The abelian group C star algebra recovers Pontryagin duality

$C^{\ast }(G)$ is commutative: $L^1(G)$ is commutative and its canonical image is dense in $C^{\ast }(G)$ by [F1], and commutativity passes to norm limits.

The Gelfand characters of $C^{\ast }(G)$ are in bijection with the continuous unitary characters of $G$, through ${\chi }_{\gamma }(f)={\int }_{G}f(g)\gamma (g)dg$ for $f\in L^1(G)$, extended by continuity to $C^{\ast }(G)$. Indeed, the unitary character $\gamma$ is a one-dimensional unitary representation, so [F3] gives its unique nondegenerate star-representation ${\chi }_{\gamma }$ of $C^{\ast }(G)$, whose restriction to $L^1(G)$ is the displayed integral. Equivalently, $\mathrm{\mid }{\chi }_{\gamma }(f)\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}$ because its integrated operator occurs in the universal supremum; this is the bound that gives an extension in the full C*-norm. conversely a character $\chi$ of $C^{\ast }(G)$ preserves the involution by [F3], hence is a one-dimensional nondegenerate star-representation of $C^{\ast }(G)$, which corresponds to a unitary representation of $G$ by [F3]; being nonzero and one-dimensional it is irreducible by [F2], so it is a continuous character $\gamma$ and $\chi ={\chi }_{\gamma }$ by density.

The Gelfand topology on the character space corresponds to the compact-open topology on $\widehat{G}$: pointwise convergence on $C^{\ast }(G)$ is equivalent to pointwise convergence on the dense subspace $L^1(G)$ by uniform boundedness of the character functionals, which is weak-* convergence of the functions ${\gamma }_i$ against $L^1(G)$; by Raikov [F4] (all $\gamma \in P_1(G)$) this is exactly uniform convergence on compact subsets, that is, convergence in $\widehat{G}$.

The Fell topology on $\widehat{G}$ agrees with compact-uniform convergence. For a character $\gamma$, all finite sums of diagonal coefficients are exactly $c\gamma$ with $c\ge 0$ (The Fell topology on the unitary dual). Given a compact $Q$ and $ϵ>0$, the Fell neighborhood testing $\gamma$ on $Q\cup \{e\}$ with tolerance $ϵ\mathrm{/}2$ is contained in $\{{\gamma }^{\mathrm{\prime }}:{\sup }_Q\mathrm{\mid }\gamma -{\gamma }^{\mathrm{\prime }}\mathrm{\mid }<ϵ\}$: its witness $c^{\mathrm{\prime }}\ge 0$ satisfies $\mathrm{\mid }1-c^{\mathrm{\prime }}\mathrm{\mid }<ϵ\mathrm{/}2$ at $e$, hence ${\sup }_Q\mathrm{\mid }\gamma -{\gamma }^{\mathrm{\prime }}\mathrm{\mid }\le {\sup }_Q\mathrm{\mid }\gamma -c^{\mathrm{\prime }}{\gamma }^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{c}^{\mathrm{\prime }}-1\mathrm{\mid }<ϵ$. Conversely, for a displayed Fell neighborhood with tests $c_1\gamma ,\dots ,c_k\gamma$ on $Q$ and tolerance $ϵ$, the compact-uniform neighborhood ${\sup }_Q\mathrm{\mid }\gamma -{\gamma }^{\mathrm{\prime }}\mathrm{\mid }<ϵ\mathrm{/}(1+{\max }_i{c}_i)$ is contained in it, using witnesses $c_i{\gamma }^{\mathrm{\prime }}$; the empty test list needs no restriction. These two refinements at every center prove equality of the topologies, hence equivalence of convergence for arbitrary nets. The compact-open topology on $\widehat{G}$ is uniform convergence on compacta (The Pontryagin dual with the compact-open topology).

By [F5] the commutative C*-algebra $C^{\ast }(G)$ is isometrically $\ast$-isomorphic to $C_0(\mathrm{\Delta }(C^{\ast }(G)))$, and by steps 1.2, 2.1 and 3.1 the character space with the Gelfand topology is homeomorphic to $\widehat{G}$ with the compact-open topology, which by step 3.1 is also the Fell topology; hence $C^{\ast }(G)\cong C_0(\widehat{G})$.

For $G=\mathbb{Z}$ every character is determined by its value at $1$, $\gamma (n)=\gamma (1{)}^n$, and $z\mapsto (n\mapsto z^n)$ is a homeomorphism $\mathbb{T}\to \widehat{\mathbb{Z}}$ for the compact-open topology, because compact subsets of $\mathbb{Z}$ are finite and pointwise convergence is convergence of the value at $1$; hence $C^{\ast }(\mathbb{Z})\cong C(\mathbb{T})$ by step 4.1. For $G=\mathbb{R}$ the continuous characters are exactly $x\mapsto e^{itx}$, $t\in \mathbb{R}$, by Continuous characters of the real line are exponentials, and $t\mapsto e_t:=e^{it(\cdot )}$ is a homeomorphism onto $\widehat{\mathbb{R}}$: it is continuous since ${\sup }_{x\in K}\mathrm{\mid }{e}^{i{t}_{k}x}-e^{itx}\mathrm{\mid }\le \mathrm{\mid }{t}_k-t\mathrm{\mid }{\sup }_{x\in K}\mathrm{\mid }x\mathrm{\mid }$ on compact $K$, and if $t_k\to ̸t$ then for some $\delta >0$ and a subnet $\mathrm{\mid }{t}_k-t\mathrm{\mid }\ge \delta$, and each compact interval $[0,\pi \mathrm{/}\delta ]$ contains $x_k=\pi \mathrm{/}\mathrm{\mid }{t}_k-t\mathrm{\mid }$ with $\mathrm{\mid }{e}^{i{t}_k{x}_k}-e^{it{x}_k}\mathrm{\mid }=\mathrm{\mid }{e}^{\pm i\pi }-1\mathrm{\mid }=2$, so compact-uniform convergence fails; thus by step 4.1 $C^{\ast }(\mathbb{R})\cong C_0(\mathbb{R})$.

The Axiom of Choice is inherited from Schur's lemma, Raikov's theorem and Gelfand–Naimark; no further choice is used in the identifications (The Axiom of Choice).