1. Let be A be the Banach algebra of strictly upper triangular 2 by 2 matrices
0 λ
, λ ∈ C.
0 0
Show that  = ∅.
2. Let A = C 1 [0, 1] be the algebra with pointwise multiplication and norm
kf k = kf k∞ + kf 0 k∞ .
(i) Prove that A is a Banach algebra. Show that any multiplicative functional on A is of the form
f 7→ f (t) for some t ∈ [0, 1]. Therefore the Gelfand transform is the embedding map C 1 [0, 1] →
C[0, 1].
(ii) Let t ∈ [0, 1]. Show that I = {f ∈ A | f (t) = f 0 (t) = 0} is a closed ideal in A of
codimension 2. Describe A/I and conclude that it is non-semisimple. Therefore the quotient of a
semisimple Banach algebra by a closed ideal is not necessarily semisimple.
3. Let K be a compact space, A = C(K). Show that there is a one-to-one correspondence
between closed ideals in A and closed subsets of K. Namely, the ideal corresponding to a closed
subset T ⊂ K is {f ∈ C(K) : f |T ≡ 0}.
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