Mathematics 442C
Exercise sheet 4
Due 12pm, Thursday 3rd December 2009
1. Consider the matrices I =
1 0
0 1
and T =
0 1
0 0
in M2 (C).
(a) Show that A = span{I, T } is a unital abelian Banach subalgebra of M2 (C).
(b) What is Ω(A)?
(c) Show that the Gelfand representation of A is not injective.
[This complements Remark 3.2.5].
2. Let X be a compact Hausdorff topological space, and let A be a Banach
subalgebra of C(X). Suppose that 1 ∈ A and that for every x, y ∈ X with
x 6= y, there is f ∈ A with f (x) 6= f (y).
If x ∈ X, let εx : A → C, f 7→ f (x) be the evaluation map for x. Let
ε : X → Ω(A), x 7→ εx .
(a) Show that ε is well-defined, continuous and injective.
(b) Deduce that X is homeomorphic to ε(X).
(c) Give an example to show that ε need not be surjective.
3. Let X = (0, 1], a half-open interval in R, and let A = BC (X).
(a) Let ε : X → Ω(A), x 7→ εx . Show that X is homeomorphic to ε(X).
(b) Show that ε(X) is dense in Ω(A).
(c) Make the following statement precise, and prove it:
If f ∈ A then the Gelfand transform fb is the unique continuous extension of f to Ω(A).
4. Let A be an abelian unital Banach algebra. For a ∈ A, let
P (a) = {p(a) : p is a polynomial}.
(a) Suppose that a ∈ A and P (a) is dense in A. Show that the map
θ : Ω(A) → σ(a),
τ 7→ τ (a)
is a homeomorphism.
(b) Deduce that if a, b ∈ A and P (a) = P (b), then σ(a) is homeomorphic
to σ(b). Must σ(a) be equal to σ(b)?
5. For sets W, Z ⊆ C, let us write W + Z = {w + z : w ∈ W, z ∈ Z} and
W Z = {wz : w ∈ W, z ∈ Z}. Let A be a unital abelian Banach algebra.
(a) Show that if a, b ∈ A then σ(a + b) ⊆ σ(a) + σ(b) and σ(ab) ⊆ σ(a)σ(b).
(b) Find an example for which both of these inclusions are strict.
(c) Show that (a) may fail if A is an arbitrary Banach algebra.
6. Let A be a unital abelian Banach algebra. Show that the Gelfand representation γ : A → C(Ω(A)) is an isometry if and only if ka2 k = kak2 for every
a ∈ A.