Dynamical and operator-algebraic setup

Reader card. This chapter fixes notation and translates the selected height direction into an iterated crossed product. Readers unfamiliar with crossed-product terminology may keep the tiling-theory dictionary open alongside it. The only computation used immediately is the height PV identification of \(K_0\) with coinvariants and \(K_1\) with invariants.

One height direction

Let \(\Sigma\) be a Cantor set. Fix commuting homeomorphisms

\[ T_h,T_1,\ldots,T_q, \qquad q=p-1, \]

and assume that \(T_h\) is minimal. Write

\[ H=\langle T_h\rangle\cong\mathbb Z, \qquad G=\langle T_1,\ldots,T_q\rangle\cong\mathbb Z^q. \]

The coefficient algebra is

\[ B_h=C(\Sigma)\rtimes_{T_h}\mathbb Z. \]

If \(U_h\) denotes its canonical unitary, our covariance convention is

\[ U_h aU_h^{\ast}=\alpha_h(a), \qquad a\in C(\Sigma). \]

Since \(K_1(C(\Sigma))=0\), the Pimsner–Voiculescu sequence gives

\[ K_0(B_h)\cong C(\Sigma,\mathbb Z)_{T_h}, \qquad K_1(B_h)\cong C(\Sigma,\mathbb Z)^{T_h}. \]

Minimality implies

\[ C(\Sigma,\mathbb Z)^{T_h}=\mathbb Z. \]

The magnetic crossed product

Let \(\Theta\in M_p(\mathbb R)\) be skew-symmetric. Splitting \(\mathbb Z^p=G\oplus H\) gives an iterated presentation

\[ A_{\Sigma,\Theta} \cong B_h\rtimes_{\alpha^\Theta,\sigma_{\Theta_G}}G. \]

The automorphism \(\alpha_i^\Theta\) acts on \(C(\Sigma)\) by \(T_i\) and rotates the height unitary by

\[ \alpha_i^\Theta(U_h) =e^{2\pi i\Theta_{ih}}U_h \]

up to the harmless sign determined by the cocycle convention.

The block-supported case is

\[ \Theta_{ih}=0 \qquad (i=1,\ldots,q). \]

Then every transverse automorphism fixes \(U_h\). This is exactly the form \(i_*\sigma\) used in Benameur–Mathai’s explicit construction.

Equivalently, the selected height vector lies in the radical of the magnetic form. Thus a nondegenerate standard symplectic cocycle on an entire even-dimensional ambient lattice is not block-supported for any height-one splitting. What is included is a standard symplectic cocycle on an even-dimensional transverse block, for example \(J_{2m}\oplus0\) on \(\mathbb Z^{2m}\oplus\mathbb Zh\).

flowchart LR
    full["Z^p"]
    G["G = Z^(p−1)<br/>transverse"]
    H["H = Z<br/>height"]
    Bh["B_h = C(Σ) ⋊ H"]
    A["A_(Σ,Θ) = B_h ⋊ G"]

    full --> G
    full --> H
    H --> Bh
    G --> A
    Bh --> A

The input class

Let

\[ f\in\bigl(C(\Sigma,\mathbb Z)_{T_h}\bigr)^G. \]

Its image in \(K_0(B_h)\) will be denoted \(x_f\). The invariance is only an equality of \(K_0\)-classes. It does not yet supply commuting module isomorphisms.

The construction problem is therefore:

Find a graded finite projective \(B_h\)-module representing \(x_f\), together with finite-propagation semilinear maps \(V_i\) satisfying \(V_iV_j=V_jV_i\).

Once this is done, equivariant descent pairs the coefficient module with any Rieffel–Heisenberg module over the transverse noncommutative torus.

Pattern equivariance

For this book, a finite PE height transport is a matrix whose entries are finite sums

\[ \sum_n 1_{C_n}U_h^n, \]

where the \(C_n\) are clopen and only finitely many integers \(n\) occur. In the regular representation on \(\ell^2(\mathbb Z)\), it is a finite-range shift whose displacement is determined by a finite clopen partition—equivalently, by a finite patch.

This broad finite-Fourier condition does not imply that a unitary or partial isometry normalizes the diagonal. Chapter 3 and the exact two-stage lemma use the stronger notion of a finite PE bisection transport: its source and range corners are diagonal corners, it normalizes those diagonals, and its support is the graph of a coefficient-free compact-open height bisection. Such a transport has a well-defined full-group symbol and GPS index. The transports constructed from the clopen orbit partitions in Chapter 1 have this stronger form; an arbitrary matrix of finite Fourier sums need not.