Abstract

This book develops a constructive method for producing magnetic \(K_0\)-classes of aperiodic and quasicrystalline crossed products. Let \(\Sigma\) be a Cantor dynamical system with a \(\mathbb Z^p\)-action and let

\[ A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^p \]

be its crossed product twisted by a constant magnetic multiplier. Magnetic gap labelling predicts that the trace range of this algebra is assembled from products of pattern frequencies with Pfaffian minors of the magnetic matrix \(\Theta\). Index theory and cohomology can constrain or identify this numerical range, but they need not exhibit finite, pattern-equivariant projective modules carrying the predicted traces. The purpose of the book is to construct such representatives and to identify the obstruction to doing so in a strict finite-propagation form.

Choose a primitive splitting

\[ \mathbb Z^p=G\oplus\mathbb Zh, \qquad G\cong\mathbb Z^{p-1}, \]

and assume that the height homeomorphism \(T_h\) is minimal. The coefficient input is an invariant height-coinvariant class

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

A clopen projection already represents the underlying positive class in the untwisted height algebra

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

For a block-supported magnetic form, the new step is to enhance this representative to a graded finite projective module \(F_f\) with strict finite-propagation, pattern-equivariant transverse transport. Thus, for every \(s,t\in G\), the construction produces semilinear operators \(V_s\) satisfying

\[ V_s(\xi b)=V_s(\xi)\alpha_s(b), \qquad V_sV_t=V_{s+t}. \]

These identities make \(F_f\) suitable for equivariant descent. If \(E_T\) is a transverse Rieffel--Heisenberg module over the noncommutative torus \(A_{\Theta_G}\), the balanced product

\[ \mathcal E_{T,f} =E_T\widehat\otimes_{A_{\Theta_G}}X_f \]

is a mixed quasicrystalline module: every transverse magnetic time--frequency shift is coupled to the finite-PE height transport determined by the local pattern. Its trace factors as

\[ \tau_\mu([\mathcal E_{T,f}]) =\mu(f)\,\tau_{\Theta_G}([E_T]). \]

Consequently, for a magnetic form with no height--transverse entries, the construction realizes the height-one lower bound

\[ \mathbb Z[\mu] +D_h(\mu)\, \tau_{\Theta_G}\!\left(K_0(A_{\Theta_G})\right) \subseteq \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right), \]

where

\[ D_h(\mu) =\mu\!\left( \left(C(\Sigma,\mathbb Z)_{T_h}\right)^G \right). \]

This includes every even Pfaffian minor supported in the transverse block, multiplied by the selected height-one pattern-frequency group. In dimension three it constructs the mixed term in the Benameur--Mathai magnetic frequency group without their finite-sheet Hypothesis (H). When combined with the known upper containment, this gives equality for the corresponding block-supported field. More generally, the result supplies a constructive lower bound in arbitrary dimension; it is not by itself an upper-bound theorem or a calculation of the full \(K\)-theory group.

The proof converts transverse invariance in height coinvariants into bounded clopen transport along height orbits. The resulting partial isometries can have square holonomy. A PV/Barlak comparison identifies its \(K_1\)-class with an integer transfer-function curl, minimality forces that curl to vanish, and a Giordano--Putnam--Skau normalizer retraction upgrades vanishing index to exact commuting transport. Rieffel descent then inserts the magnetic Heisenberg factor and yields the product trace formula.

For a general magnetic matrix, the same argument removes the integral and permutation defects but can leave a locally constant diagonal \(U(1)\)-valued cocycle. Strictification is conditional on this cocycle being a coboundary at an allowed finite stage. The book does not claim that this condition always holds or that it defines a choice-independent stable obstruction. Likewise, the present height-one theorem does not realize every coefficient layer predicted by magnetic gap labelling in higher dimensions. That extension requires a higher-rank absorption theorem coupling several coefficient directions simultaneously to a higher-dimensional Heisenberg module.

The construction therefore contributes to the broader program of understanding quasicrystal \(C^*\)-algebras by connecting three levels that are often studied separately: pattern-equivariant representatives of \(K\)-classes, their magnetic trace pairings, and the cohomological frequency groups appearing in gap labelling. It provides native mixed modules rather than passive torus pullbacks, while making explicit which parts of the expected trace range are proved, which depend on an obstruction-vanishing hypothesis, and which remain higher-rank research problems.