A short introduction to Rieffel--Heisenberg modules
Rieffel--Heisenberg modules provide the magnetic part of the construction in this book. They are explicit finite projective modules over noncommutative tori, built from translations and modulations on an ordinary phase space. Their trace supplies the Pfaffian factor in a magnetic gap label.
Reader card. This chapter is an abbreviated construction, not a replacement for Rieffel's imprimitivity theorem. It assumes elementary Fourier analysis and familiarity with lattices. The exact right-module convention, compactness proof, and trace calculation used later are recorded in Technical Engine B.
1. The noncommutative torus as magnetic translations
Let
\[ G=\mathbb Z^q \]
and let \(\sigma:G\times G\to U(1)\) be a multiplier determined by a skew-symmetric matrix \(\Theta\). The twisted group algebra
\[ D=C^*(G,\sigma)=A_\Theta \]
is generated by unitaries \(u_g\) satisfying
\[ u_gu_h=\sigma(g,h)u_{g+h}. \]
For coordinate generators this is the familiar relation
\[ u_ju_k=e^{2\pi i\Theta_{jk}}u_ku_j. \]
These are magnetic translation relations: translations commute only up to the flux through the parallelogram they span.
2. Translation and modulation on phase space
Choose a locally compact abelian group \(M\), such as
\[ M=\mathbb R^r\times\mathbb Z^{q-2r}, \]
and write \(\widehat M\) for its Pontryagin dual. A point
\[ (x,\omega)\in M\times\widehat M \]
acts on a function \(\xi\) on \(M\) by a time-frequency shift
\[ \bigl(\pi(x,\omega)\xi\bigr)(t) =\omega(t)\xi(t-x), \tag{R.1} \]
up to a harmless Weyl phase used to make the cocycle alternating.
Translations and modulations do not commute:
\[ \pi(x,\omega)\pi(y,\eta) =c\bigl((x,\omega),(y,\eta)\bigr) \pi(x+y,\omega\eta), \tag{R.2} \]
where \(c\) is the Heisenberg cocycle. Thus phase space already carries the same projective behavior as a noncommutative torus.
3. Embed the lattice
Choose an injective homomorphism
\[ T:G\longrightarrow M\times\widehat M \]
whose image is a lattice and whose Heisenberg commutator form agrees with the prescribed magnetic form \(\Theta\). Then
\[ R_g=\pi(Tg) \]
satisfies
\[ R_gR_h=\sigma(g,h)R_{g+h}. \tag{R.3} \]
Consequently the generators \(u_g\) of \(A_\Theta\) can act by the time-frequency operators \(R_g\). The choice of \(M\) and \(T\) is the geometric input. Different values of \(r\) produce Rieffel modules of different heights.
In dimension two, one may take \(M=\mathbb R\). One coordinate generator acts as translation and the other as modulation. Their failure to commute is exactly the phase \(e^{2\pi i\theta}\) defining the rotation algebra \(A_\theta\).
4. Turn test functions into a module
Start with a well-behaved space of functions on \(M\), for example a Schwartz or Feichtinger space. After choosing a consistent left- or right-module convention, the action of a Fourier series is schematically
\[ \xi\cdot\left(\sum_{g\in G}a_gu_g\right) =\sum_{g\in G}a_gR_g\xi. \tag{R.4} \]
The \(A_\Theta\)-valued inner product samples ordinary Hilbert-space coefficients along the lattice:
\[ \langle\xi,\eta\rangle_{A_\Theta} =\sum_{g\in G} \langle \xi,R_g\eta\rangle\,u_g, \tag{R.5} \]
with the order, conjugation, and possible inverse adjusted to the selected right-module convention. Formula (R.5) is the essential idea: its coefficients are samples of the short-time Fourier transform, or ambiguity function, of \(\xi\) and \(\eta\).
Completing in the norm determined by this inner product gives a Hilbert \(A_\Theta\)-module \(E_T\). The exact convention used in this book is written in formulas (B.7)--(B.8b) of Technical Engine B.
5. Why the module is finite projective
The lattice \(T(G)\) has a complementary, or adjoint, lattice in phase space. The two lattice actions commute, and Rieffel's Heisenberg imprimitivity theorem identifies the completed function space as an equivalence bimodule between the corresponding twisted group algebras.
Because the lattice quotient is compact, the module is finitely generated and projective. Concretely, one can choose finitely many vectors \(g_1,\ldots,g_N\) forming a module frame. The matrix
\[ p_{ij}=\langle g_i,g_j\rangle_{A_\Theta} \tag{R.6} \]
is a projection in \(M_N(A_\Theta)\), and
\[ E_T\cong pA_\Theta^N. \]
Under the time-frequency interpretation, a finite module frame corresponds to a multi-window Gabor frame generated by the lattice \(T(G)\). This is a powerful analytic realization of the module, but the absorption proof in this book needs only a finite projection representing \(E_T\); it does not construct new coefficient-valued Gabor windows.
6. Trace, volume, and Pfaffians
The canonical trace on \(A_\Theta\) takes the zero Fourier coefficient. Applied to the projection (R.6), Rieffel's calculation expresses the module trace through the volume or determinant associated with the lattice embedding \(T\).
For the two-dimensional rotation algebra and the basic positive module, normalized with \(0<\theta<1\),
\[ \tau_\theta(E_\theta)=\theta. \]
In higher dimensions, the trace range of \(K_0(A_\Theta)\) is generated by the Pfaffian minors
\[ \operatorname{Pf}(\Theta_I), \qquad |I|\ \text{even}. \]
An elementary Heisenberg module has positive trace determined by its embedding. Depending on the chosen embedding, it may represent a positive combination or a positive multiple of the desired component. An isolated signed Pfaffian term may therefore require a graded difference of positive modules. The book calls the output an actual projective module only when the chosen trace and \(K_0\)-class are positive; otherwise it uses a virtual class.
7. What absorption adds
The Rieffel module by itself depends only on the constant magnetic form. It does not contain nonconstant patch data from \(\Sigma\). Its trace lies in the ordinary noncommutative-torus trace group.
The absorption theorem separately constructs a pattern-dependent coefficient module \(F_f\) carrying strict finite-PE transverse transports. It then forms
\[ \mathcal E_{T,f} =E_T\widehat\otimes_{A_{\Theta_G}}X_f. \]
The lattice shift \(R_g\) is thereby coupled to the PE height transport \(V_g\). Schematically, a generator acts as
\[ \text{time-frequency shift }R_g \quad\times\quad \text{pattern-dependent transport }V_g. \]
The trace factors:
\[ \tau_\mu([\mathcal E_{T,f}]) =\mu(f)\tau_{\Theta_G}([E_T]). \tag{R.7} \]
Thus Rieffel's module supplies the magnetic topology and the Pfaffian, while twisted absorption supplies the quasicrystalline frequency and the finite-PE dependence on local patterns.
For the running Fibonacci cylinder and a basic two-dimensional module,
\[ \mu(f)=\frac1\varphi, \qquad \tau_\mu([\mathcal E_{\theta,f}]) =\frac{\theta}{\varphi}. \]
8. What to remember
- A phase-space lattice turns translations and modulations into the projective relations of \(A_\Theta\).
- Sampling time-frequency coefficients along that lattice gives the \(A_\Theta\)-valued inner product.
- Heisenberg imprimitivity makes the completed space finite projective; finite module frames are multi-window Gabor frames.
- Rieffel's trace calculation supplies the magnetic Pfaffian factor.
- The new absorption argument does not reconstruct this magnetic engine. It couples it to a strict, pattern-dependent coefficient module.
The original references and further Gabor-module literature are listed in Sources.