C*-ALGEBRAS
1. Basic C ∗ -algebra theory
The following is a short version of Section 2.1 in Murphy’s book.
References like 2.1.1 are to Murphy, while references like 1.1 is internal.
Definition 1.1. A Banach ∗ -algebra A is called a C ∗ -algebra if for all
a ∈ A we have
ka∗ ak = kak2 .
We shall always assume A 6= {0}. A may have an identity 1, if so it
follows that k1k = 1.
Problem 1.2. Show that in a C ∗ -algebra one has
kak = sup kabk.
kbk=1
Lemma. If u ∈ A is unitary (u∗ u = uu∗ = 1), then σ(u) ⊂ T.
Theorem. 2.1.1. If a = a∗ ∈ A, then kak = r(a).
Proof. See Murphy.
Problem 1.3. Show that if a ∈ A is normal ( a∗ a = aa∗ ) then also
kak = r(a).
Theorem. 2.1.6. If 1 ∈
/ A, then à is a C ∗ -algebra with norm
k(a, λ)k = sup kab + λbk = sup kba + λbk.
kbk=1
kbk=1
and adjoint
(a, λ)∗ = (a∗ , λ).
Proof. (The statement and proof is slightly different from Murphy’s.)
The equality in the definition follws from ∗ -properties.
(i) k(a, 0)k = kak. If λ 6= 0, then (a, λ) = 0 =⇒ ab + λb = 0 for all
b ∈ A =⇒ − λ1 a is an identity, contradiction. So we have a norm.
(ii) Ã is complete. (Proof?)
Date: September 26, 2016.
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C*-ALGEBRAS
(iii) Submultiplicative:
k(a, λ)(b, µ)k = k(ab + λb + µa, λµ)k
= sup kabc + λbc + µac + λµck
kck=1
=
sup
kdabc + λdbc + µdac + λµdck
kck=kdk=1
=
sup
k(da + λd)(bc + µc)k
kck=kdk=1
≤ sup k(da + λd)k · k sup k(bc + µc)k
kkdk=1
kck=1
= k(a, λ)k · k(b, µ)k
(iv) C ∗ -property:
k(a, λ)k2 = sup kab + λbk2
kbk=1
= sup kb∗ a∗ ab + λb∗ a∗ b + λb∗ ab + |λ|2 b∗ bk
kbk=1
≤ sup ka∗ ab + λa∗ b + λab + |λ|2 bk
kbk=1
= k(a∗ a + λa∗ + λa, |λ|2 )k
= k(a, λ)∗ (a, λ)k2
≤ k(a, λ)k2 .
So equality holds everywhere and we have a C ∗ -norm.
For the nest results we refer to Murphy for proofs.
Theorem. 2.1.7. A ∗-homomorphism ϕ : A 7→ B from a Banach
-algebra A to a C ∗ -algebra B is necessarily norm-decreasing.
∗
Theorem. 2.1.8. If a = a∗ in a C ∗ -algebra A, then σ(a) ⊂ R.
Definition. Recall that Ω(A) is the set of non-zero multiplicative linear
functionals on A.
Theorem. 2.1.9. If τ ∈ Ω(A) then τ (a∗ ) = τ (a).
Lemma. 2.1.10. (Gelfand) The Gelfand representation
ϕ : A 7→ C0 (Ω(A)) given by ϕ(a) = b
a,
is an isometric ∗ -isomorphism.
Lemma 1.4. Let a = a∗ ∈ A, and suppose A = C ∗ (1, a). Then
Ω(A) ∼
= σ(a) and the Gelfand representation ϕ maps a to the function
f (t) = t.
C*-ALGEBRAS
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Proof. From Thm.1.3.7 we have that b
a : Ω(A) 7→ σA (a) given by b
a(τ ) =
τ (a) is a homeomorphism.
ϕ : A 7→ C0 (Ω(A)) is given by ϕ(b) = bb. So ψ : A 7→ C0 (σ(A)) is
given by for b ∈ A by
ψ(b)(t) = ϕ(b)[b
a−1 (t)] = bb(b
a−1 (t)).
So for b = a we get ψ(a)(t) = t.
Lemma 1.5. Let B be a closed ∗ -subalgebra of a C ∗ -algebra A with
1 ∈ A ∩ B and b = b∗ ∈ B. Then σA (b) = σB (b).
Proof. Suppose b = b∗ ∈ B, clearly σA (b) ⊂ σB (b). Suppose λ ∈ σB (b),
by replacing b with b − λ1 we can assume λ = 0. So we want to show
that if b is invertible in A, then b is also invertible in B.
Suppose we have a ∈ A with ab = ba = 1. We may assume B =
C ∗ (1, b) and A = C ∗ (1, a, b).
From Thm.1.3.7 we have Ω(B) = σB (b). From Thm.1.3.4 the map
bb : Ω(A) 7→ σA (b)
is onto. Claim that bb is 1-1. Using that τ (ab) = 1 we get
bb(τ1 ) = bb(τ2 ) =⇒ τ1 (b) = τ2 (b) =⇒ τ1 (a) = τ2 (a) =⇒ τ1 = τ2 on A.
So Ω(A) = σA (b).
The inclusion ι : B 7→ A gives the map ι∗ : Ω(A) :7→ Ω(B) by
∗
ι (τ ) = τ ◦ ι. We then have
σA (b)
b
b−1
/
Ω(A)
ι∗
/
Ω(B)
b
b
/
σB (b) .
From Gelfand’s theorem we have
C(σB (b)) ∼
=B
ι
/
A∼
= C(σA (b))
This gives a map ϕ : C(σB (b)) 7→ C(σA (b)). We observe that ϕ(bb) =
bb|σ (b) and therefore ϕ(f ) = f |σ (b) for all f ∈ C(σB (b).
A
A
Now if σA (b) ( σB (b) there is a non-zero f ∈ C(σB (b)) with f (σA (b)) =
{0}, so ϕ(f ) = 0. This contradicts that ϕ is 1-1.
Theorem. 2.1.11. Let B be a closed ∗ -subalgebra of a C ∗ -algebra A
with 1 ∈ A ∩ B and b ∈ B. Then σA (b) = σB (b).
Proof. We just proved it for b = b∗ . For the general case, see Murphy.
In the above proof we used a special case of the following:
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C*-ALGEBRAS
Problem 1.6. Suppose we have a continuous map θ : Ω 7→ Ω0 between
compact spaces. Then θt (f ) = f ◦ θ defines a ∗ -homomorphism θt :
C(Ω0 ) 7→ C(Ω).
(i) Show that θ is onto ⇐⇒ θt is 1-1.
(ii) Show that θ is 1-1 ⇐⇒ θt has dense image.
Theorem. 2.1.13. Let a be a normal element of a unital C ∗ -algebra
A and suppose z is the inclusion map z : σ(a) 7→ C. Then the inverse
Gelfand transform ϕ is the unique ∗ -homomorphism C(σ(a)) 7→ A with
ϕ(z) = a. ϕ is is isometric with image equal C ∗ (1, a).
Theorem. 2.1.14. (Spectral mapping). Let a be a a normal element of a unital C ∗ -algebra A, and let f ∈ C(σ(a)). Then
σ(f (a)) = f (σ(a)).
Moreover, if g ∈ C(σ(f (a))), then
(g ◦ f )(a) = g(f (a)).
Proof. See Murphy.
Theorem. 2.1.15. Let X be a compact Hausdorff space. Then we
have Ω(C(X)) ∼
= X.
Proof. See Murphy.