Mathematics 442C
Exercise sheet 2
Due 12pm, Thursday 29th October 2009
1. Prove Proposition 1.3.3:
Let A be a unital Banach algebra and let a, b ∈ A. Show that:
(a) If 1 − ab ∈ Inv A then 1 − ba ∈ Inv A, with
(1 − ba)−1 = 1 + b(1 − ab)−1 a.
(b) σ(ab) \ {0} = σ(ba) \ {0}.
2. Show that if a is an invertible element of a unital Banach algebra, then
σ(a−1 ) = {λ−1 : λ ∈ σ(a)}.
3. Let a be an element of a unital Banach algebra A. Show that the function
S : {λ ∈ C : |λ| < 1/r(a)} → A,
λ 7→ (1 − λa)−1
is holomorphic. [Hint: use (⋆)].
4. Prove Lemma 1.3.18:
Let A be a Banach algebra. Show that if T ⊆ S ⊆ A then S ′ ⊆ T ′ and
S ⊆ S ′′ . Deduce that S ′′′ = S ′ .
5. Complete the proof of Theorem 1.5.3:
Show that if A is a unital Banach algebra with unit 1 and I is a closed
proper ideal of A, then A/I is a unital Banach algebra, with unit 1 + I.
[Remember to show that k1 + Ik = 1.]
6. Show that the only homomorphism M2 (C) → C is the zero homomorphism.
[Hint: if θ is such a homomorphism, what is θ(ab − ba)?]
7. Let D = {z ∈ C : |z| ≤ 1}, let T = {z ∈ C : |z| = 1} and let
K = {f ∈ C(D) : f (z) = 0 for z ∈ T}.
(a) Show that K is a closed ideal of C(D).
(b) Show that the map
θ : C(D)/K → C(T),
f + K 7→ f |T
is a well-defined isometric isomorphism from the quotient Banach algebra C(D)/K onto the Banach algebra C(T).
Bonus questions
8. As in Exercise 4(e) from sheet 1, let (en )n≥1 be an orthonormal basis of a
Hilbert space H and let S ∈ B(H) be defined by Sen = en+1 for n ≥ 1.
(a) What are σ(S ∗ S) and σ(SS ∗ )? [Hint: use Exercise 1].
(b) Show that if λ ∈ C and T ∈ B(H) with T (λI − S ∗ ) = I, then
T en = λ−1 en + λ−2 en−1 + · · · + λ−n e1 .
(c) Use (b) to show that σ(S ∗ ) = D.
(d) Show that T ∈ Inv B(H) ⇐⇒ T ∗ ∈ Inv B(H). Deduce that σ(S) = D.
9. For n ≥ 0, let pn be the polynomial pn (x) =
Pn
xk
k=0 k! .
(a) Explain why pn (x) → ex as n → ∞ for every real number x.
(b) Let qn (x, y) = pn (x)pn (y)−pn (x+y). Show that qn (x, y) is a polynomial
in x and y with non-negative coefficients.
[Hint: use the binomial theorem to expand pn (x + y)].
(c) Show that qn (x, y) → 0 as n → ∞ for any real numbers x, y.
(d) Let A be a unital Banach algebra. Show that if a ∈ A then pn (a)
converges as n → ∞.
(e) Show that if a, b ∈ A with ab = ba then kpn (a)pn (b) − pn (a + b)k ≤
qn (kak, kbk).
(f) Let ea = limn→∞ pn (a) for a ∈ A. Show that if a, b ∈ A with ab = ba
then ea+b = ea eb .
(g) Deduce that {ea1 ea2 . . . ean : n ≥ 1, ai ∈ A} ⊆ Inv A.
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