Dimension three: comparison and equality

Reader card. This chapter compares three logically different claims: an upper trace bound, abstract existence of classes, and explicit finite-PE representatives. It also identifies exactly what is new relative to Benameur--Mathai and when equality follows.

This chapter separates three statements that are easy to conflate:

1. an upper bound on the range of the canonical trace;
2. existence of abstract \(K_0\)-classes producing the expected mixed traces; and
3. construction of native graded finite PE projective representatives of those classes.

The first two are addressed cohomologically by Benameur--Mathai in dimension three. The absorption theorem strengthens the third. Its block-supported constructive conclusions use the proved PV/Barlak comparison (A.4). The full-field conclusion remains conditional only on the separate finite-stage diagonal phase hypothesis.

1. The magnetic frequency group

Let

\[ A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^3, \]

and let \(\mu\) be a \(\mathbb Z^3\)-invariant probability measure. Put

\[ M=C(\Sigma,\mathbb Z). \]

For a cyclic permutation \((i,j,k)\) of \((1,2,3)\), define

\[ \mathbb Z_{ij}[\mu] ={} \mu\!\left( \left(M_{\langle T_k\rangle}\right)^{ \langle T_i,T_j\rangle} \right). \]

The dimension-three magnetic frequency group is

\[ \mathcal G_\Theta(\mu) ={} \mathbb Z[\mu] +\Theta_{12}\mathbb Z_{12}[\mu] +\Theta_{13}\mathbb Z_{13}[\mu] +\Theta_{23}\mathbb Z_{23}[\mu]. \tag{7.1} \]

Benameur--Mathai prove for a minimal \(\mathbb Z^3\)-action that

\[ \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right) \subseteq \mathcal G_\Theta(\mu). \tag{7.2} \]

This is the difficult or upper-bound half: no unexpected trace values occur. Their proof uses the measured twisted foliated index theorem, the Baum--Connes assembly map, and a direct group-cohomology calculation.

2. What their cohomological calculation already detects

Choose \(k=3\). If

\[ x_f\in \left(M_{\langle T_3\rangle}\right)^{ \langle T_1,T_2\rangle}, \]

there are integer-valued transfer functions \(h_1,h_2\) satisfying

\[ (T_1-1)f=(T_3-1)h_1, \qquad (T_2-1)f=(T_3-1)h_2. \tag{7.3} \]

The boundary computed in their Lemma 7.5 is represented, up to the action and sign convention, by

\[ \kappa_{12} =(T_2-1)h_1-(T_1-1)h_2. \tag{7.4} \]

This is the same integer curvature that appears as the \(K_1\)-index of the square defect in the absorption proof. If \(T_3\) is minimal, then \(\kappa_{12}\) is constant. Its integral is zero, so

\[ \kappa_{12}=0. \]

Benameur--Mathai consequently obtain

\[ \Theta_{12}\mathbb Z_{12}[\mu] \subseteq \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right) \tag{7.5} \]

whenever \(T_3\) is minimal. Permuting the generators gives the analogous statements for the other two mixed groups. If all three generators are minimal -- their strong minimality condition in odd dimension -- they combine these inclusions with (7.2) to obtain equality.

Thus the absorption theorem should not be advertised as discovering the dimension-three trace labels that their cohomological proof missed. It constructs representatives for labels that their index theorem already detects abstractly.

3. Why Hypothesis (H) appears in their explicit construction

Section 8 of their paper asks for a particularly rigid realization. For a clopen set \(\Lambda\), they construct a homomorphism

\[ \Phi_\Lambda: C^{\ast}(\langle T_1,T_2\rangle,\sigma) \longrightarrow M_\infty\!\left(C(\Sigma)\rtimes_{i_*\sigma}\mathbb Z^3\right) \tag{7.6} \]

whose induced map multiplies the canonical trace by \(\mu(\Lambda)\). Their Hypothesis (H) supplies finitely many disjoint height sheets which the transverse generators permute with exact integer height corrections. That hypothesis is natural for a literal finite-matrix homomorphism of the form (7.6).

The absorption theorem changes the representative category. It permits:

• stabilization and graded differences;
• locally constant transfer functions rather than one finite tower;
• partial isometries between projective summands; and
• finite PE gauge corrections inside the height crossed product.

The output is a class-dependent graded finite-projective module or correspondence. For positive inputs it may be ungraded; arbitrary signed classes require a difference. It need not arise from a single global homomorphism \(\Phi_\Lambda\). This is the flexibility that removes Hypothesis (H).

4. Equality for a block-supported field

Suppose

\[ \Theta_{13}=\Theta_{23}=0, \qquad \theta=\Theta_{12}, \]

and assume \(T_3\) is minimal. The absorption theorem applies with height \(T_3\) and transverse group

\[ G=\langle T_1,T_2\rangle. \]

For every

\[ x_f\in \left(M_{\langle T_3\rangle}\right)^G \]

it constructs a graded finite PE module \(P_{f,12}\) satisfying

\[ \tau_\mu([P_{f,12}]) =\theta\,\mu(f). \tag{7.7} \]

If the selected Rieffel representative has trace \(m+\theta\), absorption first gives

\[ m\mu(f)+\theta\mu(f). \]

Subtracting \(m\) copies of the coefficient module isolates (7.7) in graded \(K_0\). Therefore

\[ \theta\mathbb Z_{12}[\mu] \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). \tag{7.8} \]

Clopen projections already give

\[ \mathbb Z[\mu] \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). \]

Combining these lower bounds with (7.2) yields

\[ \boxed{ \tau_\mu(K_0(A_{\Sigma,\Theta})) ={} \mathbb Z[\mu]+\theta\mathbb Z_{12}[\mu]. } \tag{7.9} \]

As a numerical trace theorem, (7.9) also follows from Benameur--Mathai's Corollary 7.6. The new content is that every term on the right has a native finite-propagation graded PE representative, with no Hypothesis (H).

5. Torus pullbacks and genuinely quasicrystalline classes

The constant coefficient class \(n[1]\) gives \(n\) copies of the pulled-back noncommutative-torus Rieffel class. In contrast, if

\[ x_f\notin\mathbb Z[1] \quad\text{in}\quad \left(M_{\langle T_3\rangle}\right)^G, \]

then \(P_{f,12}\) carries nonconstant pattern data. In particular, let \(\Lambda\) be a clopen pattern cylinder whose indicator satisfies the required invariance condition

\[ [1_\Lambda]\in \left(M_{\langle T_3\rangle}\right)^G. \]

Only for such a cylinder does the theorem directly construct \(P_{\Lambda,12}\), and its trace is

\[ \tau_\mu([P_{\Lambda,12}]) =\Theta_{12}\operatorname{freq}(\Lambda). \tag{7.10} \]

If this number is not in the torus trace group

\[ \mathbb Z +\Theta_{12}\mathbb Z +\Theta_{13}\mathbb Z +\Theta_{23}\mathbb Z, \]

then the class cannot lie in the torus-pullback image. If the numerical trace happens to coincide with a torus trace, one must instead use the associated-graded coefficient component or a higher cyclic pairing. The canonical trace is not generally injective on \(K_0\).

6. Minimal but not strongly minimal

There are useful examples outside strong minimality. Let \((X_r,S_r)\), \(r=0,1,2\), be Sturmian systems chosen so that the diagonal product is minimal, and set

\[ \begin{aligned} \Sigma&=X_0\times X_1\times X_2,\\ T_1&=S_0\times1\times1,\\ T_2&=1\times S_1\times1,\\ T_3&=S_0\times S_1\times S_2. \end{aligned} \]

The full action is minimal because \(T_3\) is minimal, but \(T_1\) and \(T_2\) are not minimal on the product. For a block-supported field the absorption theorem gives (7.9) and explicit labels such as

\[ \frac{\Theta_{12}}{\varphi} \]

from a Fibonacci cylinder depending only on the \(X_0\)-coordinate. Its class is \(T_2\)-fixed, and \((T_1-1)1_\Lambda=(T_3-1)1_\Lambda\), so it satisfies the required invariance in the \(T_3\)-coinvariants. This demonstrates that strong minimality is not needed for the one magnetic plane. It does not produce a trace equality beyond Benameur--Mathai's full dimension-three paper, since their one-height Corollary 7.6 already covers this numerical inclusion.

7. The conditional full-field statement

For a general \(3\times3\) magnetic matrix, repeat the construction for the three cyclic splittings. Assume first that the selected height \(T_k\) is minimal in each splitting; for all three cyclic splittings this requires \(T_1,T_2,T_3\) to be minimal, namely the strong-minimality hypothesis used here. This hypothesis is needed both in the curvature argument and in the GPS reduction. The comparison theorem and the full-field homotopy then remove the integral \(K_1\)-defects. If, in addition, for every pair \((i,j)\) and every class

\[ x\in \left(M_{\langle T_k\rangle}\right)^{ \langle T_i,T_j\rangle}, \]

the corresponding fixed-corner diagonal cocycle becomes a coboundary at an allowed finite stage, then absorption supplies all three reverse inclusions. This is an existential hypothesis for the constructed representatives, not the vanishing of a proved class invariant. Equation (7.2) then gives

\[ \boxed{ \tau_\mu(K_0(A_{\Sigma,\Theta})) =\mathcal G_\Theta(\mu). } \tag{7.11} \]

The final equality argument is formal under all of these hypotheses. A single minimal height proves only the magnetic plane complementary to that height; it does not justify the other two cyclic applications. The unresolved work is replacing strong minimality by suitable class-dependent directional hypotheses and eliminating the height--transverse magnetic phases. Equality of trace ranges also does not classify \(K_0\): torsion, extension data, and trace-zero classes remain invisible.

8. Comparison at a glance

The primary source for the first two rows is Benameur--Mathai, Gap-labelling conjecture with nonzero magnetic field.