Theorem. Positivity and trace class of the gauge transfer operator [fdip-0080]
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  • November 1, 2026

For the spatial transfer data, \(C_k\) and \(T=M_bC_kM_b\) are bounded, self-adjoint, positive, injective and trace class. Both commute with every gauge operator \(R_h\). If \[\kappa =\sum _\rho d_\rho ^2a_\rho =k(1),\] then \[\operatorname {Tr}C_k=\kappa ^{|E_s|},\qquad \operatorname {Tr}T\le \|b\|_\infty ^2\kappa ^{|E_s|}.\] Positive means \(\langle f,Tf\rangle \ge 0\); in fact this pairing is strictly positive for every nonzero \(f\). It does not assert a uniform positive lower bound on the spectrum.

Proof. 

  • November 1, 2026

For one edge, use the normalized matrix-entry basis \(\phi _{\rho ij}(U)=\sqrt {d_\rho }\rho (U)_{ij}\) of the Peter–Weyl theorem. Unitarity gives \[k(UU'^{-1})=\sum _{\rho ,i,j} a_\rho \phi _{\rho ij}(U)\overline {\phi _{\rho ij}(U')}.\] Indeed, the matrix-entry sum is \(d_\rho \operatorname {Tr}(\rho (U)\rho (U')^*)\). Schur orthogonality therefore makes convolution act by \(a_\rho \) on this block. The tensor products of these bases over the finitely many edges give an orthonormal basis \((\phi _I)\) of \(\mathcal H_{\mathrm {link}}\), with eigenvalues \[\alpha _I=\prod _{e\in E_s}a_{\rho _e}>0,\qquad \sum _I\alpha _I=\kappa ^{|E_s|}<\infty .\] The operator assertions about \(C_k\), including injectivity and its trace, follow directly from this complete diagonalization. Here trace class for a positive compact operator has the meaning used in the Gaussian spectrum: its eigenvalues are summable.

For clarity, write the rank-one operator \(|v\rangle \langle v|\) as \(f\mapsto v\langle v,f\rangle \), with the inner product linear in its second variable. Then \[T=\sum _I\alpha _I |b\phi _I\rangle \langle b\phi _I|.\] The trace norm is \(\|A\|_1=\operatorname {Tr}((A^*A)^{1/2})\); for a positive operator it equals its trace. Trace class means this norm is finite. Each summand is positive and its trace norm is \(\alpha _I\|b\phi _I\|_2^2\). Their sum is at most \(\|b\|_\infty ^2\sum _I\alpha _I\), so the series converges in trace norm and proves positivity, self-adjointness, compactness and the trace bound. For \(f\ne 0\), bounded invertibility of \(M_b\) gives \(bf\ne 0\); hence \[\langle f,Tf\rangle =\langle bf,C_kbf\rangle >0.\] This proves injectivity as well.

Under a simultaneous gauge change of \(U,U'\), \[(h\cdot U)_e(h\cdot U')_e^{-1} =h_{t(e)}(U_eU_e'^{-1})h_{t(e)}^{-1}.\] Centrality of \(k\) and invariance of product Haar measure show \(C_kR_h=R_hC_k\). Gauge invariance of \(b\) gives the same commutation for \(M_b\) and \(T\).

The coefficient calculation is a direct application of Peter–Weyl orthogonality. TauCeti's TauCeti.peterWeylBlockAveraging_eq_starProjection identifies character averaging with the corresponding orthogonal block projection. Its kernel is \(d_\rho \overline {\chi _\rho }\) and its matrix-coefficient convention is the conjugated one explained in Theorem [fdip-007C]. Reindexing by dual representations aligns the conventions; reality and inversion symmetry give \(a_{\overline \rho }=a_\rho \). This correspondence concerns the block projection, while the positive weighted sum and spatial multipliers are proved above.

The factorization proof follows the idea of [luscher1977transfer, Proposition 1, p. 288], under the explicit pure-gauge hypotheses of Definition [fdip-007Z].