Original independently reviewed H1 reconstruction by GPT-6.1 Sol (OpenAI), Ultra. Haar/Radon is now the complete actual earlier HR proof cone; H1 has no scalar Plancherel premise.
Fourier completion and the character topology
Original reviewed H1 proof, now placed after the independently proved topology and Haar/Radon foundations and L24/L25. It identifies the universal Fourier completion and compact-open character topology without a scalar Plancherel or biduality premise. Original CC0 expression retained.
The actual earlier inputs are H0: compact topology and Hilbert tensor, HR-03, HR-06, HR-07, HR-09, CF-6, CF-7, CF-9, and L24 convolution, L24 recovery, L24 translation continuity, L24 universal completion.
H1. The full Fourier completion directly from CF and L24
Let \(D=C^*(G)\), constructed in L24. Abelian convolution makes \(D\) commutative. For \(A=\mathbb C\), the left regular representation on the nonzero Haar \(L^2(G)\) space shows that \(D\ne0\). Its forced unitization \(D^+\) has compact character space \(X\), and CF Section 6 identifies it isometrically with \(C(X)\). The scalar quotient \(D^+\to\mathbb C\) is a character \(q\in X\). Its kernel \(D\) corresponds exactly to functions vanishing at \(q\). Restriction gives \[ D\cong C_0(X\setminus\{q\}). \tag{H1.1} \] For clarity, a continuous function on \(X\) vanishing at \(q\) has compact level sets away from \(q\). Conversely a function vanishing at infinity on \(X\setminus\{q\}\), extended by zero at \(q\), is continuous there because each positive level set is compact and hence closed in \(X\). This proves the asserted \(C_0\) identification.
A character of \(D\) is a nonzero one-dimensional star representation. Its restriction to \(L^1(G)\) is nondegenerate, since density prevents it from vanishing identically. L24's vector-domain recovery theorem gives a unique continuous unitary character \(s\mapsto\overline{\chi(s)}\), with integrated value \[ \psi_\chi(f)=\widehat f(\chi) =\int_G f(s)\overline{\chi(s)}\,dm(s). \tag{H1.2} \] Conversely every continuous group character gives this nonzero algebra character: integrate its unitary action, and use the shrinking mass-one bumps from L24 to see nonvanishing. These are inverse correspondences.
The topology on \(X\setminus\{q\}\) is exactly compact-open convergence of group characters. Compact-open convergence implies convergence in (H1.2) for \(f\in C_c(G)\), bounded by \(\|f\|_1\) times the uniform character error on its support. Approximate any \(L^1\) function by \(C_c\); character norms are one. Density in \(D\) extends convergence to every element of \(D\).
Conversely suppose \(\psi_i\to\psi\ne0\) on \(D\). Choose \(f_0\in L^1(G)\) with \(\psi(f_0)\ne0\). For a compact \(C\subset G\), the set \(\{L_sf_0:s\in C\}\) is norm compact by L24's translation continuity. A finite norm net and the uniform functional bound show \[ \sup_{s\in C}|\psi_i(L_sf_0)-\psi(L_sf_0)|\longrightarrow0. \] The identity \(\psi_\chi(L_sf_0)=\overline{\chi(s)}\psi_\chi(f_0)\), and denominators bounded away from zero eventually, give uniform convergence of \(\chi_i\) on \(C\). This is a net argument.
Consequently \(\widehat G\), with pointwise character multiplication, is LCH and the Fourier map extends to an isometric star isomorphism \[ C^*(G)\cong C_0(\widehat G),\qquad \|f\|_u=\|\widehat f\|_\infty. \tag{H1.3} \] The group operations are continuous for compact-open convergence by the scalar inequalities for products and conjugates of unit-modulus functions. Since \(C_c(G)\) is \(L^1\)-dense and the universal norm is at most the \(L^1\) norm, its Fourier image is uniformly dense in \(C_0(\widehat G)\). Fourier injectivity follows here from the definiteness of the universal norm already proved in L24; it is not imported from biduality.