Exercise 14, § 2

(a) Suppose that every neighbourhood of $e$ in G contains an element of infinite order. There exists a compact metrizable perfect totally disconnected subset P of G which is a Kronecker set (exercise 15 of II, p. 265), and a nonzero diffuse measure concentrated on P. (Reduce to the case where G is metrizable by using exercise 3, c). Then imitate the construction of TG, IX, p. 64, lemma 3, and use at the same time part e) of exercise 15 of II, p. 265.)

(b) If $G = (\mathbf{Z}/q\mathbf{Z})^{\mathbf{N}}$, there exists a compact perfect totally disconnected subset P of G which is of type $K_q$, and a nonzero diffuse measure concentrated on P.

(c) Suppose G is not discrete. There exists an element $\tau$ of $\mathscr{M}^1(G)$ such that $\|\mathscr{F}(\tau )\|_{\infty}< \varrho (\tau )$, where $\varrho (\tau )$ is the spectral radius in $\mathscr{M}^1(G)$. There exists a noninvertible element $\tau '$ of $\mathscr{M}^1(G)$ such that $\mathscr{F}(\tau ')\geqslant 1$ everywhere on $\widehat{G}$. There exists a nonhermitian character of $\mathscr{M}^1(G)$. (Use a), b), exercise 3, c), and part f) of exercise 15 of II, p. 265.)

(d) Deduce from c) that $\widehat{G}$, which is open in $\mathsf{X}(\mathscr{M}^1(G))$, is not dense in $\mathsf{X}(\mathscr{M}^1(G))$.