1. From a coinvariant to finite PE transports
Reader card. Input: a transverse-invariant height-coinvariant class. Output: one explicit finite-PE bisection-normalizer transport for each transverse generator, all acting on a common stabilized projection. This chapter is constructive and uses no \(KK\)-theory.
The running Sturmian example has the particularly simple transfers \(h_1=g\) and \(h_2=0\). It may be helpful to compare each general block construction below with that example.
Let
\[ M=C(\Sigma,\mathbb Z), \qquad [f]\in(M_{\alpha_h})^G. \]
The purpose of this chapter is to construct, for every transverse generator, an explicit module isomorphism between a projection and its translate. No stable-rank theorem is needed for existence.
1.1 Replace a signed class by a full positive projection
Choose \(N\) so large that
\[ g=f+N\geq1. \]
Put \(d=\lVert g\rVert_\infty\). All copies of \(P(g)\) below are regarded as projections in \(M_d(C(\Sigma))\).
For a nonnegative locally constant integer function \(a\), define
\[ P(a)=\operatorname{diag} \left( 1_{\{a\geq1\}}, \ldots, 1_{\{a\geq\lVert a\rVert_\infty\}} \right). \]
At a point \(x\), the rank of \(P(a)(x)\) is exactly \(a(x)\). Hence
\[ [P(g)]-[1_N]=[f]\in K_0(B_h). \]
Because \(g\geq1\), \(P(g)\) is nonzero everywhere and is full.
1.2 Convert coinvariant invariance into a bounded flow
The equality
\[ \alpha_i([g])=[g]\quad\text{in }M_{\alpha_h} \]
means that there is a locally constant integer function \(h_i\) with
\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]
Adding a constant does not change the right side. Choose a constant so that
\[ r_i=h_i+C_i\geq0. \]
Rearranging the preceding equation gives the pointwise equality
\[ \alpha_i(g)+r_i=g+\alpha_h(r_i). \tag{1.1} \]
Think of \(r_i(x)\) as a finite amount of auxiliary mass. Equation (1.1) says that the translated mass \(\alpha_i(g)\) can be matched with \(g\) once the unmatched part is moved by one height step.
flowchart LR
source["α_i(g) + r_i"]
perm["Clopen permutation Π_i"]
target["g + α_h(r_i)"]
shift["Unmatched free block<br/>moved by U_h"]
source --> perm --> target
source -. complement .-> shift -.-> target
1.3 The clopen permutation
Choose the constants in the preceding section for all generators, and put
\[ L=\max_i\lVert r_i\rVert_\infty. \]
Write
\[ Q_i =\operatorname{diag} \left( 1_{\{r_i\geq1\}},\ldots,1_{\{r_i\geq L\}} \right) \in M_L(C(\Sigma)). \]
Thus \(Q_i\) is \(P(r_i)\) padded by zero coordinates to the same size \(L\) for every \(i\). Refine \(\Sigma\) by a finite clopen partition on which
\[ g,\ \alpha_i(g),\ r_i,\ \alpha_h(r_i) \]
are all constant. On each atom of the partition, equation (1.1) is an equality between two finite cardinalities. Choose a permutation matching the corresponding coordinates. Since the partition is clopen, these permutations assemble to a unitary \(\Pi_i\in M_{d+L}(C(\Sigma))\) such that
\[ \Pi_i \bigl(\alpha_i(P(g))\oplus Q_i\bigr) \Pi_i^{\ast} =P(g)\oplus\alpha_h(Q_i). \tag{1.2} \]
Every matrix entry of \(\Pi_i\) is a characteristic function of a clopen set.
1.4 Complete the partial isometry with the height shift
On the complement of \(Q_i\) in the common \(L\)-dimensional free block,
\[ U_h(1_L-Q_i) \]
has initial projection \(1_L-Q_i\) and final projection
\[ \alpha_h(1_L-Q_i)=1_L-\alpha_h(Q_i). \]
Combining this complement map with (1.2) gives a partial isometry
\[ W_i ={} \Pi_i\bigl(\alpha_i(P(g))\oplus Q_i\bigr) + \bigl(0_d\oplus U_h(1_L-Q_i)\bigr). \tag{1.3} \]
The two summands have orthogonal initial projections \(\alpha_i(P(g))\oplus Q_i\) and \(0_d\oplus(1_L-Q_i)\), and orthogonal final projections \(P(g)\oplus\alpha_h(Q_i)\) and \(0_d\oplus(1_L-\alpha_h(Q_i))\). Consequently, with
\[ P=P(g)\oplus1_L. \]
Then
\[ W_i^{\ast}W_i=\alpha_i(P), \qquad W_iW_i^{\ast}=P. \tag{1.4} \]
The same matrix size and the same \(P\) work for all \(i=1,\ldots,q\). This explicit block calculation is the projectivity check; no stable-rank or cancellation theorem is being used here.
1.5 Why the transport is a finite PE bisection normalizer
There is a little more structure in (1.3) than finite Fourier support. Work in the stabilized height transformation groupoid
\[ \mathcal G_h=(\Sigma\rtimes_{T_h}\mathbb Z)\times \mathcal R_{d+L}, \]
whose matrix colour set is \(\{1,\ldots,d+L\}\). On every atom of the clopen partition used above, the first summand in (1.3) is the characteristic function of a finite union of height-zero arrows. The chosen coordinate permutation is injective on both source and range, so this union is a compact-open bisection. The second summand is the characteristic function of the union, over the free-block colours, of the height-one arrows restricted to the clopen sets on which the corresponding diagonal entry of \(1_L-Q_i\) is one (with the orientation fixed by our covariance convention). Distinct colours give disjoint sources and ranges, so this too is a compact-open bisection.
The orthogonality calculation preceding (1.4) says more precisely that the source sets of these two bisections are disjoint and that their range sets are disjoint. Their union is therefore a compact-open bisection \(B_i\) from the diagonal support of \(\alpha_i(P)\) to that of \(P\), and
\[ W_i=1_{B_i}. \]
Consequently \(W_i\) is a diagonal normalizer. If
\[ D_i=\alpha_i(P)M_{d+L}(C(\Sigma))\alpha_i(P), \qquad D_P=PM_{d+L}(C(\Sigma))P, \]
then the usual bisection-convolution formula gives
\[ W_iD_iW_i^*=D_P, \qquad W_i^*D_PW_i=D_i. \tag{1.5} \]
Thus the transports constructed here satisfy the normalizer/bisection hypothesis used in Chapter 3; this is not a property of an arbitrary finite Fourier partial isometry.
They are also finite pattern-equivariant transports. Indeed, each entry of \(W_i\) is a finite sum of terms of the form
\[ 1_C \quad\text{or}\quad 1_CU_h, \]
after block permutation and padding. More general choices of transfer function lead to finitely many powers \(U_h^n\), but never infinitely many. Thus the transport is:
- finite propagation in the height direction;
- locally constant in the Cantor variable;
- a native pattern-equivariant time shift in the regular representation.
At this point the transports exist individually. Their commutation is a different problem:
\[ W_i\alpha_i(W_j) \stackrel{?}{=} W_j\alpha_j(W_i). \]
The next chapter identifies the \(K_1\)-class of this failure.
1.6 Preserve the signed class
The stabilization does not multiply the desired class. The positive module is represented by \(P(g)\oplus1_L\); place the trivial strict action on a negative free module of rank \(N+L\). The graded class is
\[ [P(g)\oplus1_L]-[1_{N+L}] =[P(g)]-[1_N] =x_f. \]