Added some harmonic analysis.
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Bokuan Li
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\section{Pontryagin Duality}
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\label{section:lacg-dual}
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\begin{definition}[Unitary Character]
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\label{definition:unitary-character}
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Let $G$ be a locally compact Abelian group and $\phi: G \to \mathbb{T}$, then $\phi$ is a \textbf{unitary character} of $G$ if $\phi$ is a continuous homomorphism.
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The set $\wh G$ of all unitary characters of $G$, equipped with pointwise operations and the topology of compact convergence, is the \textbf{dual group} of $G$.
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For any $\phi \in \wh G$ and $x \in G$, denote $\phi(x) = \dpn{x, \phi}{G}$.
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\end{definition}
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\begin{theorem}
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\label{theorem:unitary-character-isomorphism}
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Let $G$ be a locally compact Abelian group and $\phi \in \wh G$. Under the identification that $\mathbb{T} = \partial B_\complex(0, 1)$ and $B(\complex) = \complex$, the representation of $L^1(G; \complex)$ determined by $\phi$ takes the form
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\[
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\pi_\phi: L^1(G; \complex) \to \complex \quad \dpn{f, \pi_\phi}{L^1(G; \complex)} = \int_G f(x)\dpn{x, \phi}{G} dx
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\]
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and the mapping $\wh G \to \Omega(L^1(G; \complex))$ defined by $\phi \mapsto \pi_\phi$ is a homeomorphism. Moreover, for any $f \in L^1(G; \complex)$ with $\pi_\phi(f) \ne 0$, $\dpn{x, \phi}{G} = \pi_\phi(L_xf)/\pi_\phi(f)$ for all $x \in G$.
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\end{theorem}
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\begin{proof}[Proof, {{\cite[Theorem 4.3]{FollandHarmonic}}}. ]
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(Continuous): Let $\seqf{f_j} \subset L^1(G; \complex)$ and $\eps > 0$. By inner regularity of the Haar measure, there exists $K \subset G$ compact such that $\int_{G\setminus K}|f_j| < \eps$ for all $1 \le j \le n$. In which case, for any $\phi, \psi \in \wh G$ with $\sup_{x \in K}|\phi(x) - \psi(x)| < \eps$,
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\begin{align}
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\abs{\dpn{f_j, \phi - \psi}{L^1(G; \complex)}} &\le \int_{K}|f_j \cdot (\phi - \psi)| + 2\int_{G \setminus K}|f_j| \\
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&\le \eps \norm{f_j}_{L^1(G; \complex)} + 2\eps
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\end{align}
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(Injective): By (3) of \autoref{definition:representation-l1-group-algebra}, $\phi$ is uniquely determined by $\pi_\phi$.
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(Surjective): Let $\pi \in \Omega(L^1(G; \complex))$. Since $G$ is Abelian, $L^1(G; \complex)$ is also Abelian by \autoref{proposition:convolution-measures-property}. As such, for any $x \in G$ and $f, g \in L^1(G; \complex)$, $f * (L_xg) = L_x(f * g) = (L_xf) * g$ and $\pi(L_xf)\pi(g) = \pi(f)\pi(L_xg)$. In particular, if $\pi(f), \pi(g) \ne 0$, then $\pi(L_xf)/\pi(f) = \pi(L_xg)/\pi(g)$. Let\footnote{It may be tempting to construct $\phi$ directly as an element of $L^\infty$. However, this requires either reducing to the $\sigma$-compact case, or expanding the measurable sets of the Haar measure.}
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\[
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\phi: G \to \complex \quad x \mapsto \frac{\pi(L_xf)}{\pi(f)}
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\]
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then $\phi$ is defined independently of the choice of $f$. By \hyperref[continuity of translation]{proposition:haar-translation}, $\phi$ is a continuous homomorphism from $G$ to $\complex \setminus \bracsn{0}$. Moreover, by \autoref{proposition:multiplicative-less-unit}, $\norm{\pi}_{L^1(G; \complex)^*} \le 1$, and $\phi$ is bounded. Given that $\phi$ is a homomorphism, $\phi(G) \subset \partial B_\complex(0, 1)$, and $\phi \in \wh G$.
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Since $\pi \ne 0$, there exists $\psi \in L^1(G; \complex)$ with $\pi(\psi) = 1$. Using \hyperref[Riemann sums]{lemma:convolution-distribute-functional},
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\begin{align*}
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\pi(f) &= \pi(f)\pi(\psi) = \pi(f * \psi) = \int_G f(y)\pi(L_y\psi) dy \\
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&= \int_G f(y)\frac{\pi(L_y\psi)}{\pi(\psi)}dy = \int_G f(y) \dpn{y, \phi}{G} dy
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\end{align*}
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(Embedding): By \autoref{proposition:multiplicative-less-unit}, every multiplicative linear functional on $L^1(G; \complex)$ is contractive. Hence $\Omega(L^1(G; \complex))$ is equicontinuous. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the topology of compact convergence on $\Omega(L^1(G; \complex))$ coincides with the weak* topology.
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Let $\phi, \psi \in \wh G$. Since $\phi \ne 0$, there exists $f \in L^1(G; \complex)$ such that $\pi_\phi(f) \ne 0$. If $|\pi_\phi(f) - \pi_\psi(f)|$ is sufficiently small, then $\pi_\psi(f) \ne 0$ as well. Let $K \subset G$ be compact, then $\bracsn{L_xf|x \in K}$ is compact in $L^1(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}. In which case,
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\[
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\sup_{x \in K}|(\phi - \psi)(x)| = \sup_{x \in K}\abs{\frac{\pi_\phi(L_xf)}{\pi_\phi(f)} - \frac{\pi_\psi(L_xf)}{\pi_\psi(f)}}
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\]
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Thus for any $\eps > 0$, $\bracsn{\pi_\psi|\psi \in \wh G, \sup_{x \in K}|\phi(x) - \psi(x)| \le \eps}$ is a neighbourhood of $\pi_\phi$ in the weak* topology, and the mapping $\psi \mapsto \pi_\psi$ is an embedding.
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\end{proof}
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\begin{proposition}
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\label{proposition:unitary-character-orthonormal}
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Let $G$ be a compact Abelian group with $|G| = 1$, then $\wh G$ is an orthonormal set in $L^2(G; \complex)$.
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\end{proposition}
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\begin{proof}[Proof, {{\cite[Proposition 4.4]{FollandHarmonic}}}. ]
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Let $\phi, \psi \in \wh G$ be distinct, then there exists $x_0 \in G$ with $\dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}^{-1} \ne 1$. In which case,
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\begin{align*}
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\int_G \phi \ol{\psi} &= \int_G \dpn{x, \phi}{G}\dpn{x, \psi}{G}^{-1} dx \\
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&= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}\int_G \dpn{x - x_0, \phi}{G}\dpn{x - x_0, \psi}{G} dx \\
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&= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G} \int \phi \ol{\psi}
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\end{align*}
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so $\phi \perp \psi$.
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\end{proof}
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\begin{proposition}
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\label{proposition:pontryagin-compact-discrete}
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Let $G$ be a locally compact Abelian group. If $G$ is discrete, then $\wh G$ is compact. If $G$ is compact, then $\wh G$ is discrete.
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\end{proposition}
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\begin{proof}[Proof, {{\cite[Proposition 4.5]{FollandHarmonic}}}. ]
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If $G$ is discrete, then $L^1(G; \complex)$ is a unital Banach algebra. By \autoref{definition:multiplicative-linear-functional-space}, $\wh G \iso \Omega(L^1(G; \complex))$ is compact.
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If $G$ is compact, then $\one \in L^1(G; \complex)$, and $\bracsn{f \in L^\infty(G; \complex)|\int f| > 1/2}$ is weak*-open. By \autoref{proposition:unitary-character-orthonormal}, for any $\phi \in \wh G \setminus \bracsn{\one}$, $\int \phi = 0$. Therefore $\bracsn{1}$ is open in $\wh G$, and $\wh G$ is discrete.
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\end{proof}
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\begin{proposition}
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\label{proposition:pontryagin-product}
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Let $\seqi{G}$ be locally compact Abelian groups where $G_i$ is compact for all but finitely many $i \in I$, then
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\[
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\braks{\prod_{i \in I}G_i}^\wedge \iso \bigoplus_{i \in I}\wh G_i
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\]
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\end{proposition}
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\begin{theorem}
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\label{theorem:pontryagin-dual-usual-suspects}
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\begin{enumerate}
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\item $\wh \real \iso \real$ with dual pairing $\dpn{x, \xi}{\real} = e^{2\pi i \xi x}$.
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\item $\wh \mathbb{T} \iso \integer$ with dual pairing $\dpn{x, n}{\mathbb{T}} = x^n$.
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\item $\wh \integer = \wh \mathbb{T}$ with dual pairing $\dpn{n, x}{\mathbb{T}} = x^n$.
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\end{enumerate}
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\end{theorem}
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\begin{proof}[Proof, {{\cite[Proposition 4.6]{FollandHarmonic}}}. ]
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(1): Let $\phi \in \wh \real$, then $\dpn{0, \phi}{\real} = 1$. Thus there exists $\alpha > 0$ with $A := \int_0^\alpha \dpn{x, \phi}{\real}dx \ne 0$. In which case,
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\[
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A \dpn{x, \phi}{G} = \int_0^\alpha \dpn{x + y, \phi}{\real} dy = \int_x^{x + \alpha} \dpn{y, \phi}{\real}dy
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\]
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so $\phi \in C^1(\real)$ and there exists $c \in \complex$ with $\phi' = c\phi$. Therefore $\dpn{x, \phi}{\real} = e^{cx}$ for all $x \in \real$.
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\end{proof}
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\chapter{Locally Compact Abelian Groups}
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\label{chap:lcag}
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\input{./dual.tex}
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@@ -306,6 +306,64 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
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and $f * \phi_U \to f$ in $L^p(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}.
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\end{proof}
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\begin{lemma}[Riemann Sums]
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\label{lemma:convolution-distribute-functional}
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Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$, then for any $f, g \in L^1(G; \complex)$,
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\[
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\dpn{f * g, \phi}{L^1(G; \complex)} = \int_G f(y)\dpn{L_yg, \phi}{L^1(G; \complex)} dy
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\]
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\end{lemma}
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\begin{proof}
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Let $K \subset G$ be compact and $\eps > 0$. By continuity of translation, there exists a partition $K = \bigsqcup_{j = 1}^n U_j$ of $K$ and $\seqf{y_j}$ such that:
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\begin{enumerate}
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\item For each $1 \le j \le n$, $y_j \in U_j$.
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\item For each $1 \le j \le n$ and $y \in U_j$,
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\[
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\normn{L_yg - L_{y_j}g}_{L^1(G; \complex)} \vee \normn{\dpn{L_yg, \phi}{L^1(G; \complex)} - \dpn{L_{y_j}g, \phi}{L^1(G; \complex)}} < \eps
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\]
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\end{enumerate}
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Since $g \in L^1(G; \complex)$, $\bracsn{g \ne 0}$ is $\sigma$-finite. By \hyperref[Tonelli's Theorem]{theorem:fubini-tonelli},
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\begin{align*}
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\norm{\one_K * g - \sum_{j = 1}^n|U_j| L_{y_j}g}_{L^1(G)} &= \int_G\abs{\one_K * g(x) - \sum_{j = 1}^n|U_j| L_{y_j}g(x)}dx \\
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&\le \iint_{G \times G}\abs{\one_K(y)L_yg(x) - \sum_{j = 1}^n\one_{U_j}(y) L_{y_j}g(x)} dy dx \\
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&\le \sum_{j = 1}^n \int_G\int_{U_j}|L_yg(x) - L_{y_j}g(x)|dydx \\
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&\le \sum_{j = 1}^n \eps |U_j| = \eps |K|
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\end{align*}
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Similarly,
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\[
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\abs{\int_G \one_K(y)\dpn{L_yg, \phi}{L^1(G; \complex)}dy - \sum_{j = 1}^n |U_j|\dpn{L_{y_j}g, \phi}{L^1(G; \complex)}} \le \eps |K|
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\]
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By linearity,
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\[
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\angles{\sum_{j = 1}^n |U_j|L_{y_j}g, \phi}_{L^1(G; \complex)} = \sum_{j = 1}^n |U_j| \dpn{L_{y_j}g, \phi}{L^1(G; \complex)}
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\]
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so
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\[
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\abs{\dpn{\one_K * g, \phi}{L^1(G; \complex)} - \int \one_K(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy} \le 2\eps |K| (1 \vee \norm{\phi}_{L^1(G; \complex)^*})
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\]
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As the above holds for all $\eps > 0$, $\dpn{\one_K * g, \phi}{L^1(G; \complex)} = \int_G\one_K(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy$.
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By inner regularity of the Haar measure,
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\[
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\dpn{\one_B * g, \phi}{L^1(G; \complex)} = \int_G\one_B(y)\dpn{L_y g, \phi}{L^1(G; \complex)}dy
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\]
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for all $B \in \cb_\complex$. By \hyperref[density of simple functions]{proposition:lp-simple-dense},
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\[
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\dpn{f * g, \phi}{L^1(G; \complex)} = \int_G f(y)\dpn{L_y g, \phi}{L^1(G; \complex)} dy
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\]
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for all $f \in L^1(G; \complex)$.
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\end{proof}
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\begin{theorem}
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\label{theorem:lcg-group-algebra-ideal}
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Let $G$ be a locally compact group and $J \subset L^1(G; \complex)$ be a closed subspace, then $J$ is a left ideal if and only if $L_xJ \subset J$ for all $x \in G$.
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@@ -331,3 +389,4 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
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\end{proof}
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