Mathematics 442C
Exercise sheet 5
Due Thursday 21st January 2010
1. Let A and B be Banach algebras, and let A0 be a dense subalgebra of A.
If θ0 : A0 → B is a continuous homomorphism, show that there is a unique
continuous homomorphism θ : A → B such that θ|A0 = θ0 .
2. Let a be a normal element of a unital C*-algebra. Prove the following assertions.
(a) a is hermitian if and only σ(a) ⊆ R.
(b) a is unitary if and only if σ(a) ⊆ T.
(c) a is a projection if and only if σ(a) ⊆ {0, 1}.
If a is not normal, then these assertions may all fail. Find an example to
demonstrate this.
3. Let A be a unital C*-algebra. For a ∈ A, recall that ea was defined in
Exercise 2.9 as
a
e = lim pn (a) where pn (x) =
n→∞
n
X
xk
k=0
k!
.
Show that if h is a hermitian element of A, then eih is unitary.
4. Prove Lemma 4.3.5:
Let A be a unital C*-algebra. If a is a normal element of A then C ∗ (1, a)
is a unital abelian C*-algebra.
5. Prove Lemma 4.3.8:
Suppose that X and Y are compact Hausdorff topological spaces and
ψ : X → Y is a homeomorphism. The map ψ t : C(Y ) → C(X), f 7→ f ◦ ψ is
a well-defined isometric unital ∗-isomorphism.
6. Let A be a unital C*-algebra.
(a) Show that if a, b ∈ A+ with ab = ba then ab ∈ A+ .
(b) Give an example with A = M2 (C) to show that there may be a, b ∈ A+
such that ab 6∈ A+ .
7. Prove Lemma 4.4.9:
If A is a unital C*-algebra and a ∈ A+ then a ≤ kak1.
8. Let A be a unital C*-algebra. Show that S(A) is a convex subset of A∗ .
9. Let A be a unital abelian Banach algebra. Let a be a normal element of A
and let f ∈ C(σ(a)) and g ∈ C(σ(f (a))).
(a) Check that f (a), (g ◦ f )(a) and g(f (a)) are well-defined elements of A.
Explain why it is not obvious that (g ◦ f )(a) = g(f (a)).
(b) Show that (g ◦ f )(a) = g(f (a)) if g is of the following special forms:
(i). g(λ) = λk for some integer k ≥ 0;
(ii). g(λ) = λk λj for some integers k, j ≥ 0;
(iii). g is a trigonometric polynomial, meaning that
g(λ) =
n
X
αk,j λk λj ,
λ ∈ σ(f (a))
k,j=0
for some αk,j ∈ C.
(c) Use the Stone-Weierstrass theorem to deduce that (g ◦ f )(a) = g(f (a))
for any g ∈ C(σ(f (a))). [Hint: approximate g.]
10. Let I be an index set, and for each i ∈ I, let Hi be a Hilbert space and let
Ti ∈ B(Hi ) with supi∈I kTi k < ∞. Show that the operator
M M
T =
Ti ∈ B
Hi
i∈I
i∈I
satisfies kT k = supi∈I kTi k.
[All norms in the statement of this question are operator norms.]
11. (a) Let H be a Hilbert space. A unit vector is a vector x ∈ H with kxk = 1.
Given a unit vector x, let ωx : B(H) → C, T 7→ hT x, xi. Show that ωx
is a state on B(H) for every unit vector x.
(b) Let A be a unital C*-algebra and let τ ∈ S(A), and consider the
Hilbert space H and the ∗-homomorphism π : A → B(H) constructed
in Lemma 4.5.9. Find a unit vector x ∈ H such that
τ (a) = hπ(a)x, xi for a ∈ A.
(c) The maps ωx considered in (a) are called vector states. If A is a
unital C*-algebra, consider the Hilbert space H and the isometric ∗isomorphism π : A → B(H) constructed in the GNS Theorem 4.5.10.
We call π the universal representation of A. Show that “every state is
a vector state in the universal representation”; or, more precisely, show
that
S(A) = {ωy ◦ π : y is a unit vector in H}.
[Warning: H and π are different in parts (b) and (c)!]
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