The Group Algebra of a Locally Compact Abelian Group

Prove that if $L^1(G)$ has an identity, then $G$ is discrete.

If $f \in L^\infty (G)$, show that $x \mapsto f_x$ is a continuous function from $G$ into $(L^\infty (G), \text{wk}^*)$, where $f_x(y) = f(x^{-1}y)$.

Is there a measure $\mu$ on $\mathbb {R}$ different from Lebesgue measure such that for $f$ in $L^1(\mu )$, $x \mapsto f_x$ is continuous? Is there a measure for which this map is discontinuous?

If $f \in C_0(G)$, show that $x \mapsto f_x$ is a continuous map from $G \to C_0(G)$.

If $f \in L^\infty (G)$ and $f$ is uniformly continuous on $G$, show that $x \mapsto f_x$ is a continuous function from $G \to L^\infty (G)$. Is the converse true?

If $K$ is a compact subset of $G$, $\gamma_0 \in \Gamma$, and $\varepsilon > 0$, let $U(K, \gamma_0, \varepsilon ) = \left\{ \gamma \in \Gamma : \left|\gamma (x) - \gamma_0(x)\right| < \varepsilon \text{ for all } x \text{ in } K\right\}$. Show that the collection of all such sets is a base for the topology of $\Gamma$. (This says that the topology on $\Gamma$ is the compact-open topology.)

Show that there is a discontinuous homomorphism $\gamma : \mathbb {R} \to \mathbb {T}$. If $\gamma : \mathbb {R} \to \mathbb {T}$ is a homomorphism that is a Borel function, show that $\gamma$ is continuous.

If $G$ is a compact abelian group, show that the linear span of $\Gamma$ is dense in $C(G)$.

If $G$ is a compact abelian group, show that $\Gamma$ forms an orthonormal basis in $L^2(G)$.

If $G$ is a compact abelian group, show that $G$ is metrizable if and only if $\Gamma$ is countable.

Let $\{ G_\alpha \}$ be a family of compact abelian groups and $G = \prod_\alpha G_\alpha$. If $\Gamma_\alpha = \hat{G}_\alpha$, show that the character group of $G$ is $\left\{ \{ \gamma_\alpha \} \in \prod_\alpha \Gamma_\alpha : \gamma_\alpha = e \text{ except for at most a finite number of } \alpha \right\}$.