MATH 656 - Spring 2016
Homework Assignment 3
Due: Thursday May 12, 2016
1. Let A be a unital C∗ –algebra and let a ∈ A be normal. Show that if f ∈ C(σ(a)) and
g ∈ C(σ(f (a))), then
(g ◦ f )(a) = g(f (a)).
2. Let A ⊆ B be C∗ -algebras. Let J C B be a closed 2-sided ideal of B.
(a) Show that A ∩ J C A is a closed 2-sided ideal.
(b) Show that (A + J)/J and A/(A ∩ J) are ∗-isomorphic.
(c) Show that A+J is a C∗ -subalgebra of B. (Hint: From (b) we know that (A+J)/J
is a C∗ -algebra, hence complete. Conclude from this that the ∗-algebra A+J ⊂ B
must have been complete to begin with).
3. Show that if H, K are Hilbert spaces, and x ∈ B(H) and y ∈ B(K) are normal
operators, then there is a ∗-isomorphism between C ∗ (1, x) and C ∗ (1, y) taking x to y
if and only if σ(x) = σ(y).
4. Let A ⊆ B(H) be a C∗ -algebra and let a ∈ A have polar decomposition a = u|a|. Let
f ∈ C(σ(|a|)) be such that f (0) = 0. Show that uf (|a|) ∈ A. (Hint: Approximate f
by polynomials with zero constant term.)
5. Suppose that (H, π, ξ0 ) is a cyclic representation of a C∗ -algebra A with unit cyclic
vector ξ0 ∈ H. Define ϕ ∈ S(A) by
ϕ(a) = hπ(a)ξ0 |ξ0 i
(a ∈ A).
Show that π is unitarily equivalent to the GNS construction (Hϕ , πϕ , ξϕ ) via a unitary
operator U : Hϕ → H such that U ξϕ = ξ0 . (That is, πϕ = U ∗ π(·)U .) This shows that
the GNS construction is unique, up to unitary equivalence of representations.
6. A state ϕ on a C∗ -algebra A is called faithful if ϕ(a∗ a) = 0 iff a = 0. Show that the
GNS representation πϕ of a faithful state is faithful.
7. Let A be a finite dimensional C∗ -algebra. Show that A admits a faithful representation
on a finite dimensional Hilbert space. (Hint: Show that in this situation, it suffices to
take finitely many states in the Gelfand–Naimark theorem).
8. Let ϕ : A → B be a positive map between two C∗ -algebras A and B. Show that ϕ is
bounded.
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9. Let A, B be unital C∗ -algebras and ϕ : A → B a 2-positive map.
(a) Prove the Schwarz inequality for ϕ:
kϕ(b∗ a)k ≤ kϕ(b∗ b)k1/2 kϕ(a∗ a)k1/2
(a, b ∈ A)
(b) Use the above to conclude that
kϕ(a)k2 ≤ kϕ(1)kkϕ(a∗ a)k
(a ∈ A).
(c) Use the above to conclude that kϕk = kϕ(1)k. (Note: This is not always true if
ϕ is merely positive.)
Hint: For (a), you may use the following lemma (applied to a suitable choice of X).
Lemma 0.1. Let A ⊂ B(H) be a unital C∗ -algebra and let
x y
X=
∈ M2 (A).
z w
Then X ∈ M2 (A)+ if and only if x, w ≥ 0, y = z ∗ , and |hyξ|ηi|2 ≤ hxξ|ξihwη|ηi for
every ξ, η ∈ H.
10. Let t : M2 (C) → M2 (C) denote the transpose map. In class we showed that t is
positive. Show that t is not 2-positive.
11. Let H be a separable Hilbert space with ONB (en )n∈N . Show that the map
X
d : b1 (B(H)) × b1 (B(H)) → [0, ∞);
d(x, y) =
2−n−m |h(x − y)en |em i|
1≤n,m<∞
defines a metric on the closed unit ball b1 (B(H)) of B(H). Consequently, the relative
weak operator topology on b1 (B(H)) is metrizable.
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