T η whichwefinishnow: A η forfinite A ,for Z andhadanunfinishedproofofthebijectivityof A G η isinjectiveandthat η isinjectiveonceoneacceptsthefundamentaltheoremthateverycompactgrouphasenoughfinitedimensionalcontinu-ousrepresentationstoseparatethepoints.Weveri

May 5, 2006

Introduction to Compact Groups We introduced the bidual b

Ab (resp. b

Gb ) of an abelian (resp. compact abelian) group and the evaluation mor- phisms

η_A:A → b

Ab η_G:G → b G .b

The goal is to show that these are isomorphisms. We proved that η_A is injective and that η_G is injective once one accepts the fundamental theorem that every compact group has enough finite dimensional continu- ous representations to separate the points.

We verified the bijectivity of η_A for finite A, for Z and had an unfinished proof of the bijectivity of η_T which we finish now:

The character group Tb consists of the multiples n·

for the identity character :T → T which form an infinite cyclic group, freely generated by . Thus the character group hic = Hom(hi,T) is isomorphic to T by the map Ω 7→ Ω().

Let t ∈ T, and χ = n· a character of T. Then by the definition of η_T , we have η_T(t)(χ) = η_T(t)(n·) = (n.)(t) = n·((t)) = n·t. Now t ∈ ker η_T if n·t = 0 for all n, notably for n = 1 and so η_T is injective.

If Ω:hi → T is given we set t = Ω() and have η_T(t)(χ) = n·t = n·Ω() = Ω(n·) = Ω(χ) and thus Ω = η_T(t). This shows that η_T is surjective.

Alternative proof using extra info: By the Fundamental Theorem (info!), η_T is injective and thus η_T(T) is a nonsingleton compact connected subgroup of b

Tb ∼= Zb ∼= T. But the only compact connected subgroups of T are {0} and T (info!) and so η_T(T) = b

Tb . [Dahmen]

Program for today.

Projective Limits. Character groups of abelian groups as projective limits.