Math Problems Vault

Problem V.11.1

Let $G$ be a group and a topological space. Show that $G$ is a topological group if and only if the map of $G \times G \to G$ defined by $(x,y) \mapsto x^{-1} y$ is continuous.

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Problem V.11.2

Verify the statements in Examples 11.2:

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(a)

$\mathbb {N}$ and $\mathbb {R}_{\geq 0}$ are topological semigroups under addition.

(b)

$\mathbb {Z}$ , $\mathbb {R}$ , and $\mathbb {C}$ are topological groups under addition.

(c)

$\partial \mathbb {D}$ is a topological group under multiplication.

(d)

If $X$ is a topological space and $G = \left\{ f \in C(X) : f(X) \subset \partial \mathbb {D} \right\}$ , define $(fg)(x) = f(x) g(x)$ for $f, g$ in $G$ and $x$ in $X$ . Then $G$ is a group. If $G$ is given the topology of uniform convergence on $X$ , $G$ is a topological group.

(e)

For $n \geq 1$ , let $M_n(\mathbb {C}) =$ the $n \times n$ matrices with entries in $\mathbb {C}$ ; $O(n) \equiv \left\{ A \in M_n(\mathbb {C}) : A \text{ is invertible and } A^{-1} = A^* \right\}$ ; $SO(n) \equiv \left\{ A \in O(n) : \det A = 1 \right\}$ . If $M_n(\mathbb {C})$ is given the usual topology, $O(n)$ and $SO(n)$ are compact topological groups under multiplication.

Problem V.11.3

Show that Theorems 11.4 and 11.5 are equivalent, where:

Theorem 11.4: If $G$ is a compact topological group, then there is a unique positive regular Borel measure $m$ on $G$ such that (a) $m(G) = 1$ ; (b) if $U$ is a nonempty open subset of $G$ , then $m(U) > 0$ ; (c) if $\Delta$ is any Borel subset of $G$ and $x \in G$ , then $m(\Delta ) = m(\Delta x) = m(x \Delta ) = m(\Delta^{-1})$ , where $\Delta x \equiv \left\{ ax : a \in \Delta \right\}$ , $x \Delta \equiv \left\{ xa : a \in \Delta \right\}$ , and $\Delta^{-1} \equiv \left\{ a^{-1} : a \in \Delta \right\}$ .
Theorem 11.5: If $G$ is a compact topological group, then there exists a unique positive linear functional $I : C(G) \to \mathbb {F}$ such that (a) $I(1) = 1$ ; (b) if $f \in C(G)$ , $f \geq 0$ , and $f \neq 0$ , then $I(f) > 0$ ; (c) if $f \in C(G)$ and $x \in G$ , then $I(f) = I(f_x) = I({}_x f) = I(f^\# )$ .

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Problem V.11.4

Let $G$ be a locally compact group. If $m$ is a regular Borel measure on $G$ , show that any two of the following properties imply the third: (a) $m(\Delta x) = m(\Delta )$ for every Borel set $\Delta$ and every $x$ in $G$ ; (b) $m(x \Delta ) = m(\Delta )$ for every Borel set $\Delta$ and every $x$ in $G$ ; (c) $m(\Delta ) = m(\Delta^{-1})$ for every Borel set $\Delta$ .

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Problem V.11.5

Show that the maps $S_0, L_x, R_x$ are linear isometries of $M(G)$ onto $M(G)$ .

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Problem V.11.6

Prove (11.7): if $e =$ the identity of $G$ , then

\begin{cases} L_x R_y = R_y L_x \\ L_x L_y = L_{yx} \\ R_x R_y = R_{xy} \\ S_0^2 = L_e = R_e = \text{the identity on } M(G) \\ S_0 L_x R_y = L_{y^{-1}} R_{x^{-1}} S_0 \end{cases}

for $x, y$ in $G$ .

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Problem V.11.7

Let $S$ be an abelian semigroup and show that there is a positive linear functional $L : l^\infty (S) \to \mathbb {F}$ such that (a) $L(1) = 1$ , (b) $L(f_x) = L(f)$ for every $f$ in $l^\infty (S)$ .

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Problem V.11.8

If $S = \mathbb {N}$ , what does Exercise 7 say about Banach limits?

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Problem V.11.9

If $G$ is a compact group, $f : G \to \mathbb {F}$ is a continuous function, and $\varepsilon > 0$ , show that there is a neighborhood $U$ of the identity in $G$ such that $\left|f(x) - f(y)\right| < \varepsilon$ whenever $xy^{-1} \in U$ . (Note that this says that every continuous function on a compact group is uniformly continuous.)

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Problem V.11.10

(a)(b)(c)(d)

If $G$ is a locally compact group and $f \in C_b(G)$ , let $\mathcal{O}(f) \equiv$ the closure of $\left\{ f_x : x \in G \right\}$ in $C_b(G)$ . Let $AP(G) = \left\{ f \in C_b(G) : \mathcal{O}(f) \text{ is compact} \right\}$ . Functions in $AP(G)$ are called almost periodic.

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(a)

Show that every periodic function in $C_b(\mathbb {R})$ belongs to $AP(\mathbb {R})$ .

(b)

If $G$ is compact, show that $AP(G) = C(G)$ .

(c)

Show that if $f \in C_b(\mathbb {R})$ , then $f \in AP(\mathbb {R})$ if and only if for every $\varepsilon > 0$ there is a positive number $T$ such that in every interval of length $T$ there is a number $p$ such that $\left|f(x) - f(x+p)\right| < \varepsilon$ for all $x$ in $\mathbb {R}$ .

(d)

If $G$ is not compact, then the only function in $AP(G)$ having compact support is the zero function.

For more information on this topic, see Exercise 13.5 below.