Character Groups and Elementary LCA Duals — Examples

The composite $\phi :=\chi \circ \epsilon \circ p:\mathbb{R}\to \mathbb{T}$ is a continuous group homomorphism: $p$ and $\epsilon$ are continuous homomorphisms by [F1], $\chi$ is one by hypothesis, and the composite of homomorphisms is a homomorphism; by [F2] there is a unique $\xi \in \mathbb{R}$ with $\phi (t)=\exp (2\pi i\xi t)$ for all real $t$.

The parameter $\xi$ is an integer: for every integer $k$ one has $\epsilon ([k])=\exp (2\pi ik)=1$ by [F1] and [F3] (since $\mathrm{\mid }\exp (2\pi ik)-1{\mathrm{\mid }}^2=4{\sin }^2(\pi k)=0$), so $\phi (k)=\chi (1)=1$; with $\phi (k)=\exp (2\pi i\xi k)$ this gives $\sin (\pi \xi k)=0$ for every integer $k$, in particular for $k=1$, and so $\xi \in \mathbb{Z}$ by [F3].

Consequently $\chi (z)=z^{\xi }$ for every $z\in \mathbb{T}$: write $z=\epsilon ([t])=\exp (2\pi it)$ for some real $t$, which is possible because $\epsilon$ is surjective by [F1]; then $\chi (z)=\phi (t)=\exp (2\pi i\xi t)=(\exp (2\pi it){)}^{\xi }=z^{\xi }$ by step 1.1, step 2.1 and the power laws of [F4].

Distinct integers give distinct characters: if $z^n=z^m$ for all $z\in \mathbb{T}$ with $n\mathrm{\ne }m$, then evaluating at $z=\epsilon ([t])$ gives $\exp (2\pi i(n-m)t)=1$ for all real $t$, which fails for $t=1\mathrm{/}(2\mathrm{\mid }n-m\mathrm{\mid })$ by [F3], since then $\sin (\pi \mathrm{/}2)\mathrm{\ne }0$; hence $n=m$. Each $z\mapsto z^n$ is a continuous endomorphism of $\mathbb{T}$ by [F4].

The map $n\mapsto (z\mapsto z^n)$ is a bijective homomorphism from the discrete group $\mathbb{Z}$ onto the dual: it is a homomorphism by the power laws $z^{n+m}=z^n{z}^m$ of [F4], injective by step 4.1, and surjective by steps 1.1, 2.1 and 3.1.

It is a homeomorphism: $\mathbb{Z}$ is discrete by [F5] and the dual of the compact group $\mathbb{T}$ is discrete by [F5], so a bijection between discrete spaces is a homeomorphism; hence $\mathbb{Z}\cong \widehat{\mathbb{T}}$ as topological groups, and composing with the isomorphism $\mathbb{T}\cong \mathbb{R}\mathrm{/}\mathbb{Z}$ gives the dual of the published circle.

For each $z\in \mathbb{T}$ the map ${\gamma }_z:\mathbb{Z}\to \mathbb{T}$, ${\gamma }_z(n)=z^n$, is a character: it is a homomorphism by the power laws ${\gamma }_z(m+n)=z^{m+n}=z^m{z}^n$ of [F2], and it is continuous because for every open $V\subseteq \mathbb{T}$, ${\gamma }_z^{-1}[V]$ is a subset of the discrete source $\mathbb{Z}$, hence open; at each point mapped into $V$ it is the source neighbourhood required by [F2].

The inverse $\gamma \mapsto \gamma (1)$ is continuous: it is the restriction to the subspace $\widehat{\mathbb{Z}}$ of the projection ${\pi }_1:{\mathbb{T}}^{\mathbb{Z}}\to \mathbb{T}$, which is continuous for the product topology by [F3], and a restriction of a continuous map to a subspace is continuous by the characteristic property of the subspace topology [F1].

$\mathrm{\Psi }$ is a bijective group homomorphism: it is a homomorphism because ${\gamma }_{zw}(n)=(zw{)}^n=z^n{w}^n=({\gamma }_z{\gamma }_w)(n)$ by [F2]; it is injective because ${\gamma }_z(1)=z$ recovers $z$; and it is surjective because a homomorphism $\gamma$ determines $z:=\gamma (1)$ and then $\gamma (n)=z^n$ for $n\ge 0$ by induction and $\gamma (n)=\gamma (-n{)}^{-1}=z^n$ for $n<0$ by the power laws of [F2], so $\gamma ={\gamma }_z$.

$\mathrm{\Psi }$ is continuous: the codomain carries the subspace topology from ${\mathbb{T}}^{\mathbb{Z}}$ by [F1], so by the characteristic property of the subspace it suffices that $z\mapsto (z^n{)}_{n\in \mathbb{Z}}$ is continuous into ${\mathbb{T}}^{\mathbb{Z}}$, and by [F3] it suffices that each component $z\mapsto z^n$ is continuous, which holds because $\mathbb{T}$ is a topological group by [F3].

By steps 1.1, 1.2, 2.1 and 2.2 the map $\mathrm{\Psi }$ is a continuous bijective homomorphism with continuous inverse, hence an isomorphism of topological groups $\mathbb{T}\to \widehat{\mathbb{Z}}$ by [F4]; composing with the topological group isomorphism $\mathbb{R}\mathrm{/}\mathbb{Z}\to \mathbb{T}$ gives $\widehat{\mathbb{Z}}\cong \mathbb{R}\mathrm{/}\mathbb{Z}$.

Let $\chi$ be a continuous homomorphism of $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ into $\mathbb{T}$ and put $\zeta :=\chi ([1])$. Then ${\zeta }^N=\chi ([1]{)}^N=\chi (N[1])=\chi ([0])=1$ by [F2] and the power laws of [F4], so $\zeta$ is an $N$-th root of unity and by [F3] there is $k\in \mathbb{Z}$ with $\zeta =\exp (2\pi ik\mathrm{/}N)$; then $\chi ([m])=\chi ([1]{)}^m=\exp (2\pi ikm\mathrm{/}N)$ for every $m$ by [F2] and the power laws.

The integer $k$ is unique modulo $N$, and every $k$ defines a character: if $\exp (2\pi ikm\mathrm{/}N)=\exp (2\pi i{k}^{\mathrm{\prime }}m\mathrm{/}N)$ for all $m$, then for $m=1$ the congruence $k\equiv k^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}N)$ follows from [F3]; conversely for fixed $k$ the formula ${\chi }_k([m]):=\exp (2\pi ikm\mathrm{/}N)$ is well defined by [F3] and [F2], is a homomorphism because $\exp (2\pi ik(m+m^{\mathrm{\prime }})\mathrm{/}N)=\exp (2\pi ikm\mathrm{/}N)\exp (2\pi ik{m}^{\mathrm{\prime }}\mathrm{/}N)$, and is continuous because $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ is finite and discrete by [F1].

The map $k\mapsto {\chi }_k$ from $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ to the dual is a bijective homomorphism: ${\chi }_{k+k^{\mathrm{\prime }}}={\chi }_k{\chi }_{k^{\mathrm{\prime }}}$ by the addition formula, injectivity is step 2.1, and surjectivity is step 1.1 combined with the uniqueness in step 2.1.

It is a homeomorphism: $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ is finite and discrete by [F1], its dual is discrete by [F1] because the finite discrete group is compact, and any bijection between discrete spaces is a homeomorphism by [F4]; hence $k\mapsto {\chi }_k$ is an isomorphism of topological groups.

For each $j<n$ the map ${\psi }_j(t):=\phi (t{e}_j)$ is a continuous group homomorphism $\mathbb{R}\to \mathbb{T}$, so by [F1] there is a unique ${\xi }_j\in \mathbb{R}$ with ${\psi }_j(t)=\exp (2\pi i{\xi }_{j}t)$. Define $\xi \in {\mathbb{R}}^n$ by $\xi (j):={\xi }_j$ for $j<n$. Consequently $\phi (x)={\prod }_{j<n}{\psi }_j(x_j)={\prod }_{j<n}\exp (2\pi i{\xi }_j{x}_j)=\exp \left(2\pi i{\sum }_{j<n}{\xi }_j{x}_j\right)=\exp (2\pi i\xi \cdot x)$ by [F2] and [F3].

The vector $\xi$ is unique: if $\exp (2\pi i\xi \cdot x)=\exp (2\pi i{\xi }^{\mathrm{\prime }}\cdot x)$ for all $x$, then restricting to $x=t{e}_j$ for $j<n$ shows ${\psi }_j(t)=\exp (2\pi i{\xi }_j^{\mathrm{\prime }}t)$, so the uniqueness in [F1] gives ${\xi }_j={\xi }_j^{\mathrm{\prime }}$ for every $j<n$ and hence $\xi ={\xi }^{\mathrm{\prime }}$.

Each ${\phi }_{\xi }$ is a continuous character: it is the product over $j<n$ of the one-dimensional characters $x\mapsto \exp (2\pi i{\xi }_j{x}_j)$ composed with the continuous coordinate projections, so it is continuous, and the addition formula gives its homomorphism law. The map $\xi \mapsto {\phi }_{\xi }$ is a bijective homomorphism: it is a homomorphism by [F3], injective by step 2.1, and surjective by step 1.1.

It is continuous at the identity: let $K\subseteq {\mathbb{R}}^n$ be compact and $W\subseteq \mathbb{T}$ open with $1\in W$. If $K=\mathrm{\varnothing }$, $S(K,W)$ is the whole dual and there is nothing to check; otherwise choose $\rho >0$ with $B(1,\rho )\subseteq W$, and by continuity of $u\mapsto \exp (2\pi iu)$ at $0$ by [F3] choose ${\delta }^{\mathrm{\prime }}>0$ with $\mathrm{\mid }\exp (2\pi iu)-1\mathrm{\mid }<\rho$ whenever $\mathrm{\mid }u\mathrm{\mid }<{\delta }^{\mathrm{\prime }}$. With $R:={\max }_{x\in K}\parallel x{\parallel }_2$ by [F6], put $\delta :={\delta }^{\mathrm{\prime }}\mathrm{/}(R+1)$; if $\parallel \eta {\parallel }_2<\delta$ and $x\in K$, then $\mathrm{\mid }\eta \cdot x\mathrm{\mid }\le \parallel \eta {\parallel }_2\parallel x{\parallel }_2<{\delta }^{\mathrm{\prime }}$, so $\mathrm{\mid }{\phi }_{\eta }(x)-1\mathrm{\mid }<\rho$ and ${\phi }_{\eta }[K]\subseteq B(1,\rho )\subseteq W$, that is ${\phi }_{\eta }\in S(K,W)$. Hence the map is continuous at the identity character and, being a homomorphism, continuous everywhere by [F5].

It has continuous inverse: given $\epsilon >0$, let $R:=1\mathrm{/}(2\epsilon )$ and let $K$ be the closed ball of radius $R$, compact by [F4]. If ${\phi }_{\eta }\in S(K,B(1,1))$ and $\parallel \eta {\parallel }_2\ge \epsilon$, then $x_0:=\eta \mathrm{/}(2\parallel \eta {\parallel }_2^2)$ satisfies $\parallel x_0{\parallel }_2=1\mathrm{/}(2\parallel \eta {\parallel }_2)\le R$, so $x_0\in K$, and $\eta \cdot x_0=1\mathrm{/}2$, whence $\mathrm{\mid }{\phi }_{\eta }(x_0)-1\mathrm{\mid }=\mathrm{\mid }\exp (\pi i)-1\mathrm{\mid }=2$, contradicting ${\phi }_{\eta }\in S(K,B(1,1))$; therefore $\parallel \eta {\parallel }_2<\epsilon$. So the inverse map sends the identity neighbourhood $S(K,B(1,1))$ into the ball of radius $\epsilon$, and it is continuous at the identity, hence everywhere by [F5].

By steps 3.1, 3.2 and 3.3 the map $\xi \mapsto {\phi }_{\xi }$ is a continuous bijective homomorphism with continuous inverse, hence an isomorphism of topological groups ${\mathbb{R}}^n\to \widehat{{\mathbb{R}}^n}$, and step 1.1 with step 2.1 is the stated classification with its uniqueness.