Exercise 24, § 4

A topological group G is said to be minimally almost periodic if every continuous homomorphism of G into a compact group is trivial.

a) A topological group G is minimally almost periodic if and only if G admits no nontrivial finite-dimensional unitary representation. If G is commutative, then G is minimally almost periodic if and only if every continuous unitary character of G is trivial. b) Let G be a topological group and let $g\in G$ be such that, for every integer $k\geqslant 1$, there exists an integer $m\geqslant 1$ such that $g^{km}$ is conjugate to $g$. Then $\varrho (g) = 1_E$ for every finite-dimensional unitary representation $\varrho$ of G in a Hilbert space E.

c) Let $k$ be a subfield of $\mathbf{C}$. The group $\mathbf{S}\mathbf{L}(2, k)$ endowed with the discrete topology is minimally almost periodic. (Verify that the preceding question

applies to for every $z\in k.)$

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