434 X A Character Study
thenL^1 (G)is aBanach algebraand
M(f∗g)=M(f)M(g) for allf,g∈L^1 (G).Aunitary representationofGin a Hilbert spaceH is a mapρofGinto the set
of all linear transformations ofH which maps the identity elementeofGinto the
identity transformation ofH:
ρ(e)=I,which preserves not only products inG:
ρ(st)=ρ(s)ρ(t) for alls,t∈G,but also inner products inH:
(ρ(s)u,ρ(s)v)=(u,v) for alls∈Gand allu,v∈H,andforwhichthemap(s,v)→ρ(s)vofG×H intoHis continuous (or, equiv-
alently, for which the maps→(ρ(s)v,v)ofGintoCis continuous atefor every
v∈H).
For example, any locally compact groupGhas a unitary representationρinL^2 (G),
itsregular representation,definedby
(ρ(t)f)(s)=f(t−^1 s) for allf∈L^2 (G)and alls,t∈G.Ifρis a unitary representation ofGin a Hilbert spaceH, and if a closed subspace
VofHis invariant underρ(s)for everys∈G, then so also is its orthogonal comple-
mentV⊥. The representationρis said to beirreducibleif the only closed subspaces of
Hwhich are invariant underρ(s)for everys∈GareHand{ 0 }. It has been shown
by Gelfand and Raikov (1943) that, for any locally compact groupGand anys∈G\e,
there is an irreducible unitary representationρofGwithρ(s)=I.
Consider now the case in which the locally compact group G is abelian.Then
any irreducible unitary representation ofGis one-dimensional. Hence if we define a
characterofGto be a continuous functionχ:G→Csuch that
(i)χ(st)=χ(s)χ(t)for alls,t∈G,
(ii)|χ(s)|=1foreverys∈G,
then every irreducible unitary representation is a character, and vice versa.
If multiplication and inversion of characters are defined pointwise, then the setGˆ
of all characters ofGis again an abelian group, thedual groupofG. Moreover, we
can put a topology onGˆby defining a subset ofGˆto be open if it is a union of sets of
the form
N(ψ,ε,K)={χ∈Gˆ:|χ(s)/ψ(s)− 1 |<εfor alls∈K},whereψ∈Gˆ,ε>0andKis a compact subset ofG.ThenGˆis not only abelian, but
also a locally compact topological group.