LCA group algebra and character-space results recorded externally

Assume the Axiom of Choice (The Axiom of Choice). Let $G$ be a locally compact Hausdorff abelian group, written additively, and fix a nonzero Haar measure $m$: a translation-invariant regular Borel measure finite on compact sets. Put $A=L^1(G,m;\mathbb{C})$, with functions identified when equal almost everywhere. The following results are recorded from Williams, Example 3.10; they are not proved in this library here.

1. The formulas $(f\ast g)(s)={\int }_{G}f(t)g(s-t)dm(t),f^{\ast }(s)=\overline{f(-s)}$ define, in the first formula almost everywhere and independently of representatives, a commutative Banach star algebra on $A$, with $\mathrm{\parallel }f\ast g{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1$ and $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_1=\mathrm{\parallel }f{\mathrm{\parallel }}_1$. The involution is conjugate-linear, involutive, and reverses products. The algebra $A$ has an identity if and only if $G$ is discrete.

2. Suppose $G$ is nondiscrete. Define the scalar unitization $B=\mathbb{C}\oplus A$ by

   $$(z,f)(v,g)=(zv,zg+vf+f\ast g)\mathrm{.}$$

   Let $\widehat{G}$ be the continuous homomorphisms from $G$ to $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$, with uniform convergence on compact subsets of $G$. Every character of $B$, in the sense of Character and maximal ideal space, is uniquely one of $h_w(z,f)=z+{\int }_{G}f(t)w(t)dm(t)(w\in \widehat{G}),q(z,f)=z\mathrm{.}$ With the pointwise-evaluation topology, the map $\widehat{G}\cup \{\mathrm{\infty }\}\to \mathrm{\Delta }(B)$ sending $w$ to $h_w$ and $\mathrm{\infty }$ to $q$ is a homeomorphism from the one-point compactification of the compact-open dual. This includes the local compactness of $\widehat{G}$ and the asserted agreement of topologies.

This is a recorded external prerequisite, not a local proof. In particular, no global sigma-finiteness of Haar measure and no general product-Borel identification is silently assumed. Williams's text extraction loses the conjugation bar in the displayed involution; the conjugate-reflection above is the mathematically correct star operation.