Technical Engine B: analytic descent and the product trace
Reader card. This chapter certifies the analytic part of Step 6: the balanced action, adjoints, compactness, finite projectivity, and trace factorization. For the construction and its meaning without the detailed Hilbert-module checks, first read Attaching the magnetic Heisenberg module.
The equation numbers retain the prefix \(6\) because this is the technical proof of Step 6 in the constructive argument.
This chapter gives the analytic construction in full. In particular, it fixes the cocycle convention, writes the balanced correspondence, verifies adjointability and compactness, and proves the trace formula on a finite projection.
B.1 Conventions and the equivariant coefficient module
Put
\[ B=B_h, \qquad A=B\rtimes_{\alpha,\sigma}G, \qquad D=C^\ast(G,\sigma)=A_{\Theta_G}. \]
The canonical unitaries satisfy
\[ U_gbU_g^\ast=\alpha_g(b), \qquad U_gU_h=\sigma(g,h)U_{g+h}, \]
and the generators \(u_g\) of \(D\) satisfy the same scalar cocycle relation.
First let \(F\) be an actual finitely generated projective right \(B\)-module. Its strict equivariant structure consists of semilinear unitaries \(V_g:F\to F\) such that
\[ \begin{aligned} V_g(\xi b)&=V_g(\xi)\alpha_g(b),\\ \langle V_g\xi,V_g\eta\rangle &=\alpha_g(\langle\xi,\eta\rangle),\\ V_gV_h&=V_{g+h}. \end{aligned} \tag{B.1} \]
The signed coefficient class is represented by a graded pair \(F^+\ominus F^-\), and the construction below is applied separately to its two summands.
B.2 The balanced correspondence
Define the right Hilbert \(A\)-module
\[ X_F=F\widehat\otimes_B A. \]
On the algebraic balanced tensor product its inner product is
\[ \langle\xi\otimes a,\eta\otimes c\rangle_A =a^\ast\langle\xi,\eta\rangle_Bc. \tag{B.2} \]
Positivity and completion are therefore the standard positivity and completion of an interior tensor product.
For \(g\in G\), define
\[ L_g(\xi\otimes a)=V_g\xi\otimes U_ga. \tag{B.3} \]
This is balanced. Indeed,
\[ \begin{aligned} L_g(\xi b\otimes a) &=V_g\xi\,\alpha_g(b)\otimes U_ga\\ &=V_g\xi\otimes\alpha_g(b)U_ga\\ &=V_g\xi\otimes U_gba =L_g(\xi\otimes ba). \end{aligned} \]
It is an isometry because
\[ \begin{aligned} \langle L_g(\xi\otimes a),L_g(\eta\otimes c)\rangle_A &=a^\ast U_g^\ast\alpha_g(\langle\xi,\eta\rangle)U_gc\\ &=a^\ast\langle\xi,\eta\rangle c. \end{aligned} \]
The adjoint normalization is explicit:
\[ L_g^{\ast}=L_g^{-1} =\overline{\sigma(g,-g)}\,L_{-g}, \]
because \(L_gL_{-g}=\sigma(g,-g)L_0\). Thus every \(L_g\) is adjointable. Finally,
\[ L_gL_h=\sigma(g,h)L_{g+h}. \tag{B.4} \]
Consequently \(u_g\mapsto L_g\) extends to a nondegenerate \(\ast\)-representation
\[ \lambda_F:D\longrightarrow\mathcal L_A(X_F), \]
and \(X_F\) is a \(D\)-\(A\) correspondence. This is the concrete form of twisted equivariant descent.
B.3 Compactness
Choose \(p\in M_n(B)\) with \(F\cong pB^n\). Extension of scalars gives
\[ X_F\cong pA^n. \tag{B.5} \]
Thus \(X_F\) is finitely generated projective as a right \(A\)-module, and
\[ \mathcal L_A(X_F)=\mathcal K_A(X_F) \cong pM_n(A)p. \]
In particular, the identity of \(X_F\) is compact and the entire left action \(\lambda_F(D)\) is by compact operators. This is stronger than the bare adjointability required for a correspondence.
B.4 Product with a Rieffel module
Let \(E_T\) be an elementary Rieffel--Heisenberg module over \(D\). The mixed module is the interior tensor product
\[ \mathcal E_{T,F} =E_T\widehat\otimes_DX_F. \tag{B.6} \]
To see projectivity without an abstract compactness shortcut, represent \(E_T\) by a projection \(e\in M_m(D)\). Then (B.6) is the range of the projection
\[ \lambda_F^{(m)}(e)\in M_m(\mathcal K_A(X_F)) \cong\mathcal K_A(X_F^m). \]
Since \(X_F^m\) is finitely generated projective, so is this range. For graded inputs, the output is the graded difference
\[ \mathcal E_{T,F^+}\ominus\mathcal E_{T,F^-}. \]
Likewise, an arbitrary virtual torus class is handled by a graded difference of Rieffel or standard finite-projective representatives. An ungraded projective output is asserted only when the chosen inputs are actual positive modules.
B.5 The coefficient-valued time-frequency formula
At height \(r\), the coefficients are ordinary time-frequency coefficients on
\[ M_T=\mathbb R^r\times\mathbb Z^{q-2r}. \]
With an embedding \(T=(T',T'')\), one matching normalization is
\[ (R_g\psi)(x) ={} \exp\!\left( 2\pi i \left\langle x-\tfrac12T'(g),T''(g) \right\rangle \right) \psi(x-T'(g)), \tag{B.8} \]
where Hilbert-space inner products are conjugate-linear in the first variable. Put \(a=T'(g)\), \(b=T''(g)\), \(c=T'(h)\), and \(d=T''(h)\). Direct multiplication gives
\[ R_gR_h=\sigma_T(g,h)R_{g+h}, \qquad \sigma_T(g,h) =\exp\!\left( \pi i(\langle c,b\rangle-\langle a,d\rangle) \right). \tag{B.8a} \]
Indeed, the exponent in \(R_gR_h\) minus that in \(R_{g+h}\) is \(\tfrac12(\langle c,b\rangle-\langle a,d\rangle)\). Thus \(\sigma_T\) is the alternating Weyl cocycle: \(\sigma_T(h,g)=\overline{\sigma_T(g,h)}\) and \(\sigma_T(g,-g)=1\). The embedding \(T\) is chosen to realize the fixed multiplier, so from this point \(\sigma_T=\sigma\). A cohomologous nonsymmetric normalization is obtained by the corresponding generator gauge and uses the general adjoint formula from Section B.2.
Here is the right-module convention used below. On the conjugate Hilbert space \(\overline{\mathcal H_T}\), set
\[ Q_g\overline\psi=\overline{R_g\psi}, \qquad \overline\psi\cdot u_g=Q_g\overline\psi. \tag{B.8b} \]
Since
\[ Q_hQ_g =\overline{\sigma_T(h,g)}Q_{g+h} =\sigma_T(g,h)Q_{g+h}, \]
(B.8b) is a right action of \(D=C^{\ast}(G,\sigma_T)\). Its \(D\)-valued inner product is
\[ \left\langle\overline\phi,\overline\psi\right\rangle_D =\sum_{g\in G}c_g(\overline\phi,\overline\psi)u_g, \qquad c_g(\overline\phi,\overline\psi) =\langle Q_g\overline\phi,\overline\psi\rangle_{\overline{\mathcal H_T}} =\langle\psi,R_g\phi\rangle_{\mathcal H_T}. \tag{B.7} \]
This convention has no hidden sign or inverse. For example, \(Q_h^{\ast}=Q_{-h}\) and the coefficient of \(u_k\) in \(\langle\overline\phi, \overline\psi\cdot u_h\rangle_D\) is
\[ \sigma_T(-h,k)c_{k-h} =\sigma_T(k-h,h)c_{k-h}, \]
which is exactly the coefficient of \(u_k\) in \(\langle\overline\phi,\overline\psi\rangle_Du_h\). Also \(u_g^{\ast}=u_{-g}\) for this Weyl normalization, and the coefficient formula gives \(\langle\overline\phi,\overline\psi\rangle_D^{\ast} =\langle\overline\psi,\overline\phi\rangle_D\).
For the dense vectors \(\overline\phi\otimes(\xi\otimes1)\) and \(\overline\psi\otimes(\eta\otimes1)\) in (B.6), the interior-product rule gives
\[ \left\langle \overline\phi\otimes(\xi\otimes1), \overline\psi\otimes(\eta\otimes1) \right\rangle_A {}= \sum_{g\in G} c_g(\overline\phi,\overline\psi) \langle\xi,V_g\eta\rangle_B U_g. \tag{B.9} \]
For Schwartz or Feichtinger vectors the usual Rieffel/Gabor estimates give convergence in the smooth crossed-product norm. Positivity of (B.9) is not a new assertion: it follows from (B.2) and the positive \(D\)-valued inner product (B.7).
Formula (B.9) is the native mixed Gabor picture. Each Fourier coefficient contains both the ordinary time-frequency coefficient and the finite PE height transport.
B.6 Trace factorization
Let \(\mu\) be jointly invariant. Write \(\tau_B\) for the trace on \(B\), and define the canonical crossed-product trace by
\[ \tau_A\!\left(\sum_gb_gU_g\right)=\tau_B(b_0). \]
Use the model \(F=pB^n\). There are partial unitaries \(w_g\) such that
\[ V_g\xi=w_g\alpha_g(\xi), \]
and under (B.5) the operator \(\lambda_F(u_g)\) is left multiplication by \(w_gU_g\). Therefore the induced trace on \(\mathcal K_A(X_F)\cong pM_n(A)p\) satisfies
\[ \operatorname{Tr}_{\tau_A}^{X_F}(\lambda_F(u_g))
\begin{cases} \operatorname{Tr}_{\tau_B}(p),&g=0,\\ 0,&g\ne0. \end{cases} \tag{B.10} \]
By continuity,
\[ \operatorname{Tr}\{\tau_A}^{X_F}\circ\lambda_F =\operatorname{Tr}\{\tau_B}(p)\,\tau_D \quad\text{on }D. \tag{B.11} \]
If \(e\in M_m(D)\) represents \(E_T\), apply (B.11) entrywise to the projection \(\lambda_F^{(m)}(e)\). This gives
\[ \tau_A([\mathcal E_{T,F}]) =\tau_D([E_T])\,\tau_B([F]). \tag{B.12} \]
The same equation is bilinear for graded differences. Since the graded coefficient representative constructed in Chapter 1 satisfies
\[ \tau_B([F^+]-[F^-])=\mu(f), \]
we obtain
\[ \boxed{ \tau_A([\mathcal E_{T,f}]) =\mu(f)\,\tau_{\Theta_G}([E_T]). } \tag{B.13} \]
Rieffel's trace calculation supplies the second factor. The argument did not use the height of the embedding \(T\), so every permitted Rieffel height is included.
B.7 A nontrivial diagonal phase
If the fixed-corner cocycle of Chapter 4 does not vanish, the maps \(V_g\) satisfy a coefficient-valued projective relation rather than (B.1). Then (B.4) acquires an operator-valued multiplier and the present left action of the scalar torus \(D\) is unavailable.
One could instead seek a finite-frame Heisenberg module over the corresponding Fell bundle. The regular Packer--Raeburn stabilization only gives a countably generated construction and does not prove the finite-projective or trace statement required here. That extension remains open.