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1. Let A = C(T), B ⊂ C(T) the closure of polynomials in z. Show that the spectrum of z in A
is T, while the spectrum of z in B is the unit disc.
2. Show that if r(x) = kxk for all x in a Banach algebra A then A is commutative. Hint:
show that kezx ye−zx k = kyk for all x, y ∈ A and z ∈ C, and conclude by Liouville’s theorem that
ezx ye−zx = y.
3. Show that for any f ∈ L1 (R) the map R → L1 (R), y 7→ fy , is continuous, where fy (x) =
f (x − y).
4. Recall that for a locally compact abelian group G the dual group Ĝ is the group of continuous
homomorphisms G → T with operation of pointwise multiplication.
(i) Show that if G is a finite abelian group then Ĝ ∼
= G.
(ii) Show that for any χ ∈ R̂ there exists x ∈ R such that χ(t) = eixt . Therefore R̂ ∼
= R.
(iii) Show that for any χ ∈ T̂ there exists n ∈ Z such that χ(z) = z n . Therefore T̂ ∼
= Z.
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