Mathematics 442C
Exercise sheet 1
Due 12pm, Thursday 15th October 2009
1. Prove Remark 1.1.2:
Show that if A is a Banach algebra then the product A × A → A is
continuous.
[Hint: it suffices to show that if an , a, bn , b ∈ A and an → a and bn → b as
n → ∞, then an bn → ab as n → ∞.]
2. (a) Prove that A(D) is a unital abelian Banach algebra.
(b) Prove that C0 (R) is a non-unital abelian Banach algebra.
(c) Prove that, if X is a Banach space with dim X > 1, then the Banach
algebra B(X) is non-abelian and unital.
3. Prove Remark 1.2.3:
(a) Show that an algebra can have at most one identity element.
(b) Let A be a Banach algebra and let a ∈ A. Show that La : A → A,
b 7→ ab is a bounded linear operator on A.
(c) Let A be a non-zero Banach algebra with norm k · k which contains an
identity element 1. Show that the map
| · | : A → [0, ∞),
a 7→ kLa k
is a norm on A which is equivalent to k · k such that |1| = 1.
4. Prove Remark 1.2.5:
Let A be a unital Banach algebra. Show that:
(a) Inv A forms a group under multiplication.
(b) If a ∈ A is left invertible and right invertible so that ba = 1 and ac = 1
for some b, c ∈ A, then a is invertible.
(c) If a = bc = cb then a is invertible if and only if b and c are invertible.
(d) If b1 , . . . , bn are commuting elements of A (meaning that bi bj = bj bi for
1 ≤ i, j ≤ n) then b1 b2 . . . bn is invertible if and only if b1 , . . . , bn are all
invertible.
(e) If (en )n≥1 is an orthonormal basis of a Hilbert space H and S ∈ B(H) is
defined by Sen = en+1 , n ≥ 1 and S ∗ is the adjoint of S (see [FA 4.18])
then S ∗ S is invertible although S ∗ and S are not.
5. Let X be a topological space. Show that
Inv BC (X) = {f ∈ BC (X) : 0 6∈ f (X)}
where f (X) denotes the closure of the range of f .
6. Let A be unital Banach algebra. Show that A is abelian if and only if Inv A
is an abelian group.