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Lemma 11.3.1 (Prodanov lemma). Let \( G \) be a topological abelian group, \( U \) an open set of \( G, f : U \rightarrow \mathbb{C} \) a continuous function, and \( M \) a convex closed set of \( \mathbb{C} \) . Let \( k \in {\mathbb{N}}_{ + } \) , \( {\chi }_{1},\ldots ,{\chi }_{k} \in {G}^{ * } \), and \( {c}_{1},\l... | Proof. Assume that \( {\chi }_{k} \in {G}^{ * } \) is not continuous; then it is not continuous at 0 . Consequently, there exists a net \( {\left\{ {x}_{\gamma }\right\} }_{\gamma \in A} \) in \( G \) such that \( {x}_{\gamma } \rightarrow 0 \) and \( {\left\{ {\chi }_{k}\left( {x}_{\gamma }\right) \right\} }_{\gamma \... | Yes |
Corollary 11.3.3. For every topological abelian group \( G, C\left( G\right) \cap \mathfrak{A}\left( {G}_{d}\right) = \mathfrak{A}\left( G\right) \) . | In other words, as far as continuous complex-valued functions are concerned, in the definition of \( \mathfrak{A}\left( G\right) \) it is irrelevant whether one approximates via (linear combinations of) continuous or discontinuous characters. | No |
Theorem 11.3.5 (Følner theorem). Let \( G \) be a topological abelian group. If \( k \in {\mathbb{N}}_{ + } \) and \( E \) is a subset of \( G \) such that \( k \) translates of \( E \) cover \( G \), then for every \( U \in {\mathcal{V}}_{G}\left( 0\right) \) there exist \( {\chi }_{1},\ldots ,{\chi }_{m} \in \widehat... | Proof. We can assume, without loss of generality, that \( U \) is open. By Følner lemma 11.2.5, there exist \( {\varphi }_{1},\ldots ,{\varphi }_{m} \in {G}^{ * } \) such that \( {U}_{G}\left( {{\varphi }_{1},\ldots ,{\varphi }_{m};\frac{\pi }{2}}\right) \subseteq {E}_{\left( 8\right) } \) ; our aim is to replace these... | No |
Corollary 11.3.6. Let \( G \) be a topological abelian group, \( g \in {\mathfrak{A}}_{0}\left( G\right) \), and \( M \) a convex closed set of \( \mathbb{C} \) . If \( c \in \mathbb{C} \) is such that \( g\left( x\right) + c \in M \) for every \( x \in G \), then \( c \in M \) . In particular, if \( \varepsilon \geq 0... | Proof. Assume first that \( g \in {\mathfrak{X}}_{0}\left( G\right) \) . Suppose that for every \( x \in G, g\left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{k}{c}_{j}{\chi }_{j}\left( x\right) \) for some \( {c}_{1},\ldots ,{c}_{k} \in \mathbb{C} \) and nonconstant \( {\chi }_{1},\ldots ,{\chi }_{k} \in \widehat{G} ... | Yes |
Corollary 11.3.7. Let \( G \) be an abelian group and \( {\chi }_{0},{\chi }_{1},\ldots ,{\chi }_{k} \in {G}^{ * } \) pairwise distinct characters. Then \( {\chi }_{0},{\chi }_{1},\ldots ,{\chi }_{k} \) are linearly independent. | Proof. Let \( {c}_{0},{c}_{1},\ldots ,{c}_{k} \in \mathbb{C} \) be such that \( \mathop{\sum }\limits_{{i = 0}}^{k}{c}_{i}{\chi }_{i}\left( x\right) = 0 \) for every \( x \in G \) . Then \( \mathop{\sum }\limits_{{i = 1}}^{k}{c}_{i}{\chi }_{i}\left( x\right) {\chi }_{0}{\left( x\right) }^{-1} + {c}_{0} = 0 \) for every... | Yes |
Corollary 11.3.8. Let \( G \) be an abelian group and \( {\chi }_{0},{\chi }_{1},\ldots ,{\chi }_{k} \in {G}^{ * } \) pairwise distinct characters. Then \( \begin{Vmatrix}{\mathop{\sum }\limits_{{j = 1}}^{k}{c}_{j}{\chi }_{j} - {\chi }_{0}}\end{Vmatrix} \geq 1 \) for every \( {c}_{1},\ldots ,{c}_{k} \in \mathbb{C} \) . | Proof. Let \( \varepsilon = \begin{Vmatrix}{\mathop{\sum }\limits_{{j = 1}}^{k}{c}_{j}{\chi }_{j} - {\chi }_{0}}\end{Vmatrix} \) . Then, for every \( x \in G \) ,\n\n\[ \left| {\mathop{\sum }\limits_{{j = 1}}^{k}{c}_{j}{\chi }_{j}\left( x\right) - {\chi }_{0}\left( x\right) }\right| \leq \varepsilon \]\n\n(11.25)\n\nAc... | Yes |
Corollary 11.3.9. Let \( G \) be an abelian group, \( H \) a subgroup of \( {G}^{ * } \), and let \( \chi \in {G}^{ * } \) be such that there exist \( k \in {\mathbb{N}}_{ + },{\chi }_{1},\ldots ,{\chi }_{k} \in H \), and \( {c}_{1},\ldots ,{c}_{k} \in \mathbb{C} \) with\n\n\[ \begin{Vmatrix}{\mathop{\sum }\limits_{{j ... | Proof. We assume without loss of generality that \( {\chi }_{1},\ldots ,{\chi }_{k} \) are pairwise distinct. Assume for a contradiction that \( \chi \neq {\chi }_{j} \) for all \( j \in \{ 1,\ldots, k\} \) . Then Corollary 11.3.8 applied to \( \chi ,{\chi }_{1},\ldots ,{\chi }_{k} \) yields \( \begin{Vmatrix}{\mathop{... | Yes |
Proposition 11.3.10. Let \( G \) be an abelian group. Then \( H = \left( \widetilde{G,{\mathcal{T}}_{H}}\right) \) for every subgroup \( H \) of \( {G}^{ * } \) . | Proof. Obviously, \( H \subseteq \left( \widehat{G,{\mathcal{T}}_{H}}\right) \) . Now let \( \chi \in \left( \widehat{G,{\mathcal{T}}_{H}}\right) \) . For every fixed \( \varepsilon > 0 \), the set \( O = \{ a \in \mathbb{S} : \left| {a - 1}\right| < \varepsilon \} \) is an open neighborhood of 1 in \( \mathbb{S} \) . ... | Yes |
Theorem 11.4.2. Let \( \left( {G,\tau }\right) \) be a topological abelian group. The following conditions are equivalent:\n\n(a) \( \tau \) is totally bounded;\n\n(b) the neighborhoods of 0 in \( G \) are big subsets;\n\n(c) \( \tau = {\mathcal{T}}_{\widehat{\left( G,\tau \right) }} \) . | Proof. (a) \( \Rightarrow \) (b) This is the definition of totally bounded topology.\n\n(b) \( \Rightarrow \) (c) If \( H = \left( \widehat{G,\tau }\right) \) then \( {\mathcal{T}}_{H} \leq \tau \) . Let \( U, V \) be open neighborhoods of 0 in \( \left( {G,\tau }\right) \) such that \( {V}_{\left( {10}\right) } \subse... | Yes |
Theorem 11.4.4. For an abelian group \( G \), let\n\n\[ \mathcal{D}\left( G\right) = \left\{ {H \leq {G}^{ * } : H}\right. \text{separates the points of}G\} \text{,} \]\n\n\[ \mathcal{P} = \{ \tau : \tau \text{ precompact group topology on }G\} . \]\n\nThen the following is an order-preserving bijection:\n\n\[ T : \mat... | Proof. Corollary 11.4.3 yields that \( {\mathcal{T}}_{H} \in \mathcal{P} \) for every \( H \in \mathcal{D}\left( G\right) \) and that \( T \) is surjective.\n\nBy Proposition 11.3.10, \( {\mathcal{T}}_{{H}_{1}} = {\mathcal{T}}_{{H}_{2}} \) for \( {H}_{1},{H}_{2} \in \mathcal{D}\left( G\right) \) yields \( {H}_{1} = {H}... | Yes |
Corollary 11.4.5. If \( G \) is an infinite abelian group and \( H \) a subgroup of \( {G}^{ * } \) that separates the points of \( G \), then \( w\left( {G,{\mathcal{T}}_{H}}\right) = \chi \left( {G,{\mathcal{T}}_{H}}\right) = \left| H\right| \) . | Proof. Let \( \kappa = \chi \left( {G,{\mathcal{T}}_{H}}\right) \) and note that \( \kappa \) cannot be finite. (Indeed, otherwise \( \kappa = 1 \) and \( {\mathcal{T}}_{H} \) must be discrete, so being also precompact, this would imply that \( G \) is finite, a contradiction.) According to Exercise 5.4.2, \( \kappa \l... | Yes |
Corollary 11.5.2. If \( K \) is a compact abelian group, then \( K \) is topologically isomorphic to \( a \) (closed) subgroup of \( {\mathbb{T}}^{\widehat{K}} \) . | Proof. Since \( \widehat{K} \) separates the points of \( K \) by Corollary 11.5.1, the diagonal map determined by all characters in \( \widehat{K} \) defines a continuous injective homomorphism \( {\Delta }_{\widehat{K}} : K \hookrightarrow \) \( {\mathbb{T}}^{\widehat{K}} \) . By the compactness of \( K \) and the op... | Yes |
Corollary 11.5.3. If \( \left( {K,\tau }\right) \) is a compact abelian group and \( H \) is a subgroup of \( \widehat{K} \) that separates the points of \( K \), then \( H = \widehat{K} \) . | Proof. By Corollary 11.5.1, \( \tau = {\mathcal{T}}_{\bar{K}} \) . Since \( {\mathcal{T}}_{H} \leq {\mathcal{T}}_{\bar{K}} \) by Theorem 11.4.4 and \( {\mathcal{T}}_{H} \) is Hausdorff, \( {\mathcal{T}}_{H} = {\mathcal{T}}_{\widehat{K}} = \tau \), as \( \tau \) is compact and due to the open mapping theorem (Theorem 8.... | Yes |
Proposition 11.5.4. Let \( K \) be a compact abelian group and \( U \) an open neighborhood of 0 in \( K \) . Then there exists a closed subgroup \( C \) of \( K \) such that \( C \subseteq U \) and \( K/C \) is an elementary compact abelian group. In particular, \( K \) is an inverse limit of elementary compact abelia... | Proof. By Corollary 11.5.1, the topology on \( K \) is \( {\mathcal{T}}_{\widehat{K}} \), hence there exists a finite subset \( F \) of \( \widehat{K} \) such that \( C = \mathop{\bigcap }\limits_{{\chi \in F}}\ker \chi \subseteq U \) . Define \( g = \mathop{\prod }\limits_{{\chi \in F}}\chi : K \rightarrow {\mathbb{T}... | Yes |
Corollary 11.5.5. A compact abelian group \( K \) is NSS precisely when \( K \) is an elementary compact abelian group. | Proof. If \( K \) is NSS, then \( K \) must be elementary compact, by Proposition 11.5.4. To prove the inverse implication, note first that \( \mathbb{T} \) is NSS. Moreover, the class of NSS abelian groups is stable under taking finite products and subgroups, by Exercise 11.7.4(a). Thus, all powers \( {\mathbb{T}}^{n}... | No |
Theorem 11.6.3. If \( G \) is a locally compact abelian group, then \( \widehat{G} \) separates the points of \( G \) . | Proof. Let \( V \) be a compact neighborhood of 0 in \( G \) . Take \( x \in G \smallsetminus \{ 0\} \) . Then \( {G}_{1} = \langle V \cup \{ x\} \rangle \) is an open (it has nonempty interior) compactly generated subgroup of \( G \) . In particular, \( {G}_{1} \) is locally compact. By Proposition 11.6.1, there exist... | Yes |
Corollary 11.6.4. If \( G \) is a locally compact abelian group, then every compact subgroup \( K \) of \( G \) is dually embedded. | Proof. Clearly, \( H = \left\{ {\chi \in \widehat{K} : \exists \xi \in \widehat{G},\xi { \upharpoonright }_{K} = \chi }\right\} \) is a subgroup of \( \widehat{K} \) . By Theorem 11.6.3, \( \widehat{G} \) separates the points of \( G \), so \( H \) separates the points of \( K \) . Now apply Corollary 11.5.3 to conclud... | Yes |
Corollary 11.6.5. Let \( G \) be a nontrivial locally compact abelian group.\n\n(a) Then \( G \) is connected if and only if \( \chi \left( G\right) = \mathbb{T} \) for every nontrivial \( \chi \in \widehat{G} \) . | Proof. (a) If \( G \) is connected and \( \chi \in \widehat{G} \) is nontrivial, then \( \chi \left( G\right) \) is a nontrivial connected subgroup of \( \mathbb{T} \), hence \( \chi \left( G\right) = \mathbb{T} \) (see Example 6.1.8(b)). If \( G \) is not connected, then \( G \) has a proper open subgroup \( H \), by ... | Yes |
Corollary 11.6.7. For a compact abelian group \( G \), the following are equivalent:\n\n(a) \( G \) is profinite;\n\n(b) \( G \) is hereditarily disconnected;\n\n(c) \( G \) is topologically torsion (i. e., \( G = G \) !);\n\n(d) \( \widehat{G} \) is torsion;\n\n(e) \( \chi \left( G\right) \neq \mathbb{T} \) for every ... | Proof. (a) \( \Leftrightarrow \) (b) is Corollary 8.5.7,(b) \( \Leftrightarrow \) (e) is Corollary 11.6.5(b),(b) \( \Rightarrow \) (c) was proved in Exercise 8.7.11.\n\n(b) \( \Leftrightarrow \) (d) The image \( \chi \left( G\right) \) under a continuous character \( \chi \) of \( G \) is a compact, hence closed, subgr... | No |
Corollary 11.6.8. For a compact abelian group \( G \) and a prime \( p \), the following conditions are equivalent:\n\n(a) \( G \) is pro-p-finite;\n\n(b) \( G \) is topologically p-torsion (i. e., \( G = {G}_{p} \) );\n\n(c) \( \widehat{G} \) is p-torsion;\n\n(d) \( \chi \left( G\right) \subseteq \mathbb{Z}\left( {p}^... | Proof. (a) \( \Rightarrow \) (b) This was proved in Exercise 8.7.12.\n\n(b) \( \Rightarrow \) (d) Pick any \( \chi \in \widehat{G} \) . Then every \( x \in K \) being topologically \( p \) -torsion implies that \( \chi \left( x\right) \) is a topologically \( p \) -torsion element of \( \mathbb{T} \), so \( \chi \left(... | No |
Corollary 11.6.9. For a locally compact abelian group \( G \), one has \( t{d}_{p}\left( G\right) = {G}_{p} \) for every prime \( p \) . | Proof. Fix a prime \( p \) . In view of Remark 5.3.7, always \( t{d}_{p}\left( G\right) \subseteq {G}_{p} \) . To prove the opposite inclusion, let \( x \in {G}_{p} \) ; so \( {p}^{n}x \rightarrow 0 \) in \( G \) . If \( \langle x\rangle \) is finite, then \( x \in {t}_{p}\left( G\right) \subseteq t{d}_{p}\left( G\righ... | Yes |
Proposition 11.6.10. Let \( G \) be a topological abelian group.\n\n(a) If \( G \) is connected, then \( \widehat{G} \) is torsion-free.\n\n(b) If \( G \) is compact, then \( \widehat{G} \) is torsion-free if and only if \( G \) is connected. | Proof. (a) For every nonzero \( \chi \in \widehat{G} \), the image \( \chi \left( G\right) \) is a nontrivial connected subgroup of \( \mathbb{T} \), so we deduce that \( \chi \left( G\right) = \mathbb{T} \) (see Example 6.1.8(b)). Hence, \( \widehat{G} \) is torsion-free.\n\n(b) If \( G \) is compact and not connected... | Yes |
Theorem 11.6.11. For a locally compact abelian group \( \left( {G,\tau }\right) \), its Bohr modification \( {G}^{ + } = \) \( \left( {G,{\tau }^{ + }}\right) \) has the same compact sets as \( \left( {G,\tau }\right) \) . | A proof of this theorem in the discrete case will be given in Theorem 13.4.9. | No |
Lemma 12.1.2. For a topological group \( G \), a function \( f \in {C}^{ * }\left( G\right) \) is almost periodic if and only if every infinite sequence of translates \( {\left\{ {f}_{{b}_{m}}\right\} }_{m \in \mathbb{N}} \) off admits a Cauchy subsequence. | Proof. If \( f \) is almost periodic, then \( {K}_{f} = \overline{\left\{ {f}_{a} : a \in G\right\} } \) is compact by hypothesis, so every sequence in \( {K}_{f} \) admits a convergent subsequence.\n\nAssume now that every infinite sequence of translates \( {\left\{ {f}_{{b}_{n}}\right\} }_{n \in \mathbb{N}} \) of \( ... | Yes |
Let \( f : \mathbb{R} \rightarrow \mathbb{C} \) be a function. One says that \( a \in \mathbb{R} \) is a period of \( f \) if \( f\left( {x + a}\right) = f\left( x\right) \) for every \( x \in \mathbb{R} \) (i. e., \( {f}_{a} = f \) ). Clearly, if \( a \in \mathbb{R} \) is a period of \( f \), then also \( {ka} \) is a... | It is easy to see that a periodic function \( f : \mathbb{R} \rightarrow \mathbb{C} \) has period \( a \in \mathbb{R} \smallsetminus \{ 0\} \) if and only if \( f \) factorizes through the canonical projection \( q : \mathbb{R} \rightarrow \mathbb{R}/\langle a\rangle \), that is, there exists a function \( {f}^{\prime ... | Yes |
Let \( G \) be a topological abelian group and \( \chi \in \widehat{G} \). For \( \varepsilon > 0 \), one has \( \left| {\chi \left( {x + a}\right) - \chi \left( x\right) }\right| = \left| {\chi \left( a\right) - 1}\right| \) for every \( x \in G \), hence | \[ T\left( {\chi ,\varepsilon }\right) = \{ a \in G : \left| {\chi \left( a\right) - 1}\right| \leq \varepsilon \} . \] So, there exists \( \delta > 0 \) such that \( {U}_{G}\left( {\chi ;\delta }\right) \subseteq T\left( {\chi ,\varepsilon }\right) \). Since \( {U}_{G}\left( {\chi ;\delta }\right) \) is big by Proposi... | Yes |
Theorem 12.1.9 (Bohr-von Neumann theorem). If \( G \) is a topological abelian group, then \( A\left( G\right) = \mathfrak{A}\left( G\right) \) (i. e., \( f \in {C}^{ * }\left( G\right) \) is almost periodic if and only if \( f \) can be uniformly approximated by linear combinations with complex coefficients of continu... | Proof. According to Example 12.1.8, \( \mathfrak{X}\left( G\right) \subseteq A\left( G\right) \) . By Theorem 12.1.4, \( A\left( G\right) \) is closed in \( {C}^{ * }\left( G\right) \), and \( \mathfrak{A}\left( G\right) \) is the closure of \( \mathfrak{X}\left( G\right) \) in \( {C}^{ * }\left( G\right) \), so \( \ma... | Yes |
Corollary 12.1.10. For a compact abelian group \( G \), every continuous function \( G \rightarrow \mathbb{C} \) is almost periodic (i. e., \( C\left( G\right) = A\left( G\right) = \mathfrak{A}\left( G\right) \) ). | Proof. It follows from Corollary 11.5.1 and Theorem B.5.21 that \( \mathfrak{X}\left( G\right) \) is dense in \( C\left( G\right) \) , so \( C\left( G\right) = \overline{\mathfrak{X}\left( G\right) } = \mathfrak{A}\left( G\right) = A\left( G\right) \), by Theorem 12.1.9. | Yes |
Corollary 12.1.11. Let \( \left( {G,\tau }\right) \) be a topological abelian group and \( f \in A\left( {G,\tau }\right) \) . Then \( f : \left( {G,{\tau }^{ + }}\right) \rightarrow \mathbb{C} \) is uniformly continuous. | Proof. By Theorem 12.1.9, \( f : G \rightarrow \mathbb{C} \) can be uniformly approximated by functions from \( \mathfrak{X}\left( G\right) \) . On the other hand, \( \mathfrak{X}\left( {G,\tau }\right) = \mathfrak{X}\left( {G,{\tau }^{ + }}\right) \) and these functions are uniformly continuous. Therefore, \( f : \lef... | Yes |
Theorem 12.1.12. Let \( G = \left( {G,\tau }\right) \) be a topological abelian group. Then a continuous function \( f : G \rightarrow \mathbb{C} \) is almost periodic if and only if there exists a continuous function \( \widetilde{f} : {bG} \rightarrow \mathbb{C} \) such that \( f = \widetilde{f} \circ {b}_{G} \). | Proof. Assume that there exists a continuous function \( \widetilde{f} : {bG} \rightarrow \mathbb{C} \) such that \( f = \widetilde{f} \circ {b}_{G} \) . Then \( \widetilde{f} \) is almost periodic by Corollary 12.1.10, and Exercise 12.3.1 implies that \( f \) is almost periodic, too.\n\nNow let \( f \in A\left( G\righ... | No |
Lemma 12.1.13. For every topological abelian group \( G \) , \n\n\[ \n\mathfrak{A}\left( G\right) = \mathbb{C} \cdot 1 \oplus {\mathfrak{A}}_{0}\left( G\right) \n\] \n\nMoreover, if \( f \in {C}^{ * }\left( G\right) \) is written as \( f\left( x\right) = {c}_{f} + {g}_{f}\left( x\right) \), with \( {c}_{f} \in \mathbb{... | Proof. Assume that \( c \cdot 1 = g \in {\mathfrak{A}}_{0}\left( G\right) \) for some \( c \in \mathbb{C} \) . Apply Corollary 11.3.6 to \( g - c = 0 \) and \( M = \{ 0\} \), to get \( c = 0 \) . Hence, \( {\mathfrak{A}}_{0}\left( G\right) \cap \mathbb{C} \cdot 1 = \{ 0\} \), that is,(12.4) holds.\n\nFor \( f \in \math... | Yes |
Proposition 12.2.2. Let \( G \) be a topological abelian group and \( {I}_{G} \) an arbitrary Haar integral on \( \mathfrak{A}\left( G\right) \) . If \( \varphi \in \widehat{G} \) is nontrivial, then \( {I}_{G}\left( \varphi \right) = 0 \) . | Proof. Let \( \varphi \in \widehat{G} \) and \( a \in G \) be such that \( \varphi \left( a\right) \neq 1 \) . For every \( x \in G,{\varphi }_{a}\left( x\right) = \varphi \left( a\right) \varphi \left( x\right) \) , so \( {I}_{G}\left( \varphi \right) = {I}_{G}\left( {\varphi }_{a}\right) = \varphi \left( a\right) {I}... | Yes |
Theorem 12.2.3. For every topological abelian group \( G \), the assignment\n\n\[ \n{I}_{G} : \mathfrak{A}\left( G\right) \rightarrow \mathbb{C},\;f \mapsto {c}_{f},\n\]\n\nwith \( {c}_{f} \in \mathbb{C} \) defined as in Lemma 12.1.13, gives a normed Haar integral on \( \mathfrak{A}\left( G\right) \) . | Proof. Fix \( f \in \mathfrak{A}\left( G\right) \) . Since \( {I}_{G} \) is the projection on the first component in (12.4), it is linear. Moreover, it was established in the same lemma that this linear functional is positive. To check the invariance, note that if \( f = {g}_{f} + {c}_{f} \) with \( {g}_{f} \in {\mathf... | Yes |
Corollary 12.2.4. Let \( G \) be a topological abelian group. Then the Haar integral \( {I}_{G} \) defined in Theorem 12.2.3 is the unique normed Haar integral on \( \mathfrak{A}\left( G\right) \) . | Proof. Let \( I \) be a Haar integral on \( \mathfrak{A}\left( G\right) \) . Let \( f \in {\mathfrak{A}}_{0}\left( G\right) \) . For every \( \varepsilon > 0 \), there exists \( g \in {\mathfrak{X}}_{0}\left( G\right) \) such that \( \parallel f - g\parallel \leq \varepsilon \) . Since \( I\left( g\right) = 0 \) by Pro... | No |
Lemma 12.2.6. Let \( {I}_{G} \) be a Haar integral on a locally compact abelian group \( G \) . If \( h \in \) \( {C}_{0}\left( G\right) \) is real-valued, \( h \geq 0 \) on \( G \), and \( h\left( {x}_{0}\right) > 0 \) for at least one \( {x}_{0} \in G \), then \( {I}_{G}\left( h\right) > 0 \) . | Proof. There exists a neighborhood \( V \) of 0 in \( G \) such that \( h\left( x\right) \geq a \mathrel{\text{:=}} h\left( {x}_{0}\right) /2 \) for all \( x \in {x}_{0} + V \) . \n\nThere exists \( f \in {C}_{0}\left( G\right) \) with \( {I}_{G}\left( f\right) \neq 0 \) . Since \( f = u + {iv} \) for some real-valued ... | Yes |
Lemma 12.2.7. If \( G \) is a discrete abelian group, then \( G \) admits a Haar integral. | Proof. For \( f \in {C}_{0}\left( G\right) \), which has finite support, set \( {I}_{G}\left( f\right) = \mathop{\sum }\limits_{{x \in G}}f\left( x\right) \) . One checks easily that \( {I}_{G} \) is a Haar integral on \( G \) . | No |
Theorem 12.2.9. Every locally compact abelian group G admits a Haar integral. | Proof. If \( G \) is compact or discrete, then Theorem 12.2.5 or Lemma 12.2.7 apply, respectively. In case \( G \) is compactly generated, \( G \) has a discrete subgroup \( H \) such that \( G/H \) is compact by Proposition 11.6.1. So, both \( H \) and \( G/H \) admit a Haar integral. It follows from Lemma 12.2.8 that... | Yes |
Proposition 13.1.1. Let \( G \) be a topological abelian group.\n\n(a) If \( G \) is compact, then \( \widehat{G} \) is discrete.\n\n(b) If \( G \) is discrete, then \( \widehat{G} \) is compact. | Proof. If \( G \) is compact, then \( {W}_{\widehat{G}}\left( {G,{\Lambda }_{1}}\right) = \{ 0\} \), as \( {\Lambda }_{1} \) contains no subgroup of \( \mathbb{T} \) beyond \( \{ 0\} \) . If \( G \) is discrete, then \( \widehat{G} = {G}^{ * } \) is compact, as explained above. | No |
Theorem 13.1.2. For a topological abelian group \( G \), the following hold true:\n\n(a) if \( x \in \mathbb{T} \) and \( k \in {\mathbb{N}}_{ + } \), then \( x \in {\Lambda }_{k} \) if and only if \( x,{2x},\ldots ,{kx} \in {\Lambda }_{1} \) ; | Proof. (a) For \( s \in {\mathbb{N}}_{ + },{sx} \in {\Lambda }_{1} \) if and only if \( x \in {A}_{s, t} \mathrel{\text{:=}} {q}_{0}\left( \frac{t}{s}\right) + {\Lambda }_{s} \) for some \( t \in \) \( \{ 0,\ldots, s - 1\} \) . On the other hand, \( {A}_{s,0} = {\Lambda }_{s} \) and \( {\Lambda }_{s} \cap {A}_{s + 1, t... | Yes |
Corollary 13.1.3. Let \( G \) be a locally compact abelian group. Then:\n\n(a) \( \widehat{G} \) is locally compact;\n\n(b) if \( G \) is metrizable, then \( \widehat{G} \) is \( \sigma \) -compact;\n\n(c) if \( G \) is \( \sigma \) -compact, then \( \widehat{G} \) is metrizable. | Proof. (a) follows immediately from Theorem 13.1.2( \( {\mathrm{f}}_{3} \) ).\n\n(b) Since \( G \) is metrizable, there exists a countable base \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) of \( {\mathcal{V}}_{G}\left( 0\right) \) , with \( {\bar{U}}_{n + 1} \subseteq {U}_{n} \) for every \( n \in \mathbb{N} \) .... | Yes |
Corollary 13.1.4. For a locally compact abelian group \( G \), the following conditions are equivalent:\n\n(a) the inclusion map \( j : \widehat{G} \hookrightarrow {G}^{ * } \) is an embedding;\n\n(b) \( \widehat{G} \) is compact. | Proof. (a) \( \Rightarrow \) (b) Assume that \( j : \widehat{G} \hookrightarrow {G}^{ * } \) is an embedding. By Corollary 13.1.3(a), \( \widehat{G} \) is locally compact, hence complete by Proposition 8.2.6. By Proposition 7.1.22, \( j\left( \widehat{G}\right) \cong \widehat{G} \) is closed in the compact group \( {G}... | Yes |
Lemma 13.2.1. Every continuous homomorphism \( \chi : \mathbb{T} \rightarrow \mathbb{T} \) has the form \( k{\mathrm{{id}}}_{\mathbb{T}} \), for some \( k \in \mathbb{Z} \) . | First proof of Lemma 13.2.1. Applying Lemma 9.1.3 to \( q \mathrel{\text{:=}} \chi \circ {q}_{0} : \mathbb{R} \rightarrow \mathbb{T} \) and \( {q}_{0} : \mathbb{R} \rightarrow \mathbb{T} \) , we can find a continuous homomorphism \( \eta : \mathbb{R} \rightarrow \mathbb{R} \) such that \( {q}_{0} \circ \eta = q = \chi ... | Yes |
Corollary 13.2.2. Let \( G \) be a \( \sigma \) -compact locally compact abelian group and \( \chi ,\xi : G \rightarrow \mathbb{T} \) continuous surjective characters of \( G \) . Then there exists \( m \in \mathbb{Z} \) such that \( \xi = {m\chi } \) if and only if \( \ker \chi \subseteq \ker \xi \) . If \( \ker \chi ... | Proof. Argue as in the final part of the second proof of Lemma 13.2.1. Since \( \chi \) and \( \xi \) are open by the open mapping theorem (Theorem 8.4.1), Proposition 3.2.5 can be applied with \( {H}_{1} = {H}_{2} = \mathbb{T} \) and the diagram (13.3) (with \( G \) in place of \( \mathbb{T} \) at the top) can be used... | No |
Corollary 13.2.3. Let \( G \) be a topological abelian group and \( \chi ,\xi : G \rightarrow \mathbb{T} \) continuous characters of \( G \) such that \( \ker \xi \subseteq \ker \chi \). (a) If \( \left| {\xi \left( G\right) }\right| = m \) for some \( m \in {\mathbb{N}}_{ + } \), then \( \chi = {k\xi } \) for some \( ... | Proof. (a) If \( \left| {\xi \left( G\right) }\right| = m \) for some \( m \in {\mathbb{N}}_{ + } \), then \( G/\ker \xi \cong \xi \left( G\right) = \mathbb{Z}\left( m\right) \leq \mathbb{T} \). The hypothesis \( \ker \xi \subseteq \ker \chi \) implies that \( G/\ker \chi \cong \chi \left( G\right) = \mathbb{Z}\left( n... | Yes |
Let \( p \) be a prime. Then \[ \widehat{\mathbb{Z}\left( {p}^{\infty }\right) } \cong {\mathbb{J}}_{p},\;{\widehat{\mathbb{J}}}_{p} \cong \mathbb{Z}\left( {p}^{\infty }\right) ,\;\widehat{\mathbb{T}} \cong \mathbb{Z},\;\widehat{\mathbb{Z}} \cong \mathbb{T},\;\text{ and }\;\widehat{\mathbb{R}} \cong \mathbb{R}. \] | The first isomorphism follows from our definition \( {\mathbb{J}}_{p} = \operatorname{End}\left( {\mathbb{Z}\left( {p}^{\infty }\right) }\right) = \) \( \operatorname{Hom}\left( {\mathbb{Z}\left( {p}^{\infty }\right) ,\mathbb{T}}\right) = \widehat{\mathbb{Z}\left( {p}^{\infty }\right) } \) and the fact that the topolog... | Yes |
Proposition 13.2.5. Let G be a hereditarily disconnected locally compact abelian group. Then \( \ker \chi \) is an open subgroup of \( G \) for every \( \chi \in \widehat{G} \) . | Proof. According to Theorem 8.5.2, \( G \) has a local base at 0 formed by open subgroups. Hence, there exists an open subgroup \( O \) of \( G \) such that \( \chi \left( O\right) \subseteq {\Lambda }_{1} \) . Since \( {\Lambda }_{1} \) contains no nontrivial subgroup, \( \chi \left( O\right) = \{ 0\} \), and so \( O ... | Yes |
Lemma 13.3.1. If \( G, H \) are topological abelian groups, then \( \widehat{G \times H} \cong \widehat{G} \times \widehat{H} \). | Proof. The isomorphism \( \Phi : \widehat{G} \times \widehat{H} \rightarrow \widehat{G \times H} \), defined by \( \Phi \left( {{\chi }_{1},{\chi }_{2}}\right) \left( {{x}_{1},{x}_{2}}\right) = {\chi }_{1}\left( {x}_{1}\right) + {\chi }_{2}\left( {x}_{2}\right) \) for every \( \left( {{\chi }_{1},{\chi }_{2}}\right) \i... | Yes |
Any finite abelian group \( F \) is selfdual. | Indeed, \( \widehat{F} = {F}^{ * } \) and we prove that \( {F}^{ * } \cong F \) . Recall that \( F \) has the form \( F \cong \mathbb{Z}\left( {n}_{1}\right) \times \cdots \times \mathbb{Z}\left( {n}_{m}\right) \) for suitable \( {n}_{1},\ldots ,{n}_{m} \in {\mathbb{N}}_{ + } \), by Theorem A.1.1. So, applying Lemma 13... | Yes |
Proposition 13.3.4. Let \( {P}_{1},{P}_{2},{P}_{3} \) be finite sets of primes, \( m, n, k,{k}_{p} \in \mathbb{N} \) with \( p \in {P}_{3} \) , \( {n}_{p} \in {\mathbb{N}}_{ + } \) with \( p \in {P}_{1} \) and \( {m}_{p} \in {\mathbb{N}}_{ + } \) with \( p \in {P}_{2} \) . Every group of the form \[ G = {\mathbb{T}}^{n... | Proof. Example 13.2.4 gives that \( \widehat{\mathbb{Z}} \cong \mathbb{T} \) and \( \widehat{\mathbb{T}} \cong \mathbb{Z} \), hence \( \mathbb{Z} \cong \widehat{\widehat{\mathbb{Z}}} \) and \( \mathbb{T} \cong \widehat{\widehat{\mathbb{T}}} \) . Analogously, \( \widehat{\mathbb{Z}\left( {p}^{\infty }\right) } \cong {\m... | Yes |
Theorem 13.3.5. Let \( \left\{ {{D}_{i} : i \in I}\right\} \) be a family of discrete abelian groups and let \( \left\{ {{G}_{i} : i \in I}\right\} \) be a family of compact abelian groups. Then \[ \widehat{{\bigoplus }_{i \in I}{D}_{i}} \cong \mathop{\prod }\limits_{{i \in I}}\widehat{{D}_{i}}\;\text{ and }\;\widehat{... | Proof. Let \( D = {\bigoplus }_{i \in I}{D}_{i} \), let \( \chi : D \rightarrow \mathbb{T} \) be a character and, for \( i \in I \), let \( {\chi }_{i} : {D}_{i} \rightarrow \mathbb{T} \) be its restriction to \( {D}_{i} \) . The isomorphism \( \Phi : \widehat{D} \rightarrow \mathop{\prod }\limits_{{i \in I}}\widehat{{... | Yes |
Using the isomorphism \( \mathbb{Q}/\mathbb{Z} \cong {\bigoplus }_{p \in \mathbb{P}}\mathbb{Z}\left( {p}^{\infty }\right) \), where \( \mathbb{Q}/\mathbb{Z} \) is discrete, | Example 13.2.4 and Theorem 13.3.5, we obtain \( \widehat{\mathbb{Q}/\mathbb{Z}} \cong \mathop{\prod }\limits_{{p \in \mathbb{P}}}{\mathbb{J}}_{p} \) | No |
Lemma 13.3.7. If \( G, H \) are topological abelian groups and \( f : G \rightarrow H \) is a continuous homomorphism, then \( \widehat{f} : \widehat{H} \rightarrow \widehat{G} \) is a continuous homomorphism as well. | Proof. Clearly, \( \widehat{f} \) is a homomorphism. Assume that \( K \) is a compact subset of \( G \) . Then \( f\left( K\right) \) is a compact subset of \( H \), so \( W \mathrel{\text{:=}} {W}_{\widehat{H}}\left( {f\left( K\right) ,{\Lambda }_{1}}\right) \) is a neighborhood of 0 in \( \widehat{H} \) and \( \wideh... | Yes |
Corollary 13.3.8. Let \( G \) be a locally compact abelian group, \( H \) a subgroup of \( G \), and \( i : H \rightarrow G \) the canonical inclusion of \( H \) in \( G \). Then:\n\n(a) \( \widehat{i} : \widehat{G} \rightarrow \widehat{H} \) is surjective if \( H \) is dense or open, or compact;\n\n(b) \( \widehat{i} ... | Proof. (a) If \( H \) is compact, apply Corollary 11.6.4, otherwise Lemma 13.3.7(b).\n\n(b) If \( H \) is dense, then \( \widehat{i} \) is injective by Lemma 13.3.7(a). Conversely, assume that \( \bar{H} \) is a proper subgroup of \( G \) and let \( q : G \rightarrow G/\bar{H} \) be the canonical projection. By Theorem... | Yes |
Theorem 13.3.11 (Kakutani theorem). If \( G \) is an infinite discrete abelian group, then \( \left| \widehat{G}\right| = {2}^{\left| G\right| } \) . | Proof. The inequality \( \left| \widehat{G}\right| \leq {2}^{\left| G\right| } \) is obvious since \( \widehat{G} \) is contained in the Cartesian power \( {\mathbb{T}}^{G} \) which has cardinality \( {2}^{\left| G\right| } \) . It remains to prove the inequality \( \left| \widehat{G}\right| \geq {2}^{\left| G\right| }... | Yes |
Let \( K \) denote the compact group \( \widehat{\mathbb{Q}} \). Then:\n\n(a) \( K \) contains a closed subgroup \( H \) isomorphic to \( \widehat{\mathbb{Q}/\mathbb{Z}} \) such that \( K/H \cong \mathbb{T} \); | To verify (a), consider the continuous character \( \rho : K \rightarrow \widehat{\mathbb{Z}} \cong \mathbb{T} \) obtained by the restriction to \( \mathbb{Z} \) of every \( \chi \in K \) (i. e., \( \rho = \widehat{j} \), where \( j : \mathbb{Z} \hookrightarrow \mathbb{Q} \) ). Then \( \rho \) is surjective by Lemma 13... | Yes |
For an infinite discrete abelian group \( G \) the map \( {\omega }_{{G}^{\# }} : {G}^{\# } \rightarrow \widehat{\widehat{{G}^{\# }}} = \widehat{{G}^{ * }} \) is not continuous. | Indeed, \( \widehat{{G}^{\# }} = {G}^{ * } \), since the groups \( {G}^{\# } \) and \( G \) have the same dual group, namely, \( \operatorname{Hom}\left( {G,\mathbb{T}}\right) \) . Furthermore, the only compact subsets of \( {G}^{\# } \) are the finite ones, according to Glicksberg theorem 11.6.11 (see also Theorem 13.... | Yes |
Lemma 13.4.5. If, for \( i \in \{ 1,\ldots, n\} \), the topological abelian groups \( {G}_{i} \) are reflexive, then also \( G = \mathop{\prod }\limits_{{i = 1}}^{n}{G}_{i} \) is reflexive. | Proof. To obtain a topological isomorphism \( j : \mathop{\prod }\limits_{{i = 1}}^{n}{\widehat{\widehat{G}}}_{i} \rightarrow \widehat{\widehat{G}} \), apply Lemma 13.3.1 twice. Then the product \( \pi : G \rightarrow \mathop{\prod }\limits_{{i = 1}}^{n}{\widehat{\widehat{G}}}_{i} \) of the topological isomorphisms \( ... | Yes |
Proposition 13.4.6. \( \omega \) is a natural transformation from \( {1}_{\mathcal{L}} \) to \( \overset{ \hat{} }{ \rightarrow } : \mathcal{L} \rightarrow \mathcal{L} \) . | Proof. By Proposition 13.4.1, \( {\omega }_{G} \) is continuous for every \( G \in \mathcal{L} \) . Moreover, for every continuous homomorphism \( f : G \rightarrow H \) of locally compact abelian groups, the following diagram commutes:\n\n is either compact or discrete, then \( {\omega }_{G} \) is a topological isomorphism. | Proof. If \( G \) is discrete, then \( \widehat{G} \) separates the points of \( G \) by Corollary A.2.6, and if \( G \) is compact, then \( \widehat{G} \) separates the points of \( G \) by Corollary 11.5.1. Therefore, \( {\omega }_{G} \) is injective by Remark 13.4.2.\n\nIf \( G \) is discrete, then \( \widehat{G} \)... | Yes |
Theorem 13.4.8. If \( G \) is an elementary locally compact abelian group, then \( {\omega }_{G} \) is a topological isomorphism. | Proof. According to Lemma 13.4.5 and Theorem 13.4.7, it suffices to prove that \( {\omega }_{\mathbb{R}} \) is a topological isomorphism. The mapping \( g : \mathbb{R} \rightarrow \widehat{\mathbb{R}}, r \mapsto {\chi }_{r} \), where \( {\chi }_{r}\left( x\right) = {q}_{0}\left( {rx}\right) \in \mathbb{T} \) for \( x \... | Yes |
Lemma 13.4.10. Let \( G \) be a topological abelian group and \( M \) a subset of \( G \) . Then:\n\n(a) \( {A}_{\widehat{G}}\left( M\right) \) is a closed subgroup of \( \widehat{G} \) ; | Proof. (a) Assume that \( {\left\{ {\chi }_{\alpha }\right\} }_{\alpha \in A} \) is a net in \( {A}_{\widehat{G}}\left( M\right) \) converging to \( \chi \) in \( \widehat{G} \) . Since the net converges also in the (coarser) pointwise-convergence topology, \( {\chi }_{\alpha }\left( x\right) \rightarrow \chi \left( x\... | Yes |
Proposition 13.4.11. Every closed subgroup of a locally compact abelian group \( G \) is dually closed. | Proof. Let \( H \) be a closed subgroup of \( G \) and let \( a \in G \smallsetminus H \) . By Theorem 11.6.3, \( G/H \) is MAP, that is, \( \widehat{G/H} \) separates the points of \( G/H \), and so there exists \( \xi \in \widehat{G/H} \) such that \( \xi \left( {q\left( a\right) }\right) \neq 0 \), where \( q : G \r... | Yes |
Proposition 13.4.12. Let \( G \) be a locally compact abelian group, \( H \) a closed subgroup of \( G \), i: \( H \rightarrow G \) the canonical inclusion, and \( q : G \rightarrow G/H \) the canonical projection. Then the sequence\n\n\[ 0 \rightarrow \widehat{G/H}\xrightarrow[]{\widehat{q}}\widehat{G}\overset{\wideha... | Proof. According to Lemma 13.3.7(d), \( \widehat{q} \) is an embedding, so it is proper. We have that \( \widehat{i} \circ \widehat{q} = \widehat{q \circ i} = 0 \), hence \( \operatorname{im}\widehat{q} \subseteq \ker \widehat{i} \) . If \( \xi \in \ker \widehat{i} = {A}_{\widehat{G}}\left( H\right) \), then \( \xi \le... | Yes |
Corollary 13.4.15. Let \( G \) be a locally compact abelian group and \( H \) a closed subgroup of G. If \( H \) is open (respectively, compact), then \( {A}_{\widehat{G}}\left( H\right) \) is compact (respectively, open). | Proof. If \( H \) is open, then \( G/H \) is discrete by Lemma 3.2.10(a) and so \( \widehat{G/H} \cong {A}_{\widehat{G}}\left( H\right) \) is compact by Corollary 13.4.14 and Proposition 13.1.1(b). If \( H \) is compact, then \( \widehat{H} \cong \) \( \widehat{G}/{A}_{\widehat{G}}\left( H\right) \) is discrete by Coro... | Yes |
Corollary 13.4.16. Let \( f : G \rightarrow H \) be a continuous homomorphism of locally compact abelian groups. Then the following conditions are equivalent:\n\n(a) \( f\left( G\right) \) is dense in \( H \) ;\n\n(b) \( \widehat{f} \) is injective (i. e., \( \widehat{f} \) is a monomorphism in the category \( \mathcal... | Proof. (a) \( \Rightarrow \) (b) If \( f\left( G\right) \) is dense in \( H \) then \( \widehat{f} \) is injective by Lemma 13.3.7(a). As far as the second assertion is concerned, it is easy to check the monomorphisms in \( \mathcal{L} \) are precisely the continuous injective homomorphisms (see Exercise 13.7.5).\n\n(b... | No |
Corollary 13.4.18. A locally compact abelian group \( G \) is compact (respectively, discrete) if and only if \( \widehat{G} \) is discrete (respectively, compact). | Proof. If \( \widehat{G} \) is compact (respectively, discrete), Proposition 13.1.1 implies that \( \widehat{\widehat{G}} \) is discrete (respectively, compact). So, the assertions follow from Theorem 13.4.17. | Yes |
Lemma 13.5.1. If \( G \) is a locally compact abelian group and \( H \) a closed subgroup of \( G \) , then \( H = {A}_{G}\left( {{A}_{\widehat{G}}\left( H\right) }\right) = {\omega }_{G}^{-1}\left( {{A}_{\widehat{\widehat{G}}}\left( {{A}_{\widehat{G}}\left( H\right) }\right) }\right) . | Proof. Clearly, \( H \subseteq {A}_{G}\left( {{A}_{\widehat{G}}\left( H\right) }\right) \), the equality \( H = {A}_{G}\left( {{A}_{\widehat{G}}\left( H\right) }\right) \) follows immediately from Proposition 13.4.11. The second equality follows from the first and Exercise 13.7.14. | No |
Corollary 13.5.3. Let \( G \) be a locally compact abelian group. The pair of maps\n\n\[ \n{A}_{\widehat{G}} : \mathcal{S}\left( G\right) \rightarrow \mathcal{S}\left( \widehat{G}\right) \;\text{ and }\;{A}_{G} : \mathcal{S}\left( \widehat{G}\right) \rightarrow \mathcal{S}\left( G\right) \]\n\ndefine a complete lattice... | Proof. The map \( {A}_{\widehat{G}} : \mathcal{S}\left( G\right) \rightarrow \mathcal{S}\left( \widehat{G}\right) \) is monotone decreasing by Lemma 13.4.10(b). By Lemma 13.5.1, the maps \( {A}_{\widehat{G}} : \mathcal{S}\left( G\right) \rightarrow \mathcal{S}\left( \widehat{G}\right) \) and \( {A}_{G} : \mathcal{S}\le... | Yes |
Corollary 13.5.4. Let \( G \) be a discrete abelian group. If \( H \) is a pure subgroup of \( G \), then \( {A}_{\widehat{G}}\left( H\right) \) is a pure subgroup of \( \widehat{G} \) . | Proof. Put for brevity \( K = \widehat{G} \) and pick an \( m \in {\mathbb{N}}_{ + } \) . The purity of \( H \) gives \( {mH} = {mG} \cap H \) , so taking the annihilators and making use of Corollary 13.5.3, as \( {A}_{K}\left( {mG}\right) = K\left\lbrack m\right\rbrack \) by Lemma 13.4.10(e) and Lemma 13.5.1, we get \... | Yes |
Proposition 13.5.5. Let \( G \) be a locally compact abelian group and \( H \) a closed subgroup of \( G \) . Then \( {A}_{\widehat{G}}\left( H\right) \cong \widehat{G/H} \) and \( \widehat{G}/{A}_{\widehat{G}}\left( H\right) \cong \widehat{H} \) . More precisely, \( \widehat{G}/{A}_{\widehat{G}}\left( H\right) \righta... | Proof. The first assertion follows from Corollary 13.4.14. We apply this result to \( \widehat{G}/{A}_{\widehat{G}}\left( H\right) \) in order to deduce that\n\n\[ \n{A}_{\widehat{\widehat{G}}}\left( {{A}_{\widehat{G}}\left( H\right) }\right) \rightarrow \widehat{\widehat{G}/{A}_{\widehat{G}}\left( H\right) },\;\eta \m... | Yes |
Theorem 13.6.2. The assignment \( G \mapsto {\widehat{G}}_{pw} \) defines a duality in the category of precom-pact abelian groups, more precisely \( {\gamma }_{G} : G \rightarrow {\widehat{\left( {\widehat{G}}_{pw}\right) }}_{pw} \) is a topological isomorphism for every precompact abelian group G. | Proof. By the definition of \( {\widehat{G}}_{pw} \), its topology coincides with \( {\mathcal{T}}_{G} \) . This proves that \( {\gamma }_{G} \) is surjective in view of Proposition 11.3.10. The injectivity of \( {\gamma }_{G} \) follows from Remark 13.4.2 and the precompactness of \( G \) . The continuous characters o... | Yes |
Proposition 14.1.1. For a compact abelian group \( K \) , the following are equivalent:\n\n(a) \( K \) is connected;\n\n(b) \( K \) is divisible;\n\n(c) \( \widehat{K} \) is torsion-free. | Proof. (a) \( \Leftrightarrow \) (c) is Proposition 11.6.10.\n\n\( \left( \mathrm{b}\right) \Leftrightarrow \left( \mathrm{c}\right) \) For \( m \in {\mathbb{N}}_{ + } \), since \( {A}_{\widehat{K}}\left( {mK}\right) = \widehat{K}\left\lbrack m\right\rbrack \) by Lemma 13.4.10(e), \( {mK} = K \) if and only if \( \wide... | Yes |
Theorem 14.1.3. Let \( K \) be a compact abelian group. Then \( K \) is monothetic if and only if \( \widehat{K} \) admits an injective homomorphism into \( \mathbb{T} \) . | Proof. The group \( K \) is monothetic if and only if there exists a homomorphism \( \mathbb{Z} \rightarrow K \) with dense image. According to Corollary 13.4.16, this is equivalent to the existence of an injective homomorphism \( \widehat{K} \rightarrow \mathbb{T} \) . | Yes |
Corollary 14.1.4. Let \( K \) be a compact abelian group.\n\n(a) If \( K \) is connected, then \( K \) is monothetic if and only if \( w\left( K\right) \leq \mathfrak{c} \) . | Proof. By Proposition 13.1.1(a), \( G = \widehat{K} \) is discrete.\n\n(a) By Proposition 14.1.1, \( G \) is torsion-free, so \( G \) admits an injective homomorphism into \( \mathbb{T} \) precisely when \( {r}_{0}\left( G\right) \leq \mathfrak{c} \) by Example A.2.16(b); equivalently, \( \left| G\right| \leq \mathfrak... | Yes |
Theorem 14.1.5. Let \( K \) be a compact abelian group. Then:\n\n(a) \( {M}_{K} = \left\{ {x \in K : {\omega }_{K}\left( x\right) }\right. \) is injective \( \} \) ;\n\n(b) \( {M}_{K} \) is the intersection of at most \( \left| \widehat{K}\right| \) open sets;\n\n(c) if \( K \) is connected and metrizable, \( {M}_{K} \... | Proof. (a) Let \( x \in {M}_{K} \) and \( \chi \in \widehat{K} \) . If \( {\omega }_{K}\left( x\right) \left( \chi \right) = \chi \left( x\right) = 0 \), then \( \chi \left( K\right) = \{ 0\} \), and so \( \chi \equiv 0 \) (by assumption). This shows that \( \ker {\omega }_{K}\left( x\right) = \{ 0\} \) and hence that ... | Yes |
Theorem 14.1.6. Every torsion compact abelian group \( K \) is bounded, and there exist \( {m}_{1},\ldots ,{m}_{n} \in {\mathbb{N}}_{ + } \) and cardinals \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) such that \( K \cong \mathop{\prod }\limits_{{i = 1}}^{n}\mathbb{Z}{\left( {m}_{i}\right) }^{{\alpha }_{i}} \) . | Proof. Since \( K = \mathop{\bigcup }\limits_{{n \in {\mathbb{N}}_{ + }}}K\left\lbrack {n!}\right\rbrack \) is a union of closed subgroups, we conclude, with Theorem B.5.20, that \( K\left\lbrack {n!}\right\rbrack \) is open for some \( n \in {\mathbb{N}}_{ + } \), so must have finite index, by the compactness of \( K ... | Yes |
Proposition 14.1.7. For \( K \) an infinite compact abelian group, \( d\left( K\right) = \log w\left( K\right) \) . | Proof. Let \( \kappa = \min \left\{ {\beta : w\left( K\right) \leq {2}^{\beta }}\right\} \) . Then the inequality \( w\left( K\right) \leq {2}^{d\left( K\right) } \) (see Lemma 5.1.5) implies \( \kappa \leq d\left( K\right) \) . Since \( {r}_{p}\left( {\mathbb{T}}^{\kappa }\right) = {r}_{0}\left( {\mathbb{T}}^{\kappa }... | Yes |
Proposition 14.1.8. For a connected compact abelian group \( K \), the subgroup \( t\left( K\right) \) is dense in \( K \) if and only if \( \widehat{K} \) is reduced. Consequently, every connected compact abelian group \( K \) has the form \( K \cong {K}_{1} \times {\widehat{\mathbb{Q}}}^{\alpha } \) for some cardinal... | Proof. Since \( \widehat{K} \) is discrete by Proposition 13.1.1(a) and torsion-free by Proposition 14.1.1, \( \widehat{K} \) is reduced if and only if \( \mathop{\bigcap }\limits_{{m \in {\mathbb{N}}_{ + }}}m\widehat{K} = \{ 0\} \), by Proposition A.4.6(e). Since \( {A}_{K}\left( {m\widehat{K}}\right) = \) \( K\left\l... | Yes |
Theorem 14.1.9. A compact abelian group \( K \) is hereditarily disconnected if and only if \( K = \mathop{\prod }\limits_{{p \in \mathbb{P}}}{K}_{p} \), where each \( {K}_{p} \) is a closed topological \( {\mathbb{J}}_{p} \) -module. The closed subgroups \( M \) of \( K \) are of the form \( M = \mathop{\prod }\limits... | Proof. By Corollary 11.6.5(b), \( K \) is hereditarily disconnected if and only if \( \widehat{K} \) is torsion, which is equivalent to \( \widehat{K} = {\bigoplus }_{p \in \mathbb{P}}{t}_{p}\left( \widehat{K}\right) \) . According to Corollary 11.6.8, the group \( {X}_{p} \mathrel{\text{:=}} \widehat{{t}_{p}\left( \wi... | Yes |
Proposition 14.2.5. For a locally compact abelian group \( G \) and every prime \( p \) , the subgroup \( {G}_{p} \) is contained in \( B\left( G\right) \) . If \( G \) is hereditarily disconnected, then \( B\left( G\right) \) is open and \( {G}_{p} \) is closed in \( G \) . | Proof. Pick an element \( x \in {G}_{p} \) . If \( x \) is torsion, then \( x \in B\left( G\right) \) . Otherwise, the infinite cyclic subgroup \( \langle x\rangle \) is nondiscrete, as \( {p}^{n}x \rightarrow 0 \) . By Theorem 10.2.9, \( \langle x\rangle \) is contained in \( B\left( G\right) \) .\n\nIf \( G \) is her... | Yes |
Theorem 14.2.8. For a periodic locally compact abelian group \( G \), the subgroup \( {G}_{p} \) is closed for every prime \( p \) and \( G \cong \mathop{\prod }\limits_{{p \in \mathbb{P}}}^{\text{loc }}\left( {{G}_{p},{K}_{p}}\right) \), where \( K = \mathop{\prod }\limits_{{p \in \mathbb{P}}}{K}_{p} \) is a compact o... | Proof. By van Dantzig theorem 8.5.1, \( G \) has a compact open subgroup \( K \), which splits in a direct product \( K = \mathop{\prod }\limits_{{p \in \mathbb{P}}}{K}_{p} \), as in Theorem 14.1.9.\n\nAccording to Proposition 14.2.5, the subgroup \( {G}_{p} \) is closed. Moreover, the openness of \( K \) in \( G \) yi... | Yes |
Theorem 14.2.9. Let \( G \) be a compactly generated locally compact abelian group. Then \( G \cong {\mathbb{R}}^{n} \times {\mathbb{Z}}^{m} \times K \), where \( n, m \in \mathbb{N} \) and \( K \) is a compact abelian group. | Proof. According to Proposition 11.6.2, there exists a compact subgroup \( K \) of \( G \) such that \( G/K \) is an elementary locally compact abelian group. Taking a bigger compact subgroup, one can get the quotient \( G/K \) to be of the form \( {\mathbb{R}}^{n} \times {\mathbb{Z}}^{m} \) for some \( n, m \in \mathb... | Yes |
Corollary 14.2.10. If \( G \) is a locally compact abelian group, then \( B\left( G\right) \) is closed. | Proof. Let \( x \in \overline{B\left( G\right) } \) and let \( H \) be a compactly generated open subgroup of \( G \) with \( x \in H \) . By Theorem 14.2.9, \( H \cong {\mathbb{R}}^{n} \times {\mathbb{Z}}^{m} \times K \) for some compact subgroup \( K \) of \( G \) and \( n, m \in \mathbb{N} \) . Hence, \( B\left( H\r... | Yes |
Corollary 14.2.11. Let \( G \) be a connected locally compact abelian group. Then:\n\n(a) \( G \) is compactly generated, so \( G \cong {\mathbb{R}}^{n} \times C \) for some connected compact abelian group \( C \) and \( n \in \mathbb{N} \) ;\n\n(b) \( C = B\left( G\right) \) is the largest compact subgroup of \( G \) ... | Proof. (a) If \( U \) is a compact neighborhood of 0 in \( G \), then \( U \) generates an open subgroup \( H \) of \( G \) that is obviously compactly generated. Since \( G \) is connected, \( H = G \) . From Theorem 14.2.9 we deduce that \( G \cong {\mathbb{R}}^{n} \times C \) for some compact group \( C \) that is n... | Yes |
Theorem 14.2.13. If \( G \) is a locally compact abelian group, then\n\n\[ c\left( G\right) = {A}_{G}\left( {B\left( \widehat{G}\right) }\right) \;\text{ and }\;B\left( G\right) = {A}_{G}\left( {c\left( \widehat{G}\right) }\right) . | Proof. Let \( \left\{ {{K}_{i} : i \in I}\right\} \) be the family of all compact subgroups of \( \widehat{G} \) . By definition, \( B\left( \widehat{G}\right) = \) \( \mathop{\sum }\limits_{{i \in I}}{K}_{i} \) . By Remark 13.5.2(c), \( \left\{ {{A}_{G}\left( {K}_{i}\right) :i \in I}\right\} \) is precisely the family... | Yes |
Proposition 14.2.14. Let \( G \) be a locally compact abelian group. Then \( B\left( G\right) + c\left( G\right) \) is open and \( B\left( G\right) \cap c\left( G\right) \) is the maximal connected compact subgroup of \( G \). | Proof. Let \( H \) be a compactly generated open subgroup of \( G \) . By Theorem 14.2.9, \( H = \) \( R \oplus D \oplus K \) topologically, where \( K \) is a compact subgroup of \( G, R \cong {\mathbb{R}}^{n}, D \cong {\mathbb{Z}}^{m} \) and \( n, m \in \mathbb{N} \) . Clearly, \( R \oplus K \) is an open subgroup of... | Yes |
Corollary 14.2.15. For \( K \) a compact abelian group, \( c\left( K\right) = A\left( {t\left( \widehat{K}\right) }\right) \) . | Proof. Now \( \widehat{K} \) is discrete by Proposition 13.1.1(a), hence \( B\left( \widehat{K}\right) = t\left( \widehat{K}\right) \), so Theorem 14.2.13 applies. | Yes |
Lemma 14.2.17. Let \( G \) be a locally compact abelian group and let \( K, L \) be closed subgroups of \( G \) such that algebraically \( G \) is the direct sum of the subgroups \( K \) and \( L \) and \( K \) is compactly generated. Then the topology on \( G \) coincides with the product topology of \( K \oplus L \) ... | Proof. Let \( C \) be a compact subset of \( K \) that generates \( K \) and let \( U \) be a compact neighborhood of 0 in \( G \) . Then \( H = \langle C + U\rangle \) is a compactly generated open subgroup of \( G \) containing \( K \) and algebraically \( H = K \oplus \left( {L \cap H}\right) \), as \( K \subseteq H... | Yes |
Theorem 14.2.18. If \( G \) is a locally compact abelian group, then \( G \cong {\mathbb{R}}^{n} \times {G}_{0} \), where \( n \in \mathbb{N} \) and \( {G}_{0} \) is a closed subgroup of \( G \) containing a compact open subgroup. | Proof. By Corollary 14.2.11(a), there exist connected compact subgroups \( C \) of \( G \) and \( K \) of \( \widehat{G} \) and closed subgroups \( R \cong {\mathbb{R}}^{n} \) of \( G \) and \( S \cong {\mathbb{R}}^{m} \) of \( \widehat{G} \) such that \( c\left( G\right) = R \oplus C \) and \( c\left( \widehat{G}\righ... | Yes |
Corollary 14.2.19. If \( G \) is a locally compact abelian group, then \( w\left( G\right) = w\left( \widehat{G}\right) \) . | Proof. Since this equality is obviously true for finite groups, we assume that \( G \) is infinite. By Theorem 14.2.18, \( G \cong {\mathbb{R}}^{n} \times {G}_{0} \), where \( {G}_{0} \) is a closed subgroup of \( G \) containing a compact open subgroup \( K \) . Then \( \widehat{G} \cong {\mathbb{R}}^{n} \times \wideh... | Yes |
Corollary 14.2.20. Every locally compact abelian group is topologically isomorphic to a closed subgroup of a group of the form \( {\mathbb{R}}^{n} \times D \times C \), where \( n \in \mathbb{N} \), \( D \) is a discrete divisible abelian group, and \( C \) is a compact abelian group. | Proof. Let \( G \cong {\mathbb{R}}^{n} \times {G}_{0} \) with \( n,{G}_{0} \) and \( K \) as in Theorem 14.2.18. By Corollary 11.5.2, there exist a cardinal \( \kappa \) and an embedding \( j : K \rightarrow {\mathbb{T}}^{\kappa } \). Since \( {\mathbb{T}}^{\kappa } \) is divisible, one can extend \( j \) to a homomorp... | Yes |
Theorem 14.3.6. Let \( K \) be a compact abelian group. Then \( \dim K = {r}_{0}\left( \widehat{K}\right) \) and:\n\n(a) for every hereditarily disconnected closed subgroup \( H \) of \( K \) such that \( K/H \cong {\mathbb{T}}^{K} \), one has \( \kappa = \dim K = \dim K/H \) ;\n\n(b) there exists a subgroup \( H \) of... | Proof. For \( H \) as in (a), one has \( \dim H = 0 \), by Remark 14.3.5(a). So, Theorem 14.3.4 applied to \( K \) and \( H \) gives \( \dim K = \dim K/H = \dim {\mathbb{T}}^{K} = \kappa \), where the last equality comes from Remark 14.3.5(d).\n\nNext we put \( G = \widehat{K} \) and \( \rho = {r}_{0}\left( G\right) \)... | Yes |
Corollary 14.3.7. A nontrivial finite-dimensional connected compact abelian group \( K \) is metrizable, \( \left| K\right| = \mathfrak{c} \), and \( {r}_{p}\left( K\right) \leq \dim K \) for every prime \( p \) . | Proof. Let \( n = \dim K \) and \( G = \widehat{K} \) . Then \( G \) is torsion-free by Proposition 11.6.10(b) with \( {r}_{0}\left( G\right) = n \) by Theorem 14.3.6. Hence, \( D\left( G\right) \cong {\mathbb{Q}}^{n} \) by Lemma A.2.14, and in particular \( G \) is countable. Thus, \( K \) is metrizable by Corollary 1... | Yes |
Theorem 14.3.8. Let \( K \) be a nontrivial connected compact abelian group with \( \sigma = \) \( \dim K \) .\n\n(a) There exists a continuous surjective homomorphism \( q : {\widehat{\mathbb{Q}}}^{\sigma } \rightarrow K \) with \( N = \ker q \) hereditarily disconnected and \( {N}_{p} \cong {\mathbb{J}}_{p}^{{\gamma ... | Proof. (a) Since \( K \) is connected, \( G = \widehat{K} \) is discrete and torsion-free by Proposition 13.1.1(a) and Proposition 11.6.10(a), and \( \beta \mathrel{\text{:=}} \left| G\right| = w\left( K\right) \) according to Corollary 11.4.5. By Theorem 14.3.6, \( {r}_{0}\left( G\right) = \sigma \), so \( G \) contai... | Yes |
For a connected compact abelian group \( K \), the subgroup \( {K}_{p} \) is dense and proper for every prime \( p \) . | Proof. Let us start with \( K = \widehat{\mathbb{Q}} \), which has \( \dim K = {r}_{0}\left( \mathbb{Q}\right) = 1 \) in view of Theorem 14.3.6. Since \( K \) is divisible, for every prime \( p \) the group \( {K}_{p} \) is divisible by Remark 5.3.7; so, \( H \mathrel{\text{:=}} \) \( \overline{{K}_{p}} \) is divisible... | No |
Corollary 14.3.10. A locally compact abelian group \( G \) is hereditarily disconnected if and only if \( G \) is line-free and all subgroups \( {G}_{p} \) are closed. | Proof. We can assume without loss of generality that \( G \) contains no copies of \( \mathbb{R} \), so that \( G \) has a compact open subgroup \( K \) by Theorem 14.2.18.\n\nIf \( G \) is hereditarily disconnected, then each \( {G}_{p} \) is closed by Proposition 14.2.5. If \( G \) is not hereditarily disconnected, t... | Yes |
Example 14.3.12. This theorem allows us to consider the quotient \( K = {\mathbf{A}}_{\mathbb{Q}}/Q \) as the dual of \( \mathbb{Q} \) . According to Remark 8.5.11(b), \( q\left( {a\left( {\mathbf{A}}_{\mathbb{Q}}\right) }\right) = a\left( K\right) \) . | As \( a\left( {\mathbf{A}}_{\mathbb{Q}}\right) = c\left( {\mathbf{A}}_{\mathbb{Q}}\right) = \mathbb{R} \times \{ 0\} \) and \( {\left. q\right| }_{\mathbb{R}\times \{ 0\} } \) is injective, this entails that \( a\left( K\right) = q\left( {\mathbb{R}\times \{ 0\} }\right) \) is a continuous isomorphic image of \( \mathb... | Yes |
A reduced abelian group \( K \) admits a compact group topology if and only if there exist cardinals \( \left\{ {{\sigma }_{p} : p \in \mathbb{P}}\right\} \) and \( \left\{ {{\alpha }_{n, p} : p \in \mathbb{P}, n \in \mathbb{N}}\right\} \) such that\n\n\[ K\text{is algebraically isomorphic to}\mathop{\prod }\limits_{{p... | Proof. Starting with the necessity of (14.4), assume that \( K \) is a reduced abelian group that admits a compact, necessarily hereditarily disconnected, group topology. Then \( K \cong \mathop{\prod }\limits_{{p \in \mathbb{P}}}{K}_{p} \), where each \( {K}_{p} \) is a \( {\mathbb{J}}_{p} \) -module, by Theorem 14.1.... | No |
Theorem 14.3.19. Let \( G \) be a locally compact abelian group, \( {\widehat{G}}_{d} \) its dual equipped with the discrete topology, and i: \( {\widehat{G}}_{d} \rightarrow \widehat{G} \) the continuous identity map. The Bohr compactification \( {b}_{G} : G \rightarrow {bG} \) of \( G \) coincides, up to isomorphism,... | Proof. To see that \( \phi \mathrel{\text{:=}} \widehat{i} \circ {\omega }_{G} : G \rightarrow \widehat{{\widehat{G}}_{d}} \) is the Bohr compactification of \( G \), pick a continuous homomorphism \( f : G \rightarrow K \) to a compact group \( K \) . We can assume without loss of generality that \( f\left( G\right) \... | Yes |
It was observed by Armacost [8] that, for \( A = {\left\{ {p}^{n}\right\} }_{n \in \mathbb{N}} \), where \( p \) is a prime, \( {t}_{A}\left( \mathbb{T}\right) = {t}_{p}\left( \mathbb{T}\right) = \mathbb{Z}\left( {p}^{\infty }\right) \). Armacost posed the question of describing the subgroup \( {t}_{A}\left( \mathbb{T}... | This was done in [36],[99,§4.4.2], and [66]. | Yes |
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