A polarization--Fourier criterion for regular GNS cocycle representations

Definitions and mathematical statement

Let $\Gamma$ be a countable discrete group and let $\psi:\Gamma\to{\mathbb{R}}$ be normalized conditionally negative definite: $\psi(e)=0$, $\psi(g^{-1})=\psi(g)$, and \[ \sum_{i,j}\overline{c_i}c_j\psi(g_i^{-1}g_j)\le0 \quad\text{whenever }\sum_i c_i=0. \] Its minimal cocycle construction gives a Hilbert space $H_\psi$, an orthogonal representation (or its unitary complexification) $\pi_\psi$, and a cocycle $b$ with \[ b(gh)=b(g)+\pi_\psi(g)b(h),\qquad \|b(g)\|^2=\psi(g),\qquad H_\psi=\overline{\operatorname{span}}\{b(g):g\in\Gamma\}. \] The associated affine action is $\xi\mapsto\pi_\psi(g)\xi+b(g)$. Its space is not automatically $\ell^2(\Gamma)$. The source asks when its representation is a multiple of the left regular representation and what direct conditions on $\psi$ ensure this.

The aid's precise criterion addresses containment in an amplification: an isometric intertwiner \[ H_\psi\longrightarrow\ell^2(\Gamma)\otimes K, \qquad \pi_\psi\hookrightarrow\lambda_\Gamma\otimes I_K, \] where $(\lambda_\Gamma(s)\xi)(t)=\xi(s^{-1}t)$. This is weaker than unitary equivalence to an entire amplification, if "multiple" is meant literally. Let $A(\Gamma)$ be the Fourier algebra, consisting of sums of regular coefficients $u(s)=\sum_r\langle\lambda_\Gamma(s)\xi_r,\eta_r\rangle$ with $\sum_r\|\xi_r\|_2\|\eta_r\|_2<\infty$. The reported necessary-and-sufficient condition is \[ C_{g,h}\in A(\Gamma)\quad\text{for every }g,h\in\Gamma, \] where the directly computable polarization is \[ C_{g,h}(s)=\frac{\psi(sg)-\psi(s) -\psi(g^{-1}s^{-1}h)+\psi(s^{-1}h)}2 =\langle\pi_\psi(s)b(g),b(h)\rangle. \] Thus the test concerns coefficients of the minimal linear part, not a claimed embedding of affine translations themselves as a regular linear action.

For $\Gamma={\mathbb{Z}}$, the vector $b(1)$ is cyclic and the sequence \[ q(r)=\tfrac12[\psi(r+1)-2\psi(r)+\psi(r-1)] \] is positive definite. Let its Herglotz measure $\nu$ on the unit circle satisfy $q(r)=\int z^r\,\nu({\,\mathrm{d}} z)$. The aid identifies this as the cyclic spectral measure of $\pi_\psi$, so \[ \pi_\psi\hookrightarrow\lambda_{{\mathbb{Z}}}\otimes I_K \quad\Longleftrightarrow\quad \nu\ll m_{\mathbb T}, \] where $m_{\mathbb T}$ is normalized Haar measure. Absolute continuity is a containment criterion; support and spectral multiplicity remain relevant to literal equality with a full multiple.

For $\psi(n)=|n|$, $q(r)=\mathbf1_{\{r=0\}}$ and $\nu$ is Haar, giving the regular representation. For $\psi(n)=n^2$, $q(r)=1$ and $\nu=\delta_1$, giving the trivial one-dimensional representation. Both functions are proper, and on the amenable group ${\mathbb{Z}}$ the trivial representation is weakly contained in the regular one. It is nevertheless not an actual subrepresentation of a regular amplification. Hence properness and weak containment do not suffice for the aid's stronger regular-containment conclusion.

Short English statement

Test the minimal cocycle representation of a negative-definite function by explicit Fourier-algebra polarization coefficients. On ${\mathbb{Z}}$, the second difference recovers a spectral measure whose absolute continuity characterizes containment in a regular amplification; this is distinct from weak containment or literal equality with a full multiple.

Sources

• American Institute of Mathematics workshop list Free analysis, Question 33, section 0.11, "Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)," https://aimath.org/WWN/freeanalysis/freeanalysis.pdf.
• Local UnsolvedMath registry, permanent ID 20002693, source code catalogue entry, canonical location aim-probability-notes.json, notes[134], record dated 14 August 2026. The original literature field is empty.
• Supplied research attempt 1: partial result, problem status at run partially solved, rigor labelled proved, novelty confidence low. Its polarization/Fourier criterion and cyclic special case are machine-generated research aid, not peer reviewed; independently verify before citation. No separate authoritative bibliographic reference for these refinements is supplied.