Exercise 1, § 4

Let G be a compact topological group and $\mathscr{B}$ the family of characters of the irreducible unitary representations of G.

a) The pair $(R(G),\mathscr{B})$ is a natural basic algebra with unit (cf. exercise 6 of III, p. 130).

b) Suppose that G is finite. The algebra R(G) is then of finite rank; calculate pf($\pi$ ) for every $\pi \in \widehat{G}$, and identify the representation of G corresponding to the element $r_0$ defined in question j) of loc. cit.