Continuous homomorphisms from a compact monoid into the units of a normed ring #
A continuous homomorphism f from a compact monoid into the units of a seminormed ring with
multiplicative norm and ‖1‖ = 1 (for instance a normed division ring) takes values of norm 1:
the norms ‖f (g ^ n)‖ = ‖f g‖ ^ n and ‖(f (g ^ n))⁻¹‖ = ‖(f g)⁻¹‖ ^ n stay bounded for all
n : ℕ, so both ‖f g‖ and ‖(f g)⁻¹‖ are at most 1, while their product is 1.
For 𝕜 = ℂ this says that a continuous character of a compact group is unitary.
theorem
ContinuousMonoidHom.norm_apply_eq_one_of_compactSpace
{M : Type u_1}
{R : Type u_2}
[Monoid M]
[TopologicalSpace M]
[CompactSpace M]
[SeminormedRing R]
[NormMulClass R]
[NormOneClass R]
(f : M →ₜ* Rˣ)
(g : M)
:
A continuous homomorphism from a compact monoid into the units of a seminormed ring with
multiplicative norm takes values of norm 1.