)\); let \(e_\infty=\operatorname{diag}(0,1)\).

Lemma 2.1 (the compact Bott projection). \(p\) is a smooth function \(\mathbb R^2\to M_2(\mathbb C)\) with \(p=p^*=p^2\) and \(\operatorname{Tr}p=1\) everywhere, and \(p=e_\infty\) for \(r\ge1\). The line bundle over the sphere \(\mathbb R^2\cup\{\infty\}\) given by the range of \(p\) has clutching degree \(-1\), and \[ \begin{gathered} \frac1{2\pi i}\int_{\mathbb R^2}\operatorname{Tr}(p\,dp\wedge dp)\\ =1 . \end{gathered}\tag{2.2} \]

Proof. Near \(0\), \(\theta=2r\), and \(\cos2r\) and \(\sin(2r)/r\) are power series in \(r^2\), hence smooth functions of \((t,u)\); away from \(0\) all ingredients are smooth. In polar coordinates \(t+iu=re^{i\phi}\), \(p=\frac12(1+n\cdot\sigma)\) with \(n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\) and the Pauli matrices \(\sigma=(\sigma_1,\sigma_2,\sigma_3)\); since \(|n|=1\), \((n\cdot\sigma)^2=1\), which gives \(p^2=p\), and \(p^*=p\), \(\operatorname{Tr}p=1\) are clear. For \(r\ge1\), \(n=(0,0,-1)\) and \(p=e_\infty\).

The unit vectors \[ \begin{pmatrix}\cos(\theta/2)\\e^{i\phi}\sin(\theta/2)\end{pmatrix},\qquad\begin{pmatrix}e^{-i\phi}\cos(\theta/2)\\\sin(\theta/2)\end{pmatrix} \] span the range of \(p\); the first is smooth near the origin (where \(\theta/2\) is small and \(\sin(\theta/2)\) carries the factor \(r\)), the second near the exterior region (where \(\theta\) is near \(\pi\)). On the overlap the second is \(e^{-i\phi}\) times the first, so the clutching function \(e^{-i\phi}\) has winding number \(-1\). For (2.2), \(dp=\frac12dn\cdot\sigma\), and the identity \((a\cdot\sigma)(b\cdot\sigma)=(a\cdot b)+i(a\times b)\cdot\sigma\) with \(\operatorname{Tr}\sigma_l=0\) gives \(\operatorname{Tr}(p\,dp\wedge dp)=\frac i2\,\omega\), where \(\omega=\frac12\sum\varepsilon_{ljk}n_l\,dn_j\wedge dn_k\) is the pull-back by \(n\) of the area form of the unit sphere. Since \(n\) has spherical coordinates \((\theta(r),\phi)\), \(\omega=\theta'(r)\sin\theta(r)\,dr\wedge d\phi\), and \[ \begin{aligned} &\frac1{2\pi i}\int\operatorname{Tr}(p\,dp\wedge dp)\\ &\quad=\frac1{4\pi}\int_0^{2\pi}\!\!\int_0^1\theta'(r)\sin\theta(r)\,dr\,d\phi\\ &\quad=\frac12\Bigl[-\cos\theta\Bigr]_0^1=1 . \end{aligned} \] The winding computation does not depend on the sign convention for the Chern form. \(\square\)

So \(p-e_\infty\) is a smooth compactly supported matrix function, and \([p]-[e_\infty]\) is a rank-zero class of degree one in absolute value.

3. Scaled projections and their paths

Fix \[ \begin{gathered} \alpha=\frac{\sqrt5-1}2,\quad N_i=2^{200H_i},\\ Z_i=2^{500H_i},\quad J_i=2^{50H_i}. \end{gathered} \] In a fixed chart of \(\mathbb T^2\) around \((\alpha,0)\), put \[ \begin{gathered} \mathsf p_i(\vartheta,\xi)\\ =p\bigl(Z_i(\vartheta-\alpha),N_i\xi\bigr),\\ \beta_i=\mathsf p_i-e_\infty , \end{gathered}\tag{3.1} \] extended by \(e_\infty\) outside the chart; the initial height is chosen so that all supports lie well inside the chart. Each \(\mathsf p_i\) is a projection in \(M_2(C(\mathbb T^2))\).

The paths. For \(|i-k|=1\), the rescaling path \[ \begin{gathered} p\bigl(Z_i^{1-v}Z_k^v(\vartheta-\alpha),\,N_i^{1-v}N_k^v\xi\bigr),\\ 0\le v\le1, \end{gathered} \] joins \(\mathsf p_i\) to \(\mathsf p_k\). Since \(H_k-H_i=\pm b\), the ratios \(Z_k/Z_i\) and \(N_k/N_i\) are the fixed numbers \(2^{\pm500b}\) and \(2^{\pm200b}\); in the rescaled coordinates \((Z_i(\vartheta-\alpha),N_i\xi)\) the path is one fixed compact smooth family, with all derivatives bounded independently of the height. The rotation path \[ R(v)=\begin{pmatrix}\cos\frac{\pi v}2&-\sin\frac{\pi v}2\\\sin\frac{\pi v}2&\cos\frac{\pi v}2\end{pmatrix}\otimes I_2 \] conjugates \(\mathsf p_i\oplus e_\infty\) to \(e_\infty\oplus\mathsf p_i\). For sampling, one subtracts the constant projection \(e_\infty\oplus e_\infty\) from the rotation family and \(e_\infty\) from the rescaling family.

Lemma 3.1. All the \(\beta_i\) define the same class in the \(K_0\)-group of \(C(\mathbb T^2)\). The entries of the compact differences of both paths, their rescalings, and smooth cutoffs equal to one on their supports belong to one fixed compact family of smooth compactly supported functions in the rescaled coordinates, contained in rectangles larger than the unit rectangle by a factor depending only on \(b\). In particular the paths have one common modulus of norm continuity, independent of the height.

Proof. The rescaling path is a norm-continuous path of projections in \(M_2(C(\mathbb T^2))\) from \(\mathsf p_i\) to \(\mathsf p_{i\pm1}\); homotopic projections have the same class in \(K_0\) (Corollary 3.3 of Idempotents, projections and their equivalences), and \(e_\infty\) is fixed. The remaining assertions hold because the path is a fixed smooth family in the rescaled coordinates, and the supports of all profiles at height \(i\) lie in rectangles \(|\vartheta-\alpha|\le R/Z_i\), \(|\xi|\le R/N_i\) for a constant \(R\) depending only on \(b\). \(\square\)

More generally, a profile at height \(i\) is a matrix function \[ \begin{gathered} f_i(\vartheta,\xi)\\ =F\bigl(Z_i(\vartheta-\alpha),N_i\xi\bigr) \end{gathered}\tag{3.2} \] with \(F\) in a fixed compact family in \(C_c^\infty(\mathbb R^2;M_\ell(\mathbb C))\), supported in one fixed rectangle; \(\ell\) is fixed, and zero profiles and any selection of heights are allowed. Passing between adjacent heights only enlarges the family by fixed rescalings.

4. Golden separation

Lemma 4.1 (golden separation). For every integer \(n\ne0\), \[ \|n\alpha\|_{\mathbb R/\mathbb Z}\ge\frac1{3|n|} . \]

Proof. Let \(p\) be the nearest integer to \(n\alpha\) and \(\alpha'=(-\sqrt5-1)/2\). Since \(\alpha\alpha'=-1\) and \(\alpha+\alpha'=-1\), \[ (n\alpha-p)(n\alpha'-p)=p^2+np-n^2 , \] a nonzero integer: \(p^2+np-n^2=0\) with \(n\ne0\) would make \(p/n\) a rational root of \(X^2+X-1\). Moreover \(|n\alpha'-p|\le|n||\alpha'|+|n|\alpha+\frac12\), which is \(\sqrt5|n|+\frac12\le3|n|\). \(\square\)

5. Supported spreading

Choose a nonzero real \(a\in C_c^\infty((-1,1))\) and put \[ \begin{gathered} a_{ij}=\frac{a(j/J_i)}{\bigl(\sum_{k\in\mathbb Z}a(k/J_i)^2\bigr)^{1/2}}\\ \text{and}\quad T_i=\sum_{j\in\mathbb Z}a_{ij}\lambda(y)^j . \end{gathered}\tag{5.1} \] The sums are finite, \(a_{ij}=0\) for \(|j|\ge J_i\), and \(\sum_j a_{ij}^2=1\); by Riemann sums the denominator lies between constant multiples of \(J_i^{1/2}\). Let \(E_i\) be the operator of the indicator function of the rectangle \[ |\vartheta-\alpha|\le R/Z_i,\qquad|\xi|\le R/N_i , \] with \(R\) so large that it contains all profiles and cutoffs incident to height \(i\). These Borel projections are auxiliary support operators; they need not belong to \(C^*_r(\mathsf H)\).

Lemma 5.1 (supported spreading; [OpenAI-BCI, Lemma 5.1]). For large initial height, the projections \(\lambda(y)^jE_i\lambda(y)^{-j}\) and \(\lambda(y)^kE_i\lambda(y)^{-k}\) are orthogonal whenever \(0<|j-k|\le2J_i\); in particular those with \(|j|\le J_i\) are pairwise orthogonal. Consequently \[ E_iT_i^*T_iE_i=E_i , \] \(P_i=T_iE_iT_i^*\) is a projection, and \[ \begin{gathered} \tau(P_i)=\tau(E_i)\\ \le4R^2\,2^{-700H_i} . \end{gathered}\tag{5.2} \] If \(f\) is supported in the rectangles of heights \(i\) and \(k\), and \(g\) in those of heights \(k\) and \(j\), then \[ \begin{gathered} (T_ifT_k^*)(T_kgT_j^*)\\ =T_i(fg)T_j^*,\\ (T_ifT_k^*)^*=T_kf^*T_i^*,\\ \|T_ifT_k^*\|\le\|f\| . \end{gathered}\tag{5.3} \]

Proof. By Lemma 1.1(4), \(\lambda(y)^jE_i\lambda(y)^{-j}\) is the indicator of \(\{|\vartheta-\alpha|\le R/Z_i,\ \|\xi-j\vartheta\|\le R/N_i\}\). If two of these sets, for \(0<|j-k|\le2J_i\), had a common point, then \(\|(j-k)\vartheta\|\le2R/N_i\). But by Lemma 4.1 \[ \|(j-k)\vartheta\|\ge\frac1{6J_i}-\frac{2RJ_i}{Z_i}, \] and since \(N_i=J_i^4\) and \(Z_i=J_i^{10}\), the right side exceeds \(2R/N_i\) for large heights. So the projections are orthogonal.

Expanding, \(E_iT_i^*T_iE_i=\sum_{j,k}a_{ij}a_{ik}E_i\lambda(y)^{k-j}E_i\). For \(k\ne j\), \(E_i\lambda(y)^{k-j}E_i\) equals \(E_i\bigl(\lambda(y)^{k-j}E_i\lambda(y)^{j-k}\bigr)\lambda(y)^{k-j}\), which is \(0\), because \(|k-j|<2J_i\). The diagonal terms give \(\sum_ja_{ij}^2E_i=E_i\). Hence \(P_i^2=T_iE_i(T_i^*T_i)E_iT_i^*=P_i\), and \(P_i=P_i^*\). The operator \(A=T_iE_i\) commutes with the right translations, so by Lemma 2.1(1) of Separated bands and equivariant traces, \(\tau(AA^*)=\tau(A^*A)\), that is, \(\tau(P_i)=\tau(E_i)\), which is the area \(4R^2/(Z_iN_i)\) by Lemma 1.1(3). The number of translates does not enter.

For (5.3), \(f=E_ifE_k\) and \(g=E_kgE_j\), so \(fT_k^*T_kg=fE_kT_k^*T_kE_kg=fE_kg=fg\). The adjoint formula is clear, and \(\|T_iE_i\|^2=\|E_iT_i^*T_iE_i\|=1\) gives the norm bound. \(\square\)

6. Exercises

Exercise 6.1. Check that \(p(t,u)\) has eigenvalues \(0\) and \(1\) at every point, and compute \(p\) at a point with \(\theta=\pi/2\).

Exercise 6.2. Show that Lemma 4.1 fails for some irrational numbers: find an irrational \(\gamma\) and integers \(n\to\infty\) with \(\|n\gamma\|\le n^{-2}\).

Exercise 6.3. In Lemma 5.1, explain why the trace of \(P_i\) does not grow with the number \(2J_i\) of translates, although \(P_i\) is "spread" over all of them.

Exercise 6.4. Show that \(T_i\) is not an isometry, but that \(T_i^*T_i\) restricted to the range of \(E_i\) is the identity.

7. Solutions

6.1. \(p=\frac12(1+n\cdot\sigma)\) and \(n\cdot\sigma\) has eigenvalues \(\pm1\) since \((n\cdot\sigma)^2=1\) and its trace is zero. At \(\theta=\pi/2\), \(p=\frac12\begin{pmatrix}1&e^{-i\phi}\\e^{i\phi}&1\end{pmatrix}\).

6.2. Let \(\gamma=\sum_{k\ge1}10^{-k!}\), a Liouville number, irrational because its decimal expansion is not periodic. With \(n=10^{m!}\), \(n\gamma\) differs from an integer by \(\sum_{k>m}10^{m!-k!}\). This is at most \(2\cdot10^{m!-(m+1)!}=2\cdot10^{-m\cdot m!}\), which is at most \(n^{-2}=10^{-2\cdot m!}\) for \(m\ge3\).

6.3. \(P_i=AA^*\) with \(A=T_iE_i\), and \(\tau(AA^*)=\tau(A^*A)=\tau(E_i)\). The translates of the rectangle are disjoint, but the weights \(a_{ij}\) have \(\sum_ja_{ij}^2=1\), so the total trace is that of one rectangle.

6.4. \(T_i^*T_i=\sum_{j,k}a_{ij}a_{ik}\lambda(y)^{k-j}\) has off-diagonal terms; for example \(\langle T_i^*T_i\delta_1,\delta_1\rangle=1\) but \(\langle T_i^*T_i\delta_1,\delta_y\rangle=\sum_ja_{ij}a_{i,j+1}\ne0\) in general. On the range of \(E_i\) the off-diagonal terms vanish by the orthogonality of the translates, as in the proof.

References