Haar/Radon is the complete earlier HR proof cone; H1 has no scalar Plancherel premise.
Fourier completion and the character topology
For a locally compact Hausdorff abelian group, the full group C* algebra is the algebra of continuous functions vanishing at infinity on its character group. We prove this identification and show that its spectrum topology is exactly uniform convergence of characters on compact sets. The proof uses the topology, Haar integration and representation results below. The exposition is dedicated under CC0.
The 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.