Notes on C∗-algebras

Robert Yuncken

Updated: December 14, 2015

Contents

I Fundamentals 2

I.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I.2 ∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I.3 Abelian C∗-algebras, part I . . . . . . . . . . . . . . . . . . . . 8

I.4 Unitalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

I.5 Spectra in Banach algebras . . . . . . . . . . . . . . . . . . . 15

I.6 Abelian Banach algebras and the Gelfand Transform . . . 20

I.7 Abelian C∗-algebras and the continuous functional cal-culus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

I.8 Positivity and order . . . . . . . . . . . . . . . . . . . . . . . . . 29

I.9 Representations of C∗-algebras . . . . . . . . . . . . . . . . . 36

II The Toeplitz Algebra & the Toeplitz Index Theorem 47

II.1 Matrices of operators . . . . . . . . . . . . . . . . . . . . . . . 47

II.2 Compact operators & their representations . . . . . . . . . 47

II.3 Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . 50

II.4 The Toeplitz algebra . . . . . . . . . . . . . . . . . . . . . . . . 54

III Group C∗-algebras 60

III.1 Convolution algebras . . . . . . . . . . . . . . . . . . . . . . . 60

III.2 C∗-algebras of abelian groups . . . . . . . . . . . . . . . . . 69

III.3 The C∗-algebras of the free group . . . . . . . . . . . . . . . 75

1

Chapter I

Fundamentals

I.1 Motivation

There are various possible motivations for studying C∗-algebras.

1. C∗-algebras abstract the properties of the bounded op-erators on a Hilbert space.

This is maybe not a very compelling motivation, but it is easy todescribe, so we will start there.

2. C∗-algebras generalise the properties of locally compactHausdorff topological spaces.

This is a great motivation, but it is harder to explain. We will seemore and more of this as we proceed.

3. Quantum Mechanics / Representation Theory.

These were the original motivations, but let’s avoid those fornow.

I.1.1 Bounded operators on Hilbert space.

Let H be a Hilbert space, B(H) the set of bounded operators on H.

Recall that B(H) has the following properties:

1. B(H) is a Banach space with

‖A‖ = sp‖‖≤1

‖A‖ = sp‖‖,‖y‖≤1

|⟨,Ay⟩|.

2

2. B(H) has a product satisfying

‖AB‖ ≤ ‖A‖‖B‖

3. B(H) has an involution A 7→ A∗ defined by ⟨,A∗y⟩ = ⟨A, y⟩ forall , y ∈ H. It satisfies

(a) ‖A∗‖ = ‖A‖ ∀A ∈ B(H),(b) ‖A∗A‖ = ‖A‖2 ∀A ∈ B(H),

Let’s abstract these properties.

I.2 ∗-algebras

I.2.1 Definitions

We always work over the field C.

Definition I.2.1. An algebra A (over C) is a C-vector space with anassociative bilinear product · : A× A→ A, i.e. ∀, b, c ∈ A, λ ∈ C,

(b)c = (bc)

(b+ c) = b+ c

(+ b)c = c+ bc

(λ)b = λ(b) = (λb)

If b = b for all , b ∈ A then A is called commutative or abelian.

Definition I.2.2. A ∗-algebra (or involutive algebra) is an algebra Aequipped with an involution ∗ : A→ A s.t. ∀, b ∈ A, λ ∈ C,

(∗)∗ = (involution)

(+ b)∗ = ∗ + b∗, (λ)∗ = λ∗ (conjugate-linear)

(b)∗ = b∗∗ (anti-homomorphism)

Example I.2.3. 1. A = C with z∗ = z,

2. A = Mn(C) with T∗ = Tt,

3. A = B(H) with T∗ = adjoint operator.

3

Definition I.2.4. A normed algebra is an algebra A with a vectorspace norm s.t.

‖b‖ ≤ ‖‖‖b‖, ∀, b ∈ A.

If A is complete w.r.t. ‖ · ‖ it is called a Banach algebra.

Definition I.2.5. A C∗-algebra is a Banach ∗-algebra which satisfies

‖∗‖ = ‖‖2 ∀ ∈ A. (I.2.1)

Equation (I.2.1) is called the C∗-identity. It is not clear yet why it is agood idea. In fact, it is a truly brilliant idea of Gelfand and Naimark(1943), but we will only see why later.

Example I.2.6. 1. A = C with ‖z‖ = |z|.

2. A = B(H) with operator norm ‖T‖ = sp‖‖≤1 ‖T‖.This includes A = Mn(C) = B(Cn).

3. Any norm-closed ∗-subalgebra1 of a C∗-algebra is again a C∗-algebra.

In particular, any closed subalgebra of B(H) is a C∗-algebra. Wewill refer to these as concrete C∗-algebras.

One of the major theorems about C∗-algebras (the Gelfand-NaimarkTheorem) says that any C∗-algebra is isometrically isomorphic to analgebra of operators on some Hilbert space H, i.e. a concrete C∗-algebra. But it will take some time to prove this. Often it is moreuseful to treat C∗-algebras abstractly.

Remark I.2.7. For examples of Banach algebras which are not C∗-algebras, see the exercises.

The C∗-identity can be weakened slightly without changing the defi-nition:

Lemma I.2.8. Let A be a Banach ∗-algebra s.t.

‖∗‖ ≥ ‖‖2 ∀ ∈ A.

Then A is a C∗-algebra.

Proof. For any ∈ A,

‖‖2 ≤ ‖∗‖ ≤ ‖∗‖‖‖,⇒ ‖‖ ≤ ‖∗‖.

1 A ∗-subalgebra of a ∗-algebra A is a subset which is closed with respect to allthe *-algebra operations, i.e. a linear subspace B such that bb′ ∈ B and b∗ ∈ B for allb, b′ ∈ B.

4

Exchanging and ∗ gives ‖∗‖ ≤ ‖‖, so ‖∗‖ = ‖‖. Thus,

‖‖2 ≤ ‖∗‖ ≤ ‖∗‖‖‖ = ‖‖2.

It is worth repeating the following observation from the proof.

Lemma I.2.9. In a C∗-algebra A,

‖∗‖ = ‖‖, ∀ ∈ A.

I.2.2 Homomorphisms; Representations

Definition I.2.10. A ∗-homomorphism φ : A→ B between ∗-algebrasis a map which respects all ∗-algebra operations, i.e. a linear map s.t.∀, ′ ∈ A,

φ(′) = φ()φ(′),

φ(∗) = φ()∗.

If A, B are Banach ∗-algebras, we usually demand that φ be bounded.

Remark I.2.11. Later, we will see that boundedness is automatic for∗-homomorphisms between C∗-algebras.

Definition I.2.12. 1. Let A be a Banach algebra, E a Banach space.A homomorphism π : A→ B(E) is called a representation of A onE.

2. Let A be a Banach ∗-algebra, H a Hilbert space. A ∗-homomorphismπ : A→ B(H) is called a ∗-representation of A on H.

3. If kerπ = 0 then π is called a faithful representation.

Thus, a faithful ∗-representation of a C∗-algebra A is a realization ofA as a concrete C∗-algebra of operators on H.

We will mostly be interested in Hilbert space representations. Here isone useful exception.

5

Definition I.2.13. Let A be a Banach algebra. The left multiplierrepresentation of A on itself is

L : A 7→ B(A); L()b := b.

Remark I.2.14. There is no obvious ∗-structure on B(A), so we arenot talking about a ∗-homomorphism here. Nevertheless, this maphas particularly nice properties when A is a C∗-algebra.

Proposition I.2.15. If A is a Banach algebra, L : A 7→ B(A) has ‖L‖ ≤1.

If A is a C∗-algebra, then L is an isometry, i.e. ∀ ∈ A

‖‖ = ‖L()‖B(A) = spb∈A, ‖b‖≤1

‖b‖.

Proof. Direct computation. For the C∗-algebra case, consider b =1‖‖

∗.

I.2.3 Basic terminology

Definition I.2.16. A unit in an algebra is a nonzero element 1 ∈ Asuch that 1 = = 1 for all ∈ A. An algebra with unit is calledunital.

If A,B are unital algebras, a homomorphism φ : A→ B is called unitalunital ≡ unifère

if φ(1) = 1.

Lemma I.2.17. 1. Let A be a ∗-algebra. Suppose 1 ∈ A (nonzero)is a left unit, i.e. 1 = for all ∈ A. Then it is a unit and 1 = 1∗.

2. In a unital C∗-algebra, ‖1‖ = 1.

Proof. Exercise .

Terminology for elements of a C∗-algebra A follows that of operatorson Hilbert space:

• ∈ A is normal if ∗ = ∗.

• ∈ A is unitary if ∗ = 1 = ∗.

• ∈ A is self-adjoint if = ∗.

• p ∈ A is a projection if p2 = p = p∗.

6

Remark I.2.18. Normality is a very important property. It allows us toallow =permettretalk of polynomials in and ∗, such as (∗)22 − 2∗ − 1 without

worrying about the order of and ∗.

Example I.2.19. The classic example of a non-normal element is theunilateral shift.

shift = décalage

Let H = ℓ2(N). The unilateral shift is the operator

T ∈ B(H); T(0, 1, 2, . . .) 7→ (0, 0, 1, . . .).

Its adjoint is the left shift:

T ∈ B(H); T(0, 1, 2, . . .) 7→ (1, 2, 3 . . .).

Then T∗T = d while

TT∗ ∈ B(H); T(0, 1, 2, . . .) 7→ (0, 1, 2, . . .).

The difference T∗T − TT∗ is a rank-one projection.

I.2.4 Ideals; Quotients

Definition I.2.20. An ideal / A of an algebra will always mean atwo-sided ideal, i.e. A ⊆ , A ⊆ .

If A is a ∗-algebra, we say is a ∗-ideal if ∗ = .

An ideal / A is proper if 6= A.

Remark I.2.21. In a proper ideal , no element is invertible. For if b ∈ is invertible, then 1 = b−1b ∈ and so = 1 ∈ ∀ ∈ A.

Example I.2.22. Let H be a Hilbert space and T ∈ B(H). The rank ofT is

rnk(T) = dim im(T).

Denote the set of finite-rank operators by K0(H).

It is easy to check that ∀S, T ∈ B(H),

rnk(ST) ≤ rnk(S). rnk(T),rnk(T∗) = rnk(T).

Therefore K0(H) is a ∗-ideal in B(H).

It is not a closed ∗-ideal. The closure K(H) = K0(H) is the ideal ofcompact operators.

7

Remark I.2.23. Equivalently, T ∈ B(H) is compact iff the image of theunit ball in H under T has compact closure. We won’t need this.

Lemma I.2.24. The kernel of a bounded (∗-)homomorphism is aclosed (∗-)ideal.

Proof. Direct calculation.

For a closed (∗-)ideal /A, the quotient A/ = + | ∈ A is again aBanach (∗-)algebra with the usual quotient norm: ‖+‖ =min∈ ‖+‖ (direct check).

Remark I.2.25. The analogous statement is also true for C∗-algebras:statement =énoncéif is a closed ∗-ideal in a C∗-algebra A then A/ is a C∗-algebra. But

this is harder to prove, since the C∗-identity isn’t obvious. We willprove it later.

I.3 Abelian C∗-algebras, part I

I.3.1 C0(X)

This is a very important class of examples.

Let X be a locally compact Hausdorff space.

(Recall: X is locally compact if every ∈ X has a relatively compactopen neighbourhood, i.e. ∃U 3 open s.t. U is compact.e.g. Rn, or any closed or open subset of Rn.)

Definition I.3.1. A continuous function ƒ : X→ C vanishes at ∞ if

∀ε > 0, ∃K ⊂ X compact s.t. |ƒ ()| < ε ∀ ∈ X \ K.

Notation: C0(X) = ƒ : X→ C continuous, vanishing at ∞.

Proposition I.3.2. C0(X) is a commutative C∗-algebra with

• ‖ƒ‖ = ‖ƒ‖∞ = sp∈X|ƒ ()| (ƒ ∈ C0(X)),

• Pointwise algebra operations:

(ƒ + g)() = ƒ () + g(), (λƒ )() = λƒ (),

(ƒg)() = ƒ ()g(), ƒ∗() = ƒ (),

for all ƒ , g ∈ C0(X), λ ∈ C, ∈ X.

8

Proof. From a previous course we know C0(X) is a Banach space. Theaxioms of a ∗-algebra follow from those for C, since operations arepointwise. Finally, ∀ƒ , g ∈ C0(X),

‖ƒg‖ = sp∈X|ƒ ()g()| ≤

sp∈X|ƒ ()|

sp∈X|g()|

= ‖ƒ‖‖g‖.

‖ƒ∗ƒ‖ = sp∈X|ƒ ()ƒ ()| = ‖ƒ‖2.

Remark I.3.3. If X is compact, then every function ƒ : X→ C vanishesat ∞, so C0(X) = C(X).

Exercise I.3.4. (This is just a sketch. See the exercise sheet fordetails.)

1. Let X be a locally compact Hausdorff space. Show that X iscompact if and only if C0(X) is unital.

2. Suppose X is not compact. Define the one-point compactifica-tion X and show that it is compact, Hausdorff in which X sits asan open subset.

3. Show that C0(X) sits inside C(X) as a closed ideal and that C(X) =C0(X)⊕ C1, where 1 denotes the constant function 1 on X. Hint:Show that C0(X) = ker(ev∞).

4. Show that ÝRn ' Sn, e.g. by stereographic projection.

I.3.2 Representations of C0(X)

Let X be locally compact Hausdorff space. Let μ be a Borel probabilitymeasure on X.

Then A = C(X) is represented on H = L2(X;μ) by pointwise multiplica-tion:

πμ(ƒ )h := ƒh, (ƒ ∈ C0(X), h ∈ L2(X;μ)).

Note that “functions on X” are being used in two ways:

• ƒ is a continuous function in the C∗-algebra C0(X),

• h is an L2-function in the Hilbert space L2(X;μ).

9

In fact the space H = L2(X;μ) is not really a space of functions, butof equivalence classes of functions (modulo equality almost every-where).

almosteverywhere ≡presque partoutThis process can be quite brutal. For instance if μ = δ (Dirac measure

at ∈ X) then L2(X; δ) ∼= C, since two functions are equivalent if andonly if they have the same value at . The resulting representation

π : C0(X)→ B(L2(;μ)) ∼= B(C) = C

is given by evaluation at , i.e. π(ƒ ) = ƒ () for ƒ ∈ C0(X).

The representation πμ is faithful if and only if μ is a strictly positivemeasure, i.e. μ(U) > 0 for every open set U ⊆ X (exercise).

I.3.3 *-Homomorphisms and proper maps

Definition I.3.5. A continuous map α : X → Y between locally com-pact Hausdorff spaces is called proper if the preimage of any compactset is compact.

Remark I.3.6. If X is compact this condition is automatic.

Roughly, α is proper if α(n) → ∞ whenever n → ∞. This is literallytrue if we allow (n) to be a net (see later).

Example I.3.7. Some proper maps:

• Any α : X→ Y if X is compact.

• α = id : R→ R.

• α : R→ R; α() = ||.

• α : R→ R; α() = log(1+ ||).

Not proper:

• α : R→ R; α() = c (constant map).

• α : R→ R; α() = p1+2

.

The pull-back by a proper map α is the map α∗ : C0(Y) → C0(X) isdefined by

α∗ƒ () = ƒ α(), (ƒ ∈ C0(Y), ∈ X).

10

Remark I.3.8. The pull-back α∗ can be defined even if α is not proper,but the image will be in C(X) not C0(X). See the proof of the nextlemma.

Lemma I.3.9. For any continuous proper map α : Y → X the pull-back α∗ is a ∗-homomorphism C0(X)→ C0(Y).

Proof. The first thing to prove is that α∗ is well-defined, i.e. its imagelies in C0(X). Let ε > 0. If ƒ ∈ C0(Y) then there is K ⊆ Y compact suchthat |ƒ (y)| < ε ∀y ∈ Y \K. Then |α∗ƒ ()| = |ƒ (α()| ≤ ε ∀ ∈ X\α−1(K).Since α−1(K) is compact by properness, this proves α∗ƒ ∈ C0(X).

To check that α∗ is a ∗-homomorphism is a direct calculation: e.g.α∗(ƒ + g)() = (ƒ + g)(α()) = ƒ (α()) + g(α()) = (α∗ƒ +α∗g)().

Example I.3.10. Let Y ⊆ X be a closed subspace.

The inclusion α : Y ,→ X is proper. (This needs Y to be closed. Why?)

The pull-back α∗ : C0(X) → C0(Y) is restriction α∗ : ƒ 7→ ƒ |Y . Thekernel is

kerα∗ = ƒ ∈ C0(X) | ƒ |Y ≡ 0 ∼= C0(X \ Y).

In particular, if ∈ X then restriction to is evaluation:

ev : C0(X)→ C() ∼= C; ƒ 7→ ƒ ().

The kernel ker(ev) is an ideal of codimension 1.

Remark I.3.11. A (non-zero) ∗-homomorphism A→ C is called a characterof A. Evaluation at a point on C0(X) is a very important example.

I.3.4 Noncommutative topology

The above examples (which will be completely developed in a laterlecture) could be viewed in two different ways:

1. C0(X) is a particular example of a C∗-algebra.

2. General C∗-algebras are noncommutative generalizations of C0(X).⇒ “Noncommutative geometry”

This idea goes back to the foundations of quantum mechanics, andappears throughout its history. But it was developed greatly by AlainConnes in the ’80s & ’90s.

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We will see that many algebraic properties of C∗-algebras are anal-ogous to topological properties of locally compact Hausdorff spaces.Here are some very simple examples. Let X, Y be locally compactHausdorff spaces, A = C0(X), B = C0(Y) and α : X → Y a continuousproper map.

Topology Algebra

X is compact ⇐⇒ A is unital

(For X compact) ⇐⇒ A has a non-triv. projectionX is connected 0 6= p 6= 1

α is surjective ⇐⇒ α∗ is injectiveα is injective ⇐⇒ α∗ is surjective

Proof (first two lines of the table). If ƒ ∈ C0(X) is a unit then ƒg = g∀g ∈ C0(X). So ƒ = 1 (constant function). But 1 is in C0(X) iff X iscompact.

If p ∈ C0(X) is a projection then p()2 = p() ∀ ∈ X. So p() ∈ 0,1∀ ∈ X. A 0,1-valued continuous function is constant on connectedcomponents. The result follows.

Exercise I.3.12. Prove the last two lines.

I.4 Unitalizations

Non-unital algebras are important, e.g. K(H), C0(X). But we canalways add a unit if we need to. This is called a unitalization.

Lemma I.4.1. Any (∗-)algebra A can be embedded into a unital (∗-embedded =plongé)algebra A as an ideal of codimension 1.

Proof. Put A = A⊕ C; We will write the element (, λ) ∈ A formally as + λ1. This notation suggests the following definition for a producton A:

(+ λ1)(b+ μ1) := (b+ μ+ λb) + (λμ)1.

This is associative, bilinear and A is an ideal (direct check).

If A is a ∗-algebra, then so is A with the involution

(+ λ1)∗ = ∗ + λ1.

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Proposition I.4.2. Any Banach algebra A can be isometrically em-bedded into a unital Banach algebra A as a closed ideal of codimen-

embedded =plongésion 1.

Proof. We define a Banach space norm on A by2

‖(, λ)‖A = ‖‖A + |λ|.

It remains to check:

‖(+ λ1)(b+ μ1)‖A = ‖(b+ μ+ λb)‖A + |λμ|≤ ‖‖A‖b‖A + |μ|‖‖A + |λ|‖b‖A + |λ||μ|= ‖+ λ1‖A‖b+ μ1‖A.

Proposition I.4.3. Any non-unital C∗-algebra can be isometricallyembedded in a unital C∗-algebra as a closed ideal of codimension 1.

Proof. Again, we let A = A⊕ C with the ∗-algebra structure as above.The hard part is defining a C∗-norm on A. For this we will use theleft-multiplier representation L : A→ B(H).

We can extended L to A as follows:

L : A→ B(A); L(+ λ1) = L() + λ d .

i.e. L( + λ1)b = + λb. This is an algebra homomorphism (directcheck).

We then define‖+ λ1‖A := ‖L(+ λ1)‖B(A).

Since ‖L()‖B(A) = ‖‖A (Proposition I.2.15), this agrees with the orig-inal norm on A.

To be a well-defined norm, we need to check that L is injective. Sup-pose L(+ λ1) = 0. If λ 6= 0 we get L(−λ−1) = dA so −λ−1 is a leftunit, which is absurd since A is non-unital. So λ = 0. But L is injective,so = 0.

Finally we need to prove the C∗-identity. Let ∈ A. The definition ofthe norm on B(A) gives: ∀ε > 0, ∃b ∈ A with ‖b‖ ≤ 1 s.t. ‖b‖A ≥

2 In fact, there are many possible norms that would work here, e.g. ‖ + λ1‖A :=mx‖‖A, |λ|.

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‖‖A − ε. Then

‖∗‖A ≥ ‖b∗‖A‖∗‖A‖b‖A

≥ ‖b∗(∗b)‖A= ‖(b)∗b‖A= ‖b‖2A≥ (‖‖A − ε)

2.

So ‖∗‖A ≥ ‖‖2A. By Lemma I.2.8, this completes the proof.

Remark I.4.4. We could make the construction A above even if A isunital. In that case, A is isomorphic to A⊕ C with ∗-algebra structure

(, z)(′, z′) = (′, zz′),

(, z)∗ = (∗, z), ∀, ′ ∈ A, z, z′ ∈ C.

I.4.1 The one-point compactification

Recall C0(X) is unital iff X is compact. Here we describe the unitaliza-

tion ãC0(X) topologically.

Definition I.4.5. Let X be locally compact Hausdorff. The one-pointcompactification of X is the space

X := X t ∞

where ∞ is a formal element called the point at infinity, equippedwith the topology of open sets

τ := U ⊂ X open t X \ K | K ⊂ X is compact.

This means that n →∞ iff (n) is eventually outside any given com-eventually =ultimementpact set.

Lemma I.4.6. τ is indeed a topology on X.

Proof. Exercise.

Example I.4.7. ÝRn ∼= Sn.

Recall that Sn \ N∼=→ Rn where N is the “north pole”, for instance

by stereographic projection. Denote the homeomorphism by α. Weaugment this to a map

α : Sn →ÝRn

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by sending N to ∞. This is a bijection between compact Haussdorffspaces, so it suffices to show that one of α or α−1 is continuous.

Let U ⊆ Sn be open. If U ⊂ Sn \N then α(U) is open. Otherwise, U 3 N,so K = Sn \ U is a compact subset of Sn \ N, so Rn \ α(U) = α(K) iscompact, so α(U) is open. This proves that α−1 is continuous.

Note thatC0(X) = ƒ ∈ C(X) | ƒ (∞) = 0 = ker ev∞

is an ideal of codimension 1 in C(X).

Proposition I.4.8. For any locally compact Hausdorff space X, ãC0(X) ∼=C0(X) as C∗-algebras.

Proof. Consider the map

φ :ãC0(X)→ C(X); ƒ ⊕ λ1 7→ ƒ + λ1X

where 1X denotes the constant function 1. This is a ∗-homomorphism(direct check). It is bijective, since every ƒ ∈ C(X) can be written as

ƒ = [ƒ − ƒ (∞)1X] + ƒ (∞)1X ∈ C0(X) + C1Xand this decomposition is unique.

I.5 Spectra in Banach algebras

I.5.1 The set of invertible elements

Let A be a unital Banach algebra.

As usual ∈ A is invertible if it has a two-sided inverse −1 ∈ A s.t.−1 = 1 = −1.

Remark I.5.1. A one-sided inverse is not enough: consider the unilat-eral shift.

Lemma I.5.2. Let A be a unital Banach algebra. If ∈ A with ‖‖ < 1then 1− is invertible, and

(1− )−1 = 1+ + 2 + · · · (convergence in norm).

Proof. The sum is absolutely convergent since ‖n‖ ≤ ‖‖n.

Moreover,

(1− ) limn→∞

(1+ + · · ·+ n) = limn→∞

1− n+1 = 1,

so the limit is a right-inverse to 1−. A similar calculation shows it isa left-inverse.

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Corollary I.5.3. Let ∈ A be invertible. If ‖‖ ≤ ‖−1‖−1 then − is invertible, with

(− )−1 = −1∞∑

n=0

(−1)n.

Therefore the set G(A) of invertible elements is open in A.

Proof. Write − = (1− −1).

I.5.2 Spectra

A is still a unital Banach algebra.

Definition I.5.4. The spectrum of ∈ A is

Sp() = λ ∈ C | (λ1− ) is not invertible.

The complement ρ() := C \ Sp() is called the resolvent set. Thefunction

R : ρ()→ A; λ 7→ (λ1− )−1

is called the resolvent function.

Exercise I.5.5. Let , b ∈ A. Show that (1 − b) is invertible if andonly if (1− b) is invertible. Hence show that Sp(b) and Sp(b) arethe same, except possible for 0, that is

Sp(b) ∪ 0 = Sp(b) ∪ 0.

Theorem I.5.6. For any ∈ A, Sp() is a non-empty compact subsetof C, contained in B(0; ‖‖), and the resolvent function is analytic onC \ Sp().

Proof. The resolvent set is the preimage of G(A) under the continuousmap λ 7→ λ1− , so is open. Thus Sp() is closed.

If |λ| > ‖‖ then λ1− is invertible (Lemma I.5.2) so Sp() is bounded,hence compact.

Let λ0 ∈ ρ(). Then λ1− is invertible for λ in some ball around λ0.Specifically, if |λ− λ0| < ‖(λ01− )−1‖−1 we have the formula

R(λ) := (λ1− )−1 = [(λ01− ) + (λ− λ0)1]−1

= (λ01− )−1∑

(λ− λ0)n(λ0 − )−n.

16

This is a power series in (λ− λ0) so R is an analytic function.power series =série entière

If Sp() were empty, then R would be an entire function. Also, for|λ| > ‖‖,

‖(λ1− )−1‖ ≤ |λ−1|∞∑

n=0

‖λ−nn‖ ≤ |λ|−1(1− |λ|−1‖‖)−1

= (|λ| − ‖‖)−1 → 0 as |λ| →∞,

so R is bounded. By Liouville’s Theorem, R(λ) would be constant,which is absurd.

If A is not unital, we define the spectrum of ∈ A to be its spectrumin the unitalization A.

Example I.5.7. If A = C(X) with X a compact Hausdorff space, thenSp() = im() for any ∈ C(X).

If A = C0(X) with X locally compact Hausdorff, then Sp() = im() ∪0 for any ∈ C0(X).

Theorem I.5.8 (Gelfand-Mazur). The only complex Banach algebrain which every non-zero element is invertible is C.

Proof. Suppose every element of A is invertible. Let ∈ A. SinceSp() is non-empty, there is λ ∈ Sp() s.t. λ1− is not invertible, i.e.λ1− = 0. Thus is scalar.

I.5.3 Polynomial functional calculus

This section is not profound, but it prepares us for the much morepowerful “continuous functional calculus” later on.

Let A be an algebra. Let C[z] be the algebra of polynomials in z. Fix ∈ A and consider the map

C[z]→ A

p =∑

ncnzn 7→ p() =

∑

ncnn.

This map is called the polynomial functional calculus (applied to ).

Proposition I.5.9. The map p 7→ p() is an algebra homomorphism,and satisfies the spectral mapping property:

Sp(p()) = p(Sp()).

17

Proof. Exercise

This allows us to consider polynomial functions of , e.g. 2, 3++1.

In a Banach algebra, one can extend this to define ƒ () for any ƒ :C→ C which is holomorphic on some neighbourhood of the spectrumSp() via the Cauchy integral formula:

ƒ () :=1

2π

∮

ƒ (z)(z − )−1dz,

for any curve (or union of curves) encircling the spectrum. This iscalled the holomorphic functional calculus. We won’t do this in thiscourse.

In a C∗-algebra, we will be able to define continuous functions of ,e.g.

p, ||, exp(), log(), etc. This is the continuous functional

calculus.

In a von Neumann algebra one can go even further to the Borelfunctional calculus, e.g. phase Ph(), or indicator functions 1Y() forY ⊂ Sp(). We may look at this later if we have time (but probablynot).

I.5.4 Spectral radius formula

Let A be a Banach algebra.

Definition I.5.10. The spectral radius of ∈ A isspectarl radius= rayonspectralspr() = sp|λ| | λ ∈ Sp().

Theorem I.5.11 (Spectral radius formula). For any ∈ A,

spr() = limn→∞

‖n‖1n .

Proof. Put r = spr(). So Sp() ⊆ B(0; r).

Recall that the resolvent function R : C → A is an A-valued analyticfunction on C\Sp(), in particular on |z| > r. Also it tends to 0 in normas |z| →∞. This means that the function

z 7→ R(z−1) = (z−11− )−1 =∞∑

n=0

zn+1n

18

is analytic on B(0; r−1 \ 0 with removable singularity at 0.

Cauchy’s nth root test implies

limspn→∞

‖n‖1n ≤ r.

At the same time, there is λ ∈ Sp() with |λ| = r. Then λn ∈ Sp(n)for all n. So rn = |λn| ≤ ‖n‖ for all n. Therefore, ‖n‖

1n ≥ r for all n.

The result follows.

I.5.5 Spectral radius in C∗-algebras

For normal elements in a C∗-algebra, the spectral radius formula iseven cleaner. (This is part of the power of the C∗-identity.)

Proposition I.5.12. Let A be a C∗-algebra. For any normal ∈ A,

spr() = ‖‖.

Proof. For ∈ A normal

‖‖2 = ‖∗‖ = ‖(∗)∗(∗)‖12 = ‖(2)∗(2)‖

12 = ‖2‖.

By induction,

‖2k‖ = ‖‖2

k, ∀k ∈ N.

Thus,

spr() = limk→∞

‖2k‖1

2k = ‖‖.

Example I.5.13. To understand this it helps to think of the case A =Mn(C).

A matrix is normal iff it is diagonalizable (exercise!). So PropositionI.5.12 says that the operator norm of a diagonalizable matrix is equalto the largest absolute value of its eigenvalues.

A non-normal matrix has non-diagonal Jordan form. For instance, the

matrix =

0 10 0

has only one eigenvalue, namely 0. But it does

not have operator norm 0.

Nevertheless, we can still use the spectral radius formula to calculatethe norm of a non-normal element.

19

Corollary I.5.14. For any in a C∗-algebra A,

‖‖ = spr(∗)12 .

Proof. ∗ is normal and ‖‖ = ‖∗‖12 .

This result is philosophically very important. It says that the normon a C∗-algebra is determined completely by the algebra structure(invertibility of elements)!

Corollary I.5.15. If a ∗-algebra admits a C∗-norm, it is unique.

I.6 Abelian Banach algebras and the GelfandTransform

I.6.1 Characters

Let A be a unital Banach algebra.

Definition I.6.1. A character is a nonzero homomorphism φ : A→ C.

Lemma I.6.2. Characters are automatically bounded of norm 1.

Proof. First, note that φ(1) = 1, so ‖φ‖ ≥ 1 (or φ is unbounded).

Suppose ‖φ‖ 6= 1. Then ∃ ∈ A with ‖‖ < 1 but |φ()| = 1. Putb = (1− )−1.

φ(1) = φ(b− b) = φ(b)− φ()φ(b) = 0.

Contradiction.

Note that ker(φ) is a closed ideal of codimension 1. In particular itis a maximal ideal, i.e. it is a proper ideal and there are no idealsker(φ) A.

Lemma I.6.3. Maximal ideals are automatically closed.

Proof. Let M be a maximal ideal. Recall that a proper ideal containsno invertible elements. But the open ball of radius 1 around 1 ∈ Aconsists entirely of invertible elements, so dist(1,M) = 1. Thus M is aclosed proper ideal containing M. By maximality, M = M.

20

Theorem I.6.4. Let A be an abelian unital Banach algebra. The map

φ 7→ ker(φ)

defines a bijective correspondence between multiplicative linear func-tionals and maximal ideals.

Proof. For any characer φ, ker(φ) is an ideal of codimension 1, somaximal. Moreover, φ is completely determined by ker(φ) and thevalue φ(1) = 1, so the map is injective.

Surjectivity: Let M be a maximal ideal. The quotient A/M is a simpleBanach algebra, so every element is invertible. By Gelfand-Mazur,A/M ∼= C. The quotient map is a multiplicative linear functional withkernel M.

I.6.2 The maximal ideal space

Here A is an abelian Banach algebra, possible non-unital.

Definition I.6.5. The set

A := characters of A

= maximal ideals of A.

is called the maximal ideal space or spectrum of A.

Example I.6.6. Consider A = C0(X).

Associated to each point ∈ X is a character

ev : ƒ 7→ ƒ ().

The associated maximal ideal is = ƒ ∈ C0(X) | ƒ () = 0.

Thus we have a map X ,→ A via 7→ ev. An application of the Stone-Weierstrass Theorem shows that this is a bijection.

What about the topology on A?

We have A ⊂ A∗. But A∗ has several possible topologies.

The norm topology is not a good choice. For instance, with A = C(X),given any , y ∈ X there is a continuous function ƒ ∈ C0(X) withev(ƒ ) = ƒ () = 0 and evy(ƒ ) = ƒ (y) = 1. Therefore ‖e − evy ‖ ≥ 1.Therefore the norm topology makes X ⊂ A into a discrete space.

21

On the other hand, if n → in X, then for any fixed ƒ ∈ C0(X) wehave

|(evn −ev)ƒ | = |ƒ (n)− ƒ ()| → 0.

That is evn → ev in the weak*-topology, i.e. the topology of point-wise convergence.

Proposition I.6.7. If A is a unital Banach algebra, then A is a com-pact Hausdorff space with the weak*-topology. If A is non-unital thenA is locally compact Hausdorff.

Proof. The weak∗-topology is always Hausdorff.

Let A be unital. A weak*-limit of multiplicative linear functionals isagain multiplicative (direct check) so A is a closed subset of the unitball of A∗. By Banach-Alaoglu, this is compact.

If A is not unital, embed A ,→ A.

If φ is a character of A then we can extend it to

φ : A→ C; φ(+ λ1) := φ() + λ,

which is a character of A (direct check). So we get an inclusion A ,→ ˆA.It is continuous for the weak*-topologies (direct check).

Conversely, if ψ is a character of A then we can restrict it to a linearfunctional ψ|A on A. It is multiplicative, but may be zero! But there isonly one character of A which maps to zero, since it must satisfy

ψ(+ λ1) = 0+ λψ(1) = λ, ∀+ λ1 ∈ A.

Let us denote this particular character by ψ∞. Then restriction gives

a continuous map ˆA \ ψ∞→ A, inverse to the above.

Remark I.6.8. If A is non-unital then ˆA is the one-point compactifica-tion of A, with ψ∞ being the point at infinity.

I.6.3 The Gelfand Transform

We now have:

• A loc. comp. Hausdorff space X gives us a C∗-algebra C0(X),

• A Banach algebra A gives us a loc. comp. Hausdorff space A.

22

What is the relation? For a C∗-algebra we will see that these pro-cesses are inverse to one another (up to natural isomorphism).

Here we start with an abelian Banach algebra A.

Definition I.6.9. The Gelfand transform is the map : A → C0(A)defined by 7→ where

(φ) := φ() (φ ∈ A character of A).

(We will prove that really is in C0(A) in the Theorem below.)

Theorem I.6.10. The Gelfand transform is an algebra homomor-phism A→ C0(A) of norm ‖‖ ≤ 1. The image separates the points ofA, i.e. ∀φ 6= ψ ∈ A, ∃ ∈ A st (φ) 6= (ψ).

Proof. This is an exercise in following definitions.

Well defined: For ∈ A, is continuous by definition of the weak*-topology. If A is non-unital then (ψ∞) = 0, so ∈ C0(X).

Algebra homomorphism: Direct check, e.g.

(Ø+ b)(φ) = φ(+ b) = φ() + φ(b) = (φ) + b(φ).

bounded of norm 1:

‖‖ = spφ∈A|(φ)| = sp

φ∈A|φ()| ≤ ‖‖.

Separates points: Tautology.

Corollary I.6.11. ∈ A is invertible ⇐⇒ ∈ C0(A) is invertible. Con-sequently,

Sp() = Sp() = φ() | φ ∈ A.

Proof. If is invertible, Ô−1 is an inverse to .

If is not invertible then A is a proper ideal so is contained in somemaximal ideal M. The character φ ∈ A with ker(φ) = M is s.t. (φ) =φ() = 0, so is not invertible.

Question: Is the Gelfand transform compatible with ∗-structures?

We equip C0(A) with the usual involution ∗ of pointwise conjugation.

Lemma I.6.12. Let A be a Banach ∗-algebra. : A → C0(A) is a∗-homomorphism iff ∀ ∈ A with = ∗, Sp() ⊂ R.

23

Proof. If is a ∗-homomorphism then = ∗ ⇒ = ∗ so is realvalued. Thus Sp() = Sp() ⊂ R.

Suppose Sp() ⊂ R for = ∗. Then im() ⊆ Sp() ⊂ R so ∗ = . Forgeneral b ∈ A, write b = + y with

= 12 (b+ b

∗), y = 12 (b− b

∗).

These are self-adjoint. So Ób∗ =Ù− y = − y = (b)∗.

Example I.6.13. The disk algebra is an example for which is not a∗-homomorphism (see exercises).

But C∗-algebras are always okay. . .

I.7 Abelian C∗-algebras and the continu-ous functional calculus

I.7.1 Abelian C∗-algebras, part II

Now another very important consequence of the C∗-identity.

Lemma I.7.1. Every self-adjoint element in a C∗-algebra has realspectrum.

Proof. Let ∈ A be self-adjoint. Let λ ∈ R. From the calculation

‖− λ‖2 = ‖(− λ)(− λ)∗‖ = ‖2 + λ21‖ ≤ λ2 + ‖‖2,

the spectrum of is contained in the ball of radiusp

λ2 + ‖‖2 aboutλ. The intersection of all these balls is the interval[−‖‖, ‖‖] ⊂ R.

Theorem I.7.2 (Commutative Gelfand-Naimark Theorem). Let A bea commutative C∗-algebra. Then A is isometrically ∗-isomorphic toC0(X) for some locally compact Hausdorff space X. Specifically X = A.

Proof. The Gelfand transform : A → C0(A) preserves the ∗ andall spectra. The norm on a C∗-algebra is determined completely bythese (see Corollary I.5.14). So is an isometry, hence injective.

The image of separates points, so is dense by Stone-Weierstrass. Itis also complete. So is surjective.

24

Proposition I.7.3. Every non-zero ∗-homomorphism between unitalabelian C∗-algebras is the pull-back of a continuous map between theunderlying topological spaces.

Proof. Let ψ : C(Y) → C(X) be a ∗-homomorphism. For any ∈ X,ev ψ is a character of C0(Y), so equals evα() for some α() ∈ Y.This defines α : X→ Y.

To prove continuity, we take U ⊆ Y be open and show α−1(U) is open.

Take ∈ α−1(U). By Urysohn’s Lemma, there is ƒ ∈ C(Y) with 0 ≤ ƒ ≤ 1s.t. ƒ (α()) = ψ(ƒ )() = 1 and ƒ (y) = 0 for all y /∈ U. We get an openneighbourhood of :

neighbourhood= voisinage

′ ∈ X | ψ(ƒ )(′) > 0 = ′ ∈ X | ƒ (α(′)) > 0 ⊆ α−1(U).

So α−1(U) is open.

In categorical language, the Gelfand transform yields an equivalenceof categories between the category of unital abelian C∗-algebras with∗-homomorphisms and the category of compact Hausdorff spaceswith continuous maps.

I.7.2 Invariance of spectra

A — unital C∗-algebra.

Definition I.7.4. The C∗-algebra generated by ∈ A is the smallestunital subalgebra containing .

If is normal then,

C∗() := p(, ∗) | p ∈ C[z, z],

since every C∗-subalgebra containing must contain all polynomialsin and ∗. Note that this algebra is abelian, so C∗() = C(X) forsome compact Hausdorff space X. In the next section we will seethat X = Sp().

Lemma I.7.5. Let ∈ A be normal. The spectrum of in C∗() isthe same as the spectrum of in A.

Proof. Let b ∈ C∗(). We show b is invertible in A iff it is invertible inC∗().

25

If b is invertible in C∗() then b is invertible in A.

Suppose b is invertible in A but not in C∗(). Then 0 ∈ SpC∗()(b) =im(b), where b ∈ C(X) is the Gelfand transform of b. By Urysohn’sLemma, let c ∈ C(X) be a continuous function with 0 ≤ c ≤ 1 and

c() = 1 for all ∈ b−1(0),

c() = 0 for all ∈ b−1[ε,∞),

where ε = 12‖b

−1‖−1. Then

‖c‖ = ‖c‖ = 1,

‖bc‖ = ‖bc‖ ≤ 12‖b

−1‖−1,

so1 = ‖c‖ ≤ ‖b−1‖‖bc‖ ≤ 1

2 .

Contradiction.

Corollary I.7.6. The spectrum of in any closed unital subalgebra ∈ B ⊆ A is the same.

Corollary I.7.7. Any injective ∗-homomomorphism φ : A → B be-tween C∗-algebras is an isometry.

Proof. WLOG, assume A and B are unitary. (If not, φ extends uniquelyWLOG =“without loss ofgenerality”

to the unitalizations.) Since φ preserves spectra and ∗, the resultfollows from the spectral radius formula.

I.7.3 Continuous functional calculus

Let A be a unital ∗-algebra.

If ∈ A is normal, we can extend the polynomial functional calculus

C[z]→ A; p 7→ p()

to a polynomial functional calculus on and ∗:

C[z, z]→ A; p 7→ p(, ∗).

Using the Gelfand transform on a C∗-algebra, we can extend evenfurther.

26

Proposition I.7.8. Let ∈ A be normal. Then C∗() is ∗-isomorphicto C(Sp()) via a map sending to the function z : z 7→ z.

Proof. Put X :=ÚC∗(). Consider the function : X→ C.

Any character on C∗() is completely determined by its value on thegenerating element , so is injective.

Its image is im() = SpC(X)() = Sp(). So is a homeomorphism : X→ Sp(A) ⊂ C.

Thus the Gelfand transform identifies

C∗() ∼=C(X)∗∼= C(Sp())

and under this identification

7→ ← [ z.

Definition I.7.9. Fix ∈ A normal. The map

C(Sp())∼=→ C∗() ⊆ A

is called the continuous functional calculus of . The image of ƒ ∈ Ais denoted ƒ ().

Note that we have the spectral mapping property:

Sp(ƒ ()) = SpC(Sp())ƒ = im(ƒ ) = ƒ (Sp()).

Example I.7.10. Suppose ∈ A has positive spectrum Sp() ⊂ [0,∞).

The function ƒ : z 7→ z12 is continuous on Sp(), so the functional cal-

culus gives 12 := ƒ () ∈ A.

Since ƒ .ƒ = z (the identity function on Sp()), the functional calculus

homomorphism gives z12 z

12 = .

Example I.7.11. The function exp : C→ C is continuous everywhere.Thus, for any ∈ A we can define exp() ∈ A. It inherits the usualproperties of the function exp ∈ C(Sp()), e.g.

exp(s)exp(t) = exp((s+ t)), (s, t ∈ R).

The polynomials pn(z) :=∑nk=0

zkk! converge to exp(z) uniformly on

any bounded subset of C, so the functional calculus homomorphismgives the norm convergent series

exp() =∞∑

k=0

k

k!.

27

Remark I.7.12. If A is non-unital then we have a continuous functionalcalculus on A. A continuous function ƒ ∈ C(Sp()) has

ƒ () ∈ A ⇐⇒ ƒ ∈ ⟨z⟩ ⇐⇒ ƒ (0) = 0,

where ⟨z⟩ denotes the ideal generated by z in C(Sp()). In otherwords, we have the functional calculus

C0(Sp(A) \ 0)→ A; ƒ 7→ ƒ (A).

I.7.4 Spectral projections

One particularly useful application of the functional calculus is to pro-duce spectral projections.

Definition I.7.13. An element p ∈ A s.t. p2 = p = p∗ is called aprojection.

Remark I.7.14. (Recall) in the case A = B(H),

• p2 = p ⇒ p is a projection onto a closed subspace of H parallelto some complementary subspace;

• p = p∗ ⇒ implies this projection is orthogonal.

Example I.7.15. A = C(X). If Y ⊂ X is a clopen subset, then theindicator function

1Y : →

(

1, if ∈ Y,0, if /∈ Y,

is continuous, and is a projection in C(X).

Example I.7.16. If ∈ A is normal, and Y ⊆ Sp() is a connectedclopen subset of its spectrum, then 1Y() is a projection called thespectral projection associated to Y ⊂ Sp().

Exercise I.7.17. Let A ⊆ B(H) be a concrete C∗-algebra. SupposeT ∈ A is normal and that λ is an isolated point in the spectrum of. Show that P = 1λ(T) is the orthogonal projection onto the λ-eigenspace of T.

Hint: Consider the case λ = 0 and show that P is the projection ontothe kernel of T.

28

I.8 Positivity and order

I.8.1 Positivity

Let A be a C∗-algebra. For simplicity, we will consider only unital Ahere, although all results work for non-unital C∗-algebras by passingto the unitalization.

There are several different equivalent definitions of a positive ele-ment.

Theorem I.8.1. Let ∈ A with = ∗. The following are equivalent.

1. Sp() ⊂ [0,∞).

2. For all t ≥ ‖‖, ‖t1− ‖ ≤ t.

3. For some t ≥ ‖‖, ‖t1− ‖ ≤ t.

4. = b2 for some b ∈ A self-adjoint.

5. = b∗b for some b ∈ A.

Remark I.8.2. If A ⊆ B(H) is a concrete C∗-algebra, then these areequivalent to

⟨ξ, ξ⟩ ≥ 0 ∀ξ ∈ H.

Proof of 1⇔ 2⇔ 3⇔ 4.

1⇒ 2: Recall that Sp() ⊂ [0, ‖‖].

Thus Sp(t1− ) ⊂ [t − ‖‖, t]. So ‖t1− ‖ ≤ t.

2⇒ 3: Obvious.

3⇒ 1: Suppose t has ‖t1− ‖ ≤ t. Then

t ≥ ‖t1− ‖ = ‖(t − z)‖C(Sp()).

Thus z is non-negative on Sp().

1⇒ 4: Take b =p, by the functional calculus.

4⇒ 1: Sp() = Sp(b)2 ⊂ [0,∞).

Proving 5 takes more effort. We will need some lemmas. So, for now,we will call ∈ A positive if it has positive spectrum.

29

Definition I.8.3.

A+ := ∈ A | positive ,

As := ∈ A | self-adjoint.

Remark I.8.4. The positive square root of a positive element isunique. If b is any positive square root, then functional calculus on bgives

b =p

b2 =p.

Lemma I.8.5. Any self-adjoint element ∈ A can be decomposed as

= + − −,

where ± are positive and +− = 0.

Proof. Define ƒ± : R→ C by

ƒ+() =mx(0, ),

ƒ−() =mx(0,−).

So ƒ± are positive functions with z = ƒ+ − ƒ− and ƒ+ƒ− = 0. Put ± :=ƒ±().

Corollary I.8.6. Any ∈ A is a linear combination of four positiveelements.

Proof. Recall that = Re() + m() where Re() = 12 ( + ∗) and

m() = 12 ( −

∗) are both self-adjoint. Using the lemma, we candecompose both Re() and m() as a sum of two positive elements.

Proposition I.8.7. A+ is a salient convex cone, i.e.

1. , b ∈ A+ ⇒ + b ∈ A+.

2. ∈ A+, λ ∈ R+ ⇒ λ ∈ A+.

3. ∈ A+ and − ∈ A+ ⇒ = 0.

Proof. 1. Let s ≥ ‖‖ s.t. ‖s1 − ‖ ≤ s and t ≥ ‖b‖ s.t. ‖t1 − b‖ ≤ t.Then

‖(s+ t)1− (+ b)‖ ≤ ‖s1− ‖+ ‖t1− b‖ ≤ s+ t.

2. Sp(λ) = λSp() ⊂ [0,∞).

30

3. Sp() ⊂ 0, so ‖‖ = 0 by the Spectral Radius Formula.

This means the positive elements can be used to define an order onthe subspace As of self-adjoint elements.

Definition I.8.8. Write ≥ b if − b ∈ A+.

So ≥ 0 means is positive. We say is negative if ≤ 0 ⇐⇒ − ispositive ⇐⇒ Sp() ⊂ (−∞,0].

More generally, note that

• ≤ t1⇒ Sp() ⊂ (−∞, t],

• ≥ t1⇒ Sp() ⊂ [t,∞),

Warning! The order is well-behaved with respect to addition, but notmultiplication. e.g.

0 ≤ ≤ b 6⇒ 2 ≤ b2 (in general).

But it is well-behaved for multiplication of commuting operators.

Finally, we can complete the proof of Theorem I.8.1.

Proof of Theorem I.8.1 (5⇒1). First we show = b∗b is not negativeunless = 0. Note that the spectra of b∗b and bb∗ are the same(except perhaps 0), so if one is negative, both are.

Write b = + y with , y self-adjoint. Then

b∗b+ bb∗ = 22 + 2y2 ≥ 0.

We get = 0.

Next, write = + − − as in Lemma I.8.5. Put c = b12− . Then

c∗c = 12−

12− = −

2−.

So c∗c ≤ 0, hence − = 0.

Some extra properties

Proposition I.8.9. 1. For any ∈ As, ≤ ‖‖1.

31

2. If 0 ≤ ≤ b then ‖‖ ≤ ‖b‖.

3. Let , b ∈ As. If ≤ b then ∗ ≤ ∗b∗ for all ∈ A.

4. If , b ∈ As are invertible, then 0 ≤ ≤ b ⇒ b−1 ≤ −1.

5. If 0 ≤ ≤ 1 then 2 ≤ .

Proof. 1. Sp() ⊆ [−, ] so Sp(‖‖1− ) ⊆ [0,2‖‖].

2. 0 ≤ ≤ b ≤ ‖b‖1. By the spectral radius formula, ‖‖ ∈ Sp() so‖b‖ − ‖‖ ∈ Sp(‖b‖1− ) ⊆ [0,∞].

3. Let c =pb− . Then

∗b− ∗ = ∗cc = (c)∗(c) ≥ 0.

4.

b12 b−

12 ≤ 1

=⇒ (12 b−

12 )∗(

12 b−

12 ) ≤ 1

=⇒ ‖12 b−

12 ‖ ≤ 1

=⇒ ‖(12 b−

12 )∗‖ = ‖b−

12

12 ‖ ≤ 1

=⇒ 12 b−1

12 ≤ 1

=⇒ b−1 ≤ 1.

5. Under Gelfand-Naimark, C∗()∼=→ C(Sp()) with 7→ z. But z2 ≤

z on Sp() ⊆ [0,1].

I.8.2 Moore-Smith convergence (Nets)

Sequences are insufficient to determine the topology of a generaltopological space. Note e.g. the difference between “compactness”and “sequential compactness”.

To resolve this, one trick is to introduce “generalized sequences” or“nets”.

Remark I.8.10. The French typically prefer using filters, an equivalenttechnical device.

32

Remark I.8.11. If we restricted our attention to separable C∗-algebras,everything here would be possible with ordinary sequences. But itwould be harder work.

Definition I.8.12. A directed set (“ensemble ordonné filtrant”) is aset equipped with a partial order ≤ such that any two elements havean upper bound:

∀, j ∈ , ∃k ∈ s.t. ≤ k and j ≤ k.

A net (“suite généralisée”) valued in a space X is a function : → Xfrom some directed set . We denote it by ()∈.

Example I.8.13. N is a directed set with its usual order. A net in-dexed by N is a sequence.

Example I.8.14. X – loc. compact Hausdorff.C0(X; [0,1]) = ƒ : X→ [0,1] continuous is a directed set with ƒ ≤ giff ƒ () ≤ g() ∀ ∈ X.

Definition I.8.15. Let X be a topological space, U ⊆ X a subset. Wesay a net ()∈ valued in X is eventually in U if ∃0 ∈ s.t. ∈ U∀ ≥ 0.

Say ()∈ converges to ∈ X if for every open neighbourhood U 3 ()∈ is eventually in U.

Example I.8.16. A net indexed by N converges iff it converges as asequence.

Example I.8.17. Let ƒ : [0,1] → R be piecewise cts. The Riemann

integral∫ 1

0ƒ ()d is (by definition) the limit of the net (k ƒ (k)Δk)

indexed by the directed set of partitions (0 = 0 < 1 < · · · < N = 1)of [0,1] ordered by refinement.

Example I.8.18. Let ∈ C0(X). The net (ƒ )ƒ∈C0(X;[0,1]) convergesto in C0(X).

Remark I.8.19. Nets are the correct alternative to sequences for gen-eral topological spaces: e.g. a space X is compact iff every net in Xhas a convergent subnet. The definition of subnet is subtle, though.

I.8.3 Approximate units (“Unités approchées”)

A — C∗-algebra.

33

Definition I.8.20. An approximate unit (“unités approchée”) for Ais a net ()∈ of positive elements with ‖‖ ≤ 1 which is increasingand satisfies

lim∈‖− ‖ = 0, lim

∈‖− ‖ = 0 ∀ ∈ A. (I.8.1)

Example I.8.21. The net (ƒ )ƒ∈C0(X;[0,1]) is an approximate unit forC0(X).

The same idea works for any C∗-algebra.

Theorem I.8.22. Every C∗-algebra has an approximate unit.

Proof. Let A+<1 := ∈ A+ | ‖‖ < 1 with the usual order on self-adjoint elements.

Step 1: We show A+<1 is directed.

Let , ∈ A+<1. Consider the bijection

g : [0,1)→ [0,∞), g(t) =t

1− t.

Note that

g−1 : [0,∞)→ [0,1), g−1(t) = 1−1

1+ t.

Since g and g−1 map 0 7→ 0, they will send A to A under the functionalcalculus, even if A is non-unital.

Put := g−1(g() + g()) = 1− (1+ g() + g())−1.

Then Sp() ⊂ [0,1) so ‖‖ < 1 and ∈ A+<1. Also, 1+ g() + g() ≥1+ g() implies

= 1− (1+ g() + g())−1 ≥ 1− (1− g())−1 = g−1(g()) = .

Likewise ≥ . Thus A+<1 is directed.

Step 2: Take the net ()∈A+<1 of all elements of A+<1. We need to

prove the limits (I.8.1).

Let ∈ A with = ∗. We have C∗() ∼= C0(X). The net (ƒ ())ƒ∈C0(X;[0,1))is an approximate unit. In particular, ∀ε > 0 ∃ƒ ∈ C0(X; [0,1)) s.t.‖ƒ ()− ‖ < ε.

But now, ∀ ∈ A+<1 with ≥ ƒ () we have

‖(1−)‖2 = ‖(1−)2‖ ≤ ‖(1−)‖ ≤ ‖(1−ƒ ())‖ ≤ ‖(1−ƒ ())‖ < ε.

34

(working in the unitalization A). That implies,

lim∈A+<1

‖− ‖ = 0 ∀ ∈ As.

and the other limit is similar.

For 6= ∗ we have

lim‖(1− )‖2 = lim

‖(1− )∗(1− )‖ = 0,

and similarly for lim ‖(1− )‖2.

Remark I.8.23. If A is separable, in fact it has an approximate unitwhich is a sequence: (n)n∈N.

If A is unital then the unit (1) suffices.

We note that approximate units are “approximately central”, in thefollowing sense.

Lemma I.8.24. Let () be an approximate unit for A. For any ∈ A,

lim∈‖[, ]‖ = 0,

where [, ] = − .

Proof. Follows from → and → .

I.8.4 Ideals and quotients in C∗-algebras

Ideals and quotients in C∗-algebras are nice because they are veryrigid.

Proposition I.8.25. Any closed ideal J in a C∗-algebra A is closedunder ∗, i.e. J∗ = J.

Proof. Exercise.

Theorem I.8.26. The quotient A/J is a C∗-algebra with the quotientnorm and obvious involution.

35

Proof. Denote the image of ∈ A in the quotient A/J by []. Recallthat the quotient norm is:

‖[]‖A/J := infj∈J‖+ j‖.

This is a Banach algebra (direct check) with involution (because J =J∗).

Let () be an approximate unit for J.

Claim: ∀ ∈ A,‖[]‖A/J = lim

∈‖(1− )‖.

Proof of Claim:

By defn of quotient norm: ‖(1 − )‖ ≥ ‖[]‖ ∀ ∈ . So ‖[]‖A/J ≤lim∈ ‖(1− )‖.

For the reverse, fix ε > 0. For any j ∈ J,

lim∈‖(1− )‖ ≤ lim

∈(‖(+ j)(1− )‖+ ‖j(1− )‖) ≤ ‖+ j‖.

The claim follows.

To conclude, we prove the C∗-inequality:

‖[]∗[]‖ = lim‖∗(1− )‖

≥ lim‖(1− )∗(1− )‖

= lim‖(1− )‖2 = ‖[]‖2.

I.9 Representations of C∗-algebras

I.9.1 Basic definitions

Recall that a representation of a C∗-algebra A is a ∗-homomorphismπ : A→ B(H).

Definition I.9.1. A subspace H′ ⊂ H is invariant if π(A)H′ ⊂ H′.

(Sometimes we abuse terminology and refer to H′ as a subrepresen-tation.)

36

Lemma I.9.2. Let H′ be an invariant subspace of a C∗-algebra rep-resentation.

1. The orthocomplement H′⊥ is invariant.

2. The closure H′ is invariant.

Proof. If ∈ A, η ∈ H′⊥ then ∀ξ ∈ H′,

⟨ξ, π()η⟩ = ⟨π(∗)ξ, η⟩ = 0,

so π()η ∈ H′⊥.

The second statement follows from H′ = (H′⊥)⊥.

Definition I.9.3. A closed invariant subspace H′ ⊂ H is called asubrepresentation of H. (More correctly, the restriction of π to H′

is called a subrepresentation.)

A representation is irreducible if it contains no non-trivial subrepre-sentations.

Example I.9.4.

A = M2(C)⊕M1(C) ∼=

∗ ∗ 0∗ ∗ 00 0 ∗

⊂ M3(C)

is represented on C3 in the obvious way. The subspaces H′ = spne1, e2and H′′ = spne3 are subrepresentations.

Remark I.9.5. The lemma above says that for a C∗-algebra, any sub-representation of a representation is complemented. This is not truefor general algebras, e.g.

A =

∗ ∗ ∗∗ ∗ ∗0 0 ∗

⊂ M3(C)

is represented on C3 and H′ = spne1, e2 is a subrepresentationbut it has no complementary subrepresentation. But this A is not aC∗-algebra. It is the ∗-operation which gives us complements.

Definition I.9.6. A vector ξ ∈ H is cyclic if π(A)ξ = π()ξ | ∈ Ais dense in H.

cyclic vector =vecteurtotalisateurA representation π : A→ B(H) is cyclic if it has a cyclic vector.

37

NB: Every nonzero vector ξ in an irreducible representation is cyclic,since π(A)ξ is a subrepresentation.

Example I.9.7. For the representation of Example I.9.4, the vectorse1 and e3 are not cyclic, since π(A)e1 = spne1, e2 and π(A)e3 =spne3. But e1 + e3 is cyclic. So π is a cyclic representation.

The zero representation π() = 0 ∀ ∈ A is not interesting. Nor do wewant π(A) acting by 0 on some non-trivial closed subspace. We makethe following definition.

Definition I.9.8. A representation π : A→ B(H) is non-degenerate ifπ(A)H is dense in H.

In fact, this is equivalent to the a priori stronger statement π(A)H = H.

Theorem I.9.9 (Cohen Factorization Theorem). If π is a non-degeneraterepresentation of a C∗-algebra A, then π(A)H = H.

Proof. Exercise.

Remark I.9.10. The name is because every ∈ H can be factorizedas = for some ∈ A, ∈ H.

If A is unital, non-degeneracy is equivalent to the representation π :A→ B(H) being unital.

I.9.2 The GNS Representation Theorem(statement & idea of proof)

Definition I.9.11. An representation is called faithful if it is injective.faithful ≡ fidèle

Thus a faithful representation π : A ,→ B(H) is a realization of A asa concrete C∗-algebra of bounded operators on H. A major result inC∗-algebras says that every C∗-algebra is isomorphic to a concreteC∗-algebra.

Theorem I.9.12 (Gelfand-Naimark-Segal). Every C∗-algebra A ad-mits a faithful representation π : A → B(H) on some Hilbert spaceH.

The proof will be given in the next few sections. To motivate it, let usrecall the abelian case.

38

X — compact Hausdorff space,μ — Radon probability measure on X.

Then A = C(X) is represented on H = L2(X;μ) by pointwise multiplica-tion:

π()ƒ := ƒ , ( ∈ C0(X), ƒ ∈ L2(X;μ)).

This already contains many of the key ideas of the proof. Recall thatthe Hilbert space H = L2(X;μ) needs to be constructed from A = C0(X)by

1. quotienting by the functions with (∫

ƒ∗ƒ dμ) = 0,

2. completing w.r.t. ‖ƒ‖ := (∫

ƒ∗ƒ dμ)12 .

w.r.t. ≡ withrespect to ≡ parrapport àWe recall that this does not usually produce a faithful representation

(think of μ = δ). But to make a faithful representation, we can takea direct sum of a collection of non-faithful representations, as long aseach element of A is not killed by at least one of them.

The main difference for non-abelian C∗-algebras is that we replaceprobability measures by states.

I.9.3 States

Definition I.9.13. A linear functional φ;A → C on a C∗-algebra islinear functional= formelinéaire.

positive if φ maps A+ to R+.

Equivalently, φ is positive if φ(∗) > 0 for all ∈ A.

Lemma I.9.14. Any positive linear functional on a C∗-algebra isbounded.

Proof. Suppose φ was positive but not bounded. Then we could findn ∈ A with ‖n‖ = 1 but |φ(n)| > 4n for all n ∈ N. We can write nas a sum of four positive elements, so for at least one of these, call itbn, we have φ(n) > 4n−1.

Consider

b =∑

n2−(n−1)bn (absolutely convergent).

Then b ≥ 2−(n−1)bn so φ(b) > 2−(n−1)φ(bn) > 2n−1 for all n, which isa contradiction.

39

Definition I.9.15. A positive linear functional of norm 1 is called astate.

state = état.

The set of all states of A is called the state space and denoted S(A).

A state σ is called faithful if φ(∗) > 0 for all 6= 0.

Example I.9.16. A regular Borel probability measure μ on a compactHausdorff space X gives a state φμ on C(X) by

φμ : ƒ 7→∫

Xƒ dμ.

Every state on C(X) is of this form by the Riesz Representation Theo-rem.

In particular, if μ = δ is Dirac measure at a point ∈ X, we get thestate ev.

Remark I.9.17. The name state comes from quantum physics.

Definition I.9.18. Let π : A → B(H) be a representation of A on aHilbert space H, and let ξ ∈ H, ‖ξ‖ = 1. The functional

Functional =forme linéaireφ() := ⟨ξ, π()ξ⟩

is a state (direct check). It is called the vector state associated to ξ.

Example I.9.19. The state φμ on C(X) above is a vector state for theconstant function 1 ∈ L2(X;μ), since

φμ(ƒ ) =∫

Xƒ dμ = ⟨1, ƒ1⟩ ∀ƒ ∈ C(X).

I.9.4 Inner products from states

A — C∗-algebra.

Definition I.9.20. Let φ : A→ C be a state. We define a sesquilinearform on A by

⟨, b⟩σ := σ(∗b).

Remark I.9.21. ⟨ · , · ⟩σ is positive semidefinite (i.e. ⟨, ⟩σ ≥ 0 ∀ ∈A). But it is definite (i.e. ⟨, ⟩σ > 0 ∀ 6= 0) only if σ is faithful.

Semidefiniteness is enough to prove the Cauchy-Schwarz Inequality:

Lemma I.9.22 (Cauchy-Schwarz Inequality).

|⟨, b⟩σ | ≤ ⟨, ⟩12σ ⟨b, b⟩

12σ ∀, b ∈ A.

40

Proof. Same as the classical proof.

Corollary I.9.23. Let ∈ A. We have ⟨, ⟩σ = 0 iff ⟨, b⟩σ = 0 for allb ∈ A.

Moreover, the set of null vectors

Nσ := ∈ A | ⟨, ⟩σ = 0

is a left ideal in A

Proof. The first statement follows immediately from Cauchy-Schwartz.

If ∈ Nσ then

⟨c, b⟩ = σ(∗c∗b) = ⟨, c∗b⟩ = 0, ∀b, c ∈ A,

so c ∈ Nσ.

I.9.5 A characterization of states

The next proposition is extremely useful for recognizing states.

Proposition I.9.24. Let A be a C∗-algebra and (j) an approximateunit. A bounded linear functional φ : A → C is positive iff limj φ(j) =‖φ‖. In particular, if A is unital then φ is positive iff φ(1) = ‖φ‖.

Thus a state on a unital C∗-algebra is a linear functional of norm 1with φ(1) = 1.

Proof. After rescaling φ we may assume that ‖φ‖ = 1.rescale = faireunehomothétie.(⇒): Let φ be positive.

Then φ(j) is an increasing net of positive numbers bounded by 1, soconverges. Write

r = limjφ(j).

Note that r ≤ ‖φ‖ = 1.

For any ∈ A with ‖‖ ≤ 1, we have φ() = limj φ(j). But, byCauchy-Schwarz,

|φ(j)|2 ≤ φ(∗)φ(2j ) ≤ φ(2j ) ≤ φ(j) ≤ r.

So |φ()|2 ≤ r for all ‖‖ ≤ 1. Thus r ≥ ‖φ‖2 = 1.

41

We thus have r = 1 as desired.

(⇐): Suppose limj φ(j) = 1. First we prove φ : As → R (this is thehard part), then we prove φ : A+ → R+.

Let ∈ A be self-adjoint with ‖‖ ≤ 1. Write φ() = + y with , y ∈ R.Suppose y 6= 0. WLOG we assume y > 0 (otherwise replace by −).

WLOG =without loss ofgenerality =sans perte degénéralité.

To get the idea of the proof, let us first the case where A is unital.Consider the elements n1− where n ∈ N. Note that

‖n1− ‖2 = ‖(n1− )∗(n1− )‖ = ‖n21+ 2‖ ≤ n2 + 1.

Therefore|φ(n1− )|2 ≤ ‖n1− ‖2 ≤ n2 + 1.

But

|φ(n1− )|2 = |n− + y|2 = (n+ y)2 + 2 = n2 + ny+ (2 + y2).

This is greater than n2 + 1 for sufficiently large n ⇒ contradiction.

If A is not unital, we must use an approximate unit (j) instead of 1.Let us write φ(j) = ξj+ ηj with ξj, ηj ∈ R. Fix n ∈ N. Since ξj+ ηj → 1,we may choose j “sufficiently large” s.t.

|(nξj + y) + (nηj − )|2 − |(n+ y)− |2

< 1

and also (by Lemma I.8.24)

‖[, j]‖ <1

n.

We then have

‖nj − ‖2 = ‖n22j + n[, j] + 2‖ ≤ n2 + 2.

Therefore|φ(nj − )|2 ≤ n2 + 2.

But

|φ(nj − )|2 = |(nξj + y) + (nηj − )|2

≥ |(n+ y)− |2 − 1

= n2 + ny+ (2 + y2).

This is greater than n2 + 2 for n sufficiently large ⇒ contradiction.Thus φ : As → R.

42

Finally, we need to prove φ : A+ → R+. Let ∈ A+ with 0 ≤ ≤ 1.Then,

− 1 ≤ j − ≤ 1 ∀j

=⇒ ‖j − ‖ ≤ 1 ∀j

=⇒ φ(j − ) ≤ 1 ∀j

=⇒ 1− φ() ≤ 1 (by taking limj)

=⇒ φ() ≥ 0.

Corollary I.9.25. Any state σ on a non-unital C∗-algebra A extendsuniquely to a state σ on the unitalization A.

Proof. By the proposition above, if σ is an extension of σ to a stateon A, we must have σ(1) = 1. This gives uniqueness.

Conversely, given σ on A, we can define σ by

σ(+ λ1) := σ() + λ.

For any + λ1 ∈ A we get

σ((+ λ1)∗(+ λ1)) = limσ((+ λj)∗(+ λj)) ≥ 0.

Therefore σ is positive, so bounded.

The next corollary shows the real power of the above characterizationof states.

Corollary I.9.26. Let A be a C∗-algebra. For every self-adjoint ∈ A,there is a state σ on A s.t. |σ()| = ‖‖.

Proof. By passing to A we may assume that A is unital.

Consiser the C∗-subalgebra C∗() ⊂ A. Recall that C∗() ∼= C(Sp()).By the spectral radius formula, there is z ∈ Sp() with |z| = ‖‖. Putφ = evz : C(Sp()) → C. It is bounded and φ(1) = ‖φ‖ = 1 so it is astate on C∗(). It has |φ()| = |z| = ‖‖.

By the Hahn-Banach Theorem φ can be extended to a bounded linearfunctional σ on A with the same norm: ‖σ‖ = 1. It is therefore a stateon A with |σ()| = ‖‖.

In particular, for every 0 6= b ∈ A there is a state σ such that σ(b∗b) 6=0.

43

I.9.6 The GNS construction

The Gelfand-Naimark-Segal (GNS) construction produces a Hilbertspace representation of A from a state on A, in the same way thatwe produce L2(X;μ) from A = C(X).

Definition I.9.27. A — C∗-algebra.σ — state on A.Let Nσ := ∈ A | σ(∗) = 0 be the set of null vectors for ⟨ · , · ⟩σ.

By Corollary I.9.23, if , b ∈ A, m,n ∈ Nσ then

⟨+m,b+ n⟩σ = ⟨, b⟩σ ,

so ⟨ · , · ⟩σ descends to a it positive definite inner product on A/Nσ.

Define Hσ to be the completion of A/Nσ with this inner product. It is aHilbert space, called the GNS space of A for the state σ. Let us write[]σ = + Nσ for the class of in A/Nσ. Sometimes we will drop σfrom the notation.

Proposition I.9.28. The map

πσ : A→ End(A/Nσ); πσ()[b]σ = [b]σ

is well-defined and extends to a ∗-representation

πσ : A→ B(Hσ).

Moreover πσ is cyclic, with a unit cyclic vector Ωσ ∈ Hσ whose vectorstate is σ, i.e.

⟨Ωσ , π()Ωσ⟩ = σ(), ∀ ∈ A.

Definition I.9.29. πσ is called the GNS representation associated toσ.

Proof. Recall that Nσ is a left ideal. Thus left multiplication πσ() iswell-defined on A/Nσ for any ∈ A.

It is bounded for the operator norm, since ∀b ∈ A,

‖πσ()[b]‖2 = ⟨πσ()[b], πσ()[b]⟩σ= σ(b∗∗b)

≤ ‖‖2σ(b∗b)

= ‖‖2⟨[b], [b]⟩σ ,

where we have used b∗∗b ≤ b∗‖‖2b. Therefore π() extendsto a bounded linear map on the completion Hσ. Moreover, it is a∗-homomorphism: e.g.

⟨πσ(∗)[b], [c]⟩σ = σ(b∗c) = ⟨[b], πσ()[c]⟩ ∀, b, c ∈ A

44

so πσ(∗) = πσ()∗; the other identities are similar.

If A is unital then Ωσ := [1] is a unit vector, and ⟨[1], πσ()[1]⟩ =σ().

If A is not unital we must use an approximate unit (j). Note that if ≥ j

‖[]− [j]‖2 = σ(( − j)2) ≤ σ( − j)→ 1− 1 = 0

shows that ([j]) is a Cauchy net in Hσ so admits a limit which wedenote Ωσ. Then ∀ ∈ A,

⟨Ω, πσ()Ω⟩ = limj⟨[j], πσ()[j]⟩ = lim

jσ(jj) = σ().

Example I.9.30. Let A = C(X) and σ be the state associated to aRadon probability measure μ, i.e. σ(ƒ ) :=

∫

X ƒ dμ. Then ⟨ · , · ⟩σ is

the usual L2-inner product (with respect to μ) and the GNS space isHσ = L2(X;μ).

The GNS representation is the usual multiplication representation:

πσ(ƒ )[g] = [ƒg]

where here [g] denotes the L2-class of a continuous function g ∈C(X).

The cyclic vector is Ωσ = 1X, the constant function 1.

Following this example, for general A, the space Hσ could be reason-ably denoted by L2(A;σ).

Theorem I.9.31. Every C∗-algebra is isomorphic to a concrete C∗-algebra of operators on some Hilbert space.

Proof. For every 0 6= b ∈ A there is a state σ s.t.

σ(b∗b) 6= 0=⇒ πσ(b)Ωσ 6= 0 in Hσ=⇒ πσ(b) 6= 0.

Thus if we take the enormous direct sum of representations,

=⊕

σ∈S(A)πσ : A→ B(

⊕

σ∈S(A)Hσ).

it is faithful (i.e. injective).

45

Remark I.9.32. The Hilbert space H =⊕

σ∈S(A)Hσ is really enormous(typically non separable). Usually, a much smaller selection of statesσ suffices. For instance, if A admits a faithful state σ then the GNSrepresentation πσ is already faithful.

The GNS construction shows that every state can be realized as avector state of some cyclic representation. There is a uniquenessresult too:

Proposition I.9.33. Let π1 : A→ B(H1) and π2 : A→ B(H2) be cyclicrepresentations with cyclic vectors Ω1 and Ω2, resp. If the vectorstates σ() = ⟨Ω, π()Ω⟩ are equal then the representations areisomorphic, i.e., there is a unitary operator U : H1 → H2 such thatπ2() = Uπ1()U−1 for all ∈ A.

Proof. Let σ = σ1 = σ2. Note that

π1()Ω1 = 0

⇐⇒ ⟨π1(b)Ω1, π1()Ω1⟩ = 0 ∀b ∈ A⇐⇒ σ(b∗) = 0 ∀b ∈ A⇐⇒ π2()Ω2 = 0

So the map

U0 : π1(A)Ω1 → π2(A)Ω2; π1()Ω1 7→ π2()Ω2

is well-defined between dense subspaces of H1 and H2, respectively.It is also isometric, so extends to an isometry U : H1 → H2. The restis a direct check.

46

Chapter II

The Toeplitz Algebra &the Toeplitz IndexTheorem

II.1 Matrices of operators

If H = H1⊕H2 and H′ = H′1⊕H′

2are (orthogonal) direct sums of Hilbert

spaces, then any bounded linear operator T : H → H′ can be writtenas

T =

bc d

where : H1 → H′1, b : H2 → H′

1, c : H1 → H′

2, d : H2 → H′

2are

bounded linear operators. Specifically, one can write = pH′1TpH1 ,where pH1 : H→ H1 is the orthogonal projection, etc.

This extends in an obvious way to higher-rank decompositions. Theusual laws of matrix multiplication hold.

II.2 Compact operators & their represen-tations

II.2.1 Compact operators

Here we recall some of the basic facts about compact operators. HereH is a Hilbert space.

47

1. The rank of an operator T ∈ B(H) is

rnk(T) = dim im(T).

An operator is finite-rank if rnk(T) <∞.

2. Any finite rank operator is a sum of rank-one operators. Everyrank-one operator on H has the form

7→ ⟨, ⟩y

for some nonzero , y ∈ H.

3. The finite rank operators form a *-algebra, and in fact a ∗-idealin B(H), since rnk(ST) ≤ min(rnk(S), rnk(T). But it is not anorm-closed ideal (unless H is finite dimensional).

4. The closure of the finite-rank operators is the C∗-ideal of compactoperators K(H).

(The usual definition of a compact operator is an operator K ∈B(H) such that K(B(0; 1)) has compact closure, where B(0; 1).But for C∗-algebras, the definition as limits of finite-rank opera-tors is often more useful.)

5. The Spectral Theorem for Compact Normal operators: LetT ∈ K(H) be normal. Then T admits an orthonormal eigen-basis (ek). Moreover, the corresponding eigenvalues tend tozero: λk → 0. (If the Hilbert space is uncountable dimensional,i.e. non-separable, then only countably many eigenvalues arenonzero, and these converge to zero.)

Otherwise stated: Sp(T) is discrete, except for a possible accu-mulation point at 0 ∈ C, and all nonzero λ ∈ Sp(T) are eigenval-

accumulationpoint = valeurd’adhérence

ues with finite dimensional eigenspaces.

6. Still with T ∈ K(H) normal, note that for any nonzero λ ∈ Sp(T)the indicator function 1λ is continuous on Sp(T) and satisfies12λ = 1∗

λ = 1λ. Therefore, under functional calculus the ele-

ment pλ := 1λ(T) is a projection. It is the orthogonal projectiononto the λ-eigenspace of T (exercise).

This is called a spectral projection.

II.2.2 Representations of K(H)

The C∗-algebra of compact operators comes with a canonical repre-sentation π : K(H) ,→ B(H).

48

Lemma II.2.1. Let π : A→ B(H) be an irreducible representation of aC∗-algebra. If π(A) contains a single nonzero compact operator, thenit contains all compact operators.

Proof. It is no loss of generality to suppose A ⊆ B(H). Note that Acontains a finite-rank projection, namely any spectral projection ofa nonzero compact operator. Let P ∈ A be a projection of minimalnonzero rank.

For any self-adjoint T ∈ A, PTP is a self-adjoint compact operator. Anyspectral projection of PTP (for a nonzero spectral value) is a projectiononto a subspace of P.H. By minimality, it must be a scalar multiple ofP, i.e. PTP = cTP for some cT ∈ C.

Now suppose rnk(P) > 1. Then there are unit vectors , y ∈ P.H with⟨, y⟩ = 0. For any T ∈ A we have

⟨, Ty⟩ = ⟨, PTPy⟩ = cT ⟨, y⟩ = 0.

Therefore Ay is a non-trivial closed invariant subspace. This contra-dicts irreducibility.

Therefore rnk(P) = 1. In other words, there is ∈ H, ‖‖ = 1 suchthat

P = ⟨, ⟩ for all ∈ H.

Let y, z ∈ H. By irreducibility, A.H is dense in H, so for any ε > 0 thereare S, T ∈ A such that ‖y− S‖, ‖z − T‖ < ε. Then

SPT∗ = ⟨T, ⟩S for all nH.

One then has ‖SPT∗ − ⟨z, · ⟩y‖ < 2ε. Thus, A contains all rank oneoperators, and hence all compact operators.

Remark II.2.2. We used here that A is dense in H. In fact, thereis a stronger theorem—the Kadison Transitivity Theorem—which saysthat for any irreducible representation of a C∗-algebra A on a Hilbertspace H, A acts transitively on H, i.e. for all , y ∈ H there is T ∈ Asuch that T = y. This is similar to the Cohen Factorization Theorem.We won’t prove it.

From this, we can also prove the simplicity of K(H).

Corollary II.2.3. K(H) is a simple C∗-algebra, i.e. has no non-trivialclosed ideals.

49

Proof. Let J ⊂ K(H) be a nonzero ideal. Suppose H′ ⊆ H is a closedinvariant subspace for J. By the Cohen Factorization Theorem, any ∈ H′ can be written as = T for some j ∈ J, ∈ H′. Then forany S ∈ K(H) we have S = ST ∈ H′ since ST ∈ J. Therefore H′ isinvariant for K(H), so H′ = H. That is, H is irreducible for J. So byLemma II.2.1, J = K(H).

II.3 Fredholm operators

Recall the definition of the kernel and cokernel of a linear operatorT : V →W on a vector space.

ker(T) = ∈ V | T = 0coker(T) =W/ im(T)

From now on T : H1 → H2 will be a bounded operator between Hilbertspaces.

Definition II.3.1. A linear operator T : H1 → H2 is Fredholm if itskernel and its cokernel are both finite dimensional.

The index of a Fredholm operator is

ind(T) = dimker(T)− dimcoker(T).

We will write Fred(H) for the space of bounded Fredholm operatorson H.

The following observation is useful.

Lemma II.3.2. If T : H1 → H2 is Fredholm then ker(T) and im(T) areclosed subspaces.

Proof. The kernel of a bounded operator is always closed.

Note that T restricts to a bounded linear bijection from ker(T)⊥ toim(T).

Let W ⊂ H2 be a complementary subspace for im(T) in H2. Note thatW is finite dimensional, since the quotient map H2 → H2/ im(T) =coker(T) restricts to a bijection W ∼= coker(T).

Now define

T : ker(T)⊥ ⊕W→ H2; T(,) = T() +.

This is a bounded linear bijection, so is a topological isomorphism. Itfollows that im(T) = T(ker(T)⊥ ⊕ 0) is closed.

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Remark II.3.3. It follows that one could take the complementary sub-space W of im(T) to be im(T)⊥. In other words, if we wish, we couldidentify coker(T) ∼= im(T)⊥ when T is Fredholm.

Recall also that im(T)⊥ = ker(T∗).

Example II.3.4. Every linear map between finite dimensional Hilbertspaces T : V1 → V2 is Fredholm. The Rank-Nullity Theorem shows that

Rank-NullityTheorem =Théorème durang

ind(T) = dimker(T)− (dimV2 − dim im(T)) = dimV1 − dimV2,

i.e. in this case the index depends only on the spaces V1 and V2, notthe operator T.

In particular, any endomorphism of a finite dimensional Hilbert spacehas index 0.

Example II.3.5. The right shift T : ℓ2(N) → ℓ2(N) is Fredholm withindex −1. The left shift T∗ is Fredholm with index 1.

Remark II.3.6. If T : H1 → H2 and T′ : H′1→ H′

2are both Fredholm,

then the operator

T ⊕ T′ =

T 00 T′

: H1 ⊕H′1 → H2 ⊕H′2

is Fredholm and

ind(T ⊕ T′) = ind(T) + ind(T′).

Theorem II.3.7 (Atkinsons’ Theorem). Let T ∈ B(H). The followingare equivalent:

1. T is Fredholm;

2. T is invertible modulo finite rank operators, i.e. there exists S ∈B(H) s.t. − ST and − TS are finite rank operators;

3. T is invertible modulo compact operators i.e. there exists S ∈B(H) s.t. − ST and − TS are compact operators.

Proof. (1)⇒ (2): As noted in the previous proof, T restricts to a linear

homeomorphism T′ : H1 = (kerT)⊥∼=→ H2 = im(T). Let S : H → H be

defined by

S : H = H2 ⊕H⊥2 → H; (,) 7→ T′−1().

It is easy to check that ST is the orthogonal projection onto H1 andTS is the orthogonal projection onto H2. Since both have finite codi-mension, the result follows.

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(2) ⇒ (3): Immediate.

(3) ⇒ (1): Suppose − ST and − TS are compact. Then there areF1, F2 finite rank such that ‖ − ST − F1‖ < 1 and ‖ − TS − F2‖ < 1.Hence ST + F1 and TS+ F2 are invertible.

The restriction of ST + F1 to ker(T) is finite rank. So the restriction of = (ST + F1)−1(ST + F1) to ker(T) is finite rank. Thus dimker(T) isfinite.

Similarly, writing

= (TS+ F2)(TS+ F2)−1 = TS(TS+ F2)−1 + F2(TS+ F2)−1

shows that im(T)+im(F2) = H. Thus im(T) has finite codimension.

II.3.1 The Calkin algebra and essential spectra

Definition II.3.8. The Calkin algebra is the quotient C∗-algebra Q(H) =B(H)/K(H). We denote the quotient map by q : B(H)→ Q(H).

Atkinson’s Theorem immediate gives the following.

Corollary II.3.9. An operator T ∈ B(H) is Fredholm iff q(T) is invert-ible in Q(H).

Definition II.3.10. The essential spectrum of a bounded operatorT ∈ B(H) is

Spess(T) = SpQ(H)(q(T))

By Atkinson’s Theorem, this is equivalent to

Spess(T) = λ ∈ C | λ dH−T is not Fredholm.

II.3.2 Stability of the index

Theorem II.3.11. The space Fred(H) of Fredholm operators is openin B(H) (with the norm topology) and the index is constant on eachconnected component of Fred(H).

Proof. Fred(H) is the preimage by q of the set of invertibles in Q(H)which is open.

52

Let T ∈ Fred(H). Put V1 = ker(T), V2 = im(T)⊥ and H = V⊥ so that wehave decompositions H = H1 ⊕V1 = H2 ⊕V2. With respect to these, Tdecomposes as

T =

T 00 0

: H1 ⊕ V1 → H2 ⊕ V2,

where T′ : H1 → H2 is just the restriction of T. Note that T′ is anisomorphism.

Let A ∈ B(H) with ‖A‖ < ‖T′−1‖−1. Decompose A as

A =

bc d

: H1 ⊕ V1 → H2 ⊕ V2.

Since ‖‖ < ‖T′−1‖, we have T′ + is invertible. Set

N =

−(T′ + )−1b0

: H1 ⊕ V1 → H1 ⊕ V1,

N′ =

0−c(T′ + )−1

: H2 ⊕ V2 → H2 ⊕ V2.

These are both invertible (with inverses given by the usual 2 × 2-matrix law), and

N′(T + A)N =

T′ + 00 −c(T′ + )−1b+ d

.

From the finite dimensionality of V1 and V2, we get

ind(T + A) = indN′(T + A)N

= ind(−c(T′ + )−1b+ d)= dim(V1)− dim(V2)= ind(T).

We have thus proven that Fred(T) is open and ind(T) is constant onsome open ball around any T ∈ Fred(T). It is therefore constant onconnected components.

Corollary II.3.12. Let T ∈ Fred(H) and K ∈ K(H). Then T + K isFredholm and ind(T + K) = ind(T).

Proof. The linear path Tt = T + tK for t = [0,1] is norm-continuousand every every Tt is Fredholm by Atkinson’s Theorem. The resultfollows from the previous theorem.

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II.4 The Toeplitz algebra

II.4.1 The unilateral shift

Consider H = ℓ2 = ℓ2(N).

Define the unilateral shift:

T : ℓ2 → ℓ2; T(0, 1, 2, . . .) = (0, 0, 1 . . .).

Then ker(T) = 0 and coker(T) ∼= im(T)⊥ = spne0 where e0 =(1,0,0, . . .) is the first canonical basis vector. In particular,

nd(T) = −1.

Remark II.4.1. • T∗ is the left-shift, i.e.

T∗(0, 1, 2, . . .) = (1, 2, 3 . . .).

It has nd(T∗) = 1.

• T∗T = 1 but TT∗ = 1− P0 where

P0 = projection onto spne0,

with e0 = (1,0,0, . . .). In particular, T is not normal.

Definition II.4.2. The Toeplitz algebra T is the C∗-algebra gener-ated by the unilateral shift T.

Remark II.4.3. The Toeplitz algebra is not abelian because T is notnormal.

By definition T ,→ B(ℓ2) and we shall call this the canonical representationof the Toeplitz algebra. (It is not the only possible representation.)

Lemma II.4.4. The canonical representation of T is irreducible, non-degenerate and contains the compact operators.

Proof. T∗(ℓ2) = ℓ2, which is enough to show that the representationis non-degenerate.

Let H′ ⊆ ℓ2 be a nonzero subrepresentation. Let y ∈ H′ nonzero.By applying T∗ sufficiently many times, we get = (0, 1, . . .) =(T∗)ny ∈ H′ with 0 6= 0. Then P0 ∈ H′, so e0 ∈ H′. ThereforeTne0 = en ∈ H′ for all n ∈ N. It follows that H′ = H. This provesirreducibility.

Now T contains all compact operators by Lemma II.2.1.

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Thus we have K / T . What it the quotient C∗-algebra?

Note that T /K is a subalgebra of the Calkin algebra Q. It is generatedby a single unitary element

U = q(T) ∈ Q.

Therefore T /K ∼= C(X) where X = SpQ(U) = Spess(T).

Lemma II.4.5. The essential spectrum of the unilateral shift T is S1

(the unit circle in C).

Proof. We repeat that Spess(T) = SpQ(U) where U is unitary. Thespectrum of any unitary U in a C∗-algebra is always contained in S1,because under the Gelfand Transform C∗(U) ∼= C(Sp(U)), U maps tothe inclusion function z : Sp(U) ,→ C, which must satisfy z∗z = |z|2 =1.

It remains to show that Spess(T) is all of S1. Suppose ω ∈ S1 were notin the essential spectrum. Then the operators

Tt := tω− T (t ∈ [0,2])

is a continuous path of Fredholm operators. But

nd(T0) = nd(T) = 1,

while T2 = 2ω − T is invertible (since ‖T‖ < 2), so nd(T2) = 0. Thiscontradicts the stability of the index.

Corollary II.4.6. We have T /K ∼= C(S1). That is, there is a shortexact sequence of C∗-algebras

0→ K ,→ T C(S1)→ 0

where the quotient map sends the unilateral shift T to the functionz : S1 → C.

II.4.2 Toeplitz operators

Here is another point of view on the Toeplitz algebra, using Fourierseries.

Consider L2(S1). Here, we use S1 = z ∈ C | |z| = 1, but also R/Z ∼= S1via t 7→ e2πt. The measure is Lebesgue measure λ = dt, so that S1

has total measure 1.

55

As usual, put z ∈ C(S1), z(z) = z.

NB: Via R/Z ∼= S1 we have zn(t) = e2πnt.

Fourier Theory ⇒ (zn)n∈Z is an orthonormal basis for L2(S1). In otherwords, the map

L2(S1) 7→ ℓ2(Z)

ƒ 7→ (ƒn);

is an isometric isomorphism, where

ƒn := ⟨zn, ƒ ⟩ =∫

S1e−2πntƒ (t)dt.

Definition II.4.7. The Hardy space is

H2(S1) = ƒ ∈ L2(S1) | ƒn = 0 ∀n < 0.

By Fourier transform, H2(S1) ∼= ℓ2(N).

Let P be the orthogonal projection of L2(S1) onto H2(S1). On Fourierseries, P acts by

P(. . . , ƒ−2, ƒ−1, ƒ0, ƒ1, ƒ2, . . .) = (. . . ,0,0, ƒ0, ƒ1, ƒ2, . . .).

The C∗-algebra C(S1) is represented on L2(S1) by pointwise multipli-cation. We denote the representation by 7→ M, i.e.

M : ƒ 7→ ƒ , ∈ C(S1), ƒ ∈ L2(S1).

In particular z acts byMz : zn 7→ zn+1.

That is, z acts as the bilateral shift with respect to the basis (zn)n∈Z.

Remark II.4.8. This is a special case of the fact that the Fourier trans-form converts multiplication into convolution: If ∈ C(S1) then

F(ƒ ) =

(∗ ƒ )n

n∈N .

where ∗ ƒ denotes convolution of Fourier coefficients:

(∗ ƒ )n =∑

m∈Zm ƒn−m.

Definition II.4.9. The Toeplitz operator with symbol ∈ C(S1) is theoperator

T := PMP : H2(S1)→ H2(S1).

56

Remark II.4.10. Note that ‖T‖ ≤ ‖‖∞. In fact, ‖T‖ = ‖‖∞, but wewill prove this later.

In particular,

• Tz = unilateral shift on (one-sided) Fourier series,

• Tzn = shift right by n places.

The map 7→ T is not a representation of C(S1). For instance,TzTz−1 6= T1 = d since

TzTz−1 : (ƒ0, ƒ1, ƒ2, . . .)Tz−17−→ (ƒ1, ƒ2, ƒ3, . . .)

Tz7−→ (0, ƒ1, ƒ2, . . .).

Note though that T1−TzTz−1 is the rank-one projection onto spnz0.In fact T is a representation modulo compacts, in the following sense.

Proposition II.4.11. Write H = H2(S1).

1. For any , b ∈ C(S1) we have TTb − Tb ∈ K(H).

2. The map C(S1)→ Q(H); 7→ [T] is a ∗-homomorphism.

Proof. 1. First consider the case = zn. We get

Tznb − TznTb = PMznMbP− PMznPMbP

= PMzn(d−P)MbP.

Put d−P = P⊥, the projection onto the orthocomplement ofH2(S1), i.e.

P⊥(. . . , ƒ−2, ƒ−1, ƒ0, ƒ1, ƒ2, . . .) = (. . . , ƒ−2, ƒ−1,0,0,0, . . .).

Then

im(PMznP⊥) =

(

spnz0, . . . ,zn−1, if n > 0,

0, if n ≤ 0.

In particular, it is finite rank. Therefore Tznb−TznTb is finite-rank.

It follows that Tpb − TpTb is finite-rank for any “trigonometric

polynomial” p =∑Nn=−M nz

n.

Finally, these polynomials are dense in C(S1) (Stone-Weierstrass).That is, for any ∈ C(S1) and any ϵ > 0, ∃p trigonometric poly-nomial s.t. ‖− p‖ < ϵ. We have

Tb − TTb = (Tb − Tpb) + (Tpb − TpTb) + (TpTb − TTb)

where

57

• ‖Tb − Tpb‖ = ‖Tb−pb‖ =≤ ‖b− pb‖∞ ≤ ϵ‖b‖,• ‖TpTb − TTb‖ = ‖Tp−Tb‖ ≤ ϵ‖b‖,• Tpb − TpTb is finite-rank.

Since ϵ was arbitrary, Tb − TTb is compact.

2. Direct calculation ⇒ T∗ = T∗, T + Tb = T+b, and TTb ≡ Tbmod K(H).

Corollary II.4.12. The quotient map T → C(S1) of Corollary II.4.6sends T to .

Corollary II.4.13. For any ∈ C(S1), ‖T‖ = ‖‖∞.

Proof. Recall that T = PMP. So ‖T‖ ≤ ‖P‖‖M‖‖P‖ = ‖‖∞.

Also, any C∗-homomorphism has norm ≤ 1, so Corollary II.4.12 gives‖‖∞ ≤ ‖T‖.

II.4.3 The Toeplitz Index Theorem

Proposition II.4.14. The Toeplitz operator T is Fredholm iff its sym-bol ∈ C(S1) is invertible (i.e. nowhere zero).

Proof. This follows from Corollary II.4.12 and Atkinson’s Theorem.

Natural question: What is the index of T when Fredholm?

The answer, surprisingly, is calculated by topology, not by analysis.

Theorem II.4.15 (Toeplitz Index Theorem). For : S1 → C× continu-ous,

nd(T) = −Winding Nmber().

The winding number1 of is the number of times (t) turns around 0(anticlockwise) as t passes once around the circle (anticlockwise).

The winding number is most elegantly defined via algebraic topology:the fundamental group of C× is π1(C×) ∼= Z with the generator givenby the inclusion z : S1 → C× as the unit circle. Let’s give a quickoverview of all this.

Let X, Y be topological spaces. Two continuous maps ƒ0, ƒ1 : X → Yare homotopic to one another if ƒ0 can be continuously deformed intoƒ1. Precisely:

1 In French, the winding number is called l’indice, which leads to an unfortunatestatement of the Toeplitz Index Theorem: nd(T) = − nd().

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Definition II.4.16. The continuous maps ƒ0, ƒ1 : X→ Y are homotopicif there is a continuous map

F : X × [0,1]→ Y

with restrictions F( · ,0) = ƒ0 and F( · ,1) = ƒ1.

If we put ƒt = F( · , t) for all t ∈ [0,1] then the family of functions ƒt isthe continuous deformation of ƒ0 into ƒ1.

Homotopy is an equivalence relation on the set of continuous mapsX→ Y, since they can be inverted and concatenated.

Theorem II.4.17. Any continuous function : S1 → C× is homotopicto one, and only one, of the functions zn for n ∈ Z. The number n isthe winding number of .

I’ll give a rapid proof of this in class, but leave the details for thecourse on algebraic topology.

Proof of the Toeplitz Index Theorem. The key is to show that the in-dex of T depends only on the homotopy class of .

Let F : S1 × [0,1] → C× be continuous, and put ƒt = F( · , t). Thent 7→ ƒt is a continuous function [0,1]→ C(S1). since ‖T‖ = ‖‖∞, themap t 7→ Tƒt is continuous. Since Tƒt is Fredholm for all t ∈ [0,1], theirindices are all equal.

It therefore suffices to calculate the index of Tzn for each n ∈ Z.But Tzn is the right unilateral shift by n places if n ≥ 0 and the leftunilateral shift by −n places if n ≤ 0. An easy calculation givesnd(Tzn) = −n = −Winding Nmber(zn).

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Chapter III

Group C∗-algebras

III.1 Convolution algebras

III.1.1 Topological groups; Haar measure

Definition III.1.1. A locally compact group is a group G equippedwith a locally compact Hausdorff topology such that the group oper-ations

m :G×G→ G; (g, h) 7→ gh

ι :G→ G; g 7→ g−1

are continuous maps. (Here G×G has the product topology.)

Any locally compact group G has a unique (up to scalar multiple)nonzero Borel measure μ which is finite on compact sets and invariantunder left translations, i.e.

μ(gE) = μ(E) for any E ⊂ G (measurable), g ∈ G,

or equivalently∫

∈Gƒ ()dμ() =

∫

∈Gƒ (y−1)dμ() for any ƒ ∈ L1(G), y ∈ G.

This μ is called Haar measure. Usually we will write d fo dμ().

NB: Existence and uniqueness of Haar measure on a general l.c.group is a rather difficult theorem in measure theory. But in prac-tice, there is often an obvious left-invariant measure, so the abstracttheorem is not necessary.

If G is compact, then the Haar measure is finite, and we normalize isso that μ(G) = 1.

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Example III.1.2. 1. Any group G with the discrete topology. Inparticular, finite groups or countable groups such as Z, Zn, Fn(the free group on n generators).

The Haar measure is counting measure.

2. The groups Rn with their usual topology. The Haar measure isLebesgue measure.

3. The tori Tn = Rn/Zn with their usual topology. Haar measure isLebesgue measure.

4. Matrix groups G ⊆ Mn(C) with the subspace topology from Mn(C) ∼=Cn

2. For instance:

• GL(n,C), SL(n,C), GL(n,R), SL(nR), U(n), SU(n), O(n), SO(n),. . .

• The Heisenberg group

H =

1 b0 1 c0 0 1

: , b, c ∈ R

,

with the usual topology on R3 = (, b, c). Haar measureis Lebesgue measure on R3, i.e. dμ = ddbdc.

• The “+ b”-group

G =

¨

b0 1

: ∈ R×+, b ∈ R«

.

In general, the Haar measure is not right-invariant. (Although it oftenis: of all the groups in the above list, the only one for which Haarmeasure is not right-invariant is the + b-group.)

In any case, for any fixed g ∈ G, the measure g∗μ defined by

g∗μ(E) = μ(Eg)

is again left-invariant, so is a multiple of Haar measure. Therefore,there is a function

Δ : G→ R×+

such that

μ(Eg) = Δ(g)μ(E) for all E ⊂ G (measurable), g ∈ G.

It is an exercise to check that Δ is a continuous group homomorphismG→ R×+. It is called the modular function of G.

Lemma III.1.3. Let g ∈ G be fixed. We have the following change-of-variable formulas for the Haar measure:

61

1. dμ(g) = dμ(),

2. dμ(g) = Δ(g)dμ(),

3. dμ(−1) = Δ()−1 dμ().

Proof. (1) is by definition of the Haar measure. (2) is by definition ofthe modular function.

The second implies that Δ−1μ is a right-invariant measure, since ∀ƒ ∈Cc(G),

∫

∈Gƒ ()Δ()−1 dμ() =

∫

∈Gƒ (g)Δ(g)−1 dμ(g)

=∫

∈Gƒ (g)Δ()−1 dμ().

Now, dμ(−1) is also right-invariant, so by uniqueness it is a scalarmultiple of Δ−1μ. Considering dμ((−1)−1) = dμ() one sees thescalar multiple must be 1.

III.1.2 The L1 algebra

Let G be a l.c.group, μ its Haar measure.

Lemma III.1.4. The Banach space L1(G;μ) is a Banach ∗-algebrawith convolution product and involution as follows:

ƒ ∗ g() =∫

y∈Gƒ (y)g(y−1)dy

ƒ∗() = Δ()−1ƒ (−1)

for ƒ , g ∈ L1(G;μ), ∈ G. Also, ‖ƒ∗‖L1 = ‖ƒ‖L1 .

Proof. The fact that L1 is an algebra is a direct consequence of ba-sic properties of Lebesgue integrals: linearity, change of variables,Fubini’s Theorem.

To show that it is a Banach algebra: ∀ƒ , g ∈ L1(G),

‖ƒ ∗ g‖1 =∫

∈G

∫

y∈Gƒ (y)g(y−1)dy

d

≤∫

y∈G

∫

∈G|ƒ (y)| |g(y−1)|ddy (Fubini)

=∫

y∈G

∫

∈G|ƒ (y)| |g()|ddy (left-invariance)

= ‖ƒ‖1 ‖g‖1.

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The ∗-algebra properties are left as an exercise.

Remark III.1.5. L1(G;μ) is not a C∗-algebra, because ‖ƒ∗ƒ‖L1 6= ‖ƒ‖2L1 .

But (as we will soon see), we can put a different norm on L1(G;μ)which does satisfy the C∗-identity, so the completion will be a C∗-algebra.

In fact, in general there are several possible C∗-norms. But we shouldleave that discussion until later.

If G is discrete then L1(G) = ℓ1(G). Let us write [] for the deltafunction δ supported at ∈ G (which is continuous because G isdiscrete). Then [] | ∈ G is basis of a dense subspace of ℓ1(G).

Convolution of these elements is

[]∗ [y] = [y].

Thus, convolution is just the linear extension of the group multiplica-tion law.

The delta function at the identity [e] is an unit for the algebra ℓ1(G).

If G is not discrete, then the delta functions [] are not L1-functions(or rather, they are trivial as L1-functions). The elements ƒ ∈ L1(G)are not “sums of group elements” but “integrals of group elements:formally we might write

ƒ = “

∫

Gƒ ()[]d”.

In this purely formal sense,

ƒ ∗ g = “

∫

G

∫

Gƒ (y)g(y−1)dy

[]d”

“

∫

G×Gƒ (y)g(′)[y′]dyd′”.

That is, convolution is an ”integrated” linear extension of the grouplaw. This motivates the definition of convolution.

In this case L1(G) is not unital. However, it does have an approximateunit consisting of positive continuous functions of norm 1: take a net() of bump functions of norm 1 supported in decreasing neighbour-hoods of e.

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III.1.3 Operator topologies

For a discrete group G (e.g. a finite group) a unitary representationis simply a group homomorphism from G to the group of unitary op-erators on some Hilbert space H. For a topological group, we shoulddemand the representations are continuous homomorphisms. Butthe kind of continuity should be chosen carefully. There are severaltopologies on B(H).

The norm topology on B(H) is the usual metric topology defined bythe operator norm. It is a bad choice:

Example III.1.6. Let G = R. It is a topological group with its usualtopology.

The left regular representation is the representation λ of R on H =L2(R) by translations:

[λ()ƒ](y) = ƒ (−1y), , y ∈ R.

It is not continuous in the norm topology. For, given any ∈ R (verysmall) there is a function ƒ ∈ L2(R) with ‖ƒ‖2 = 1 but spp(ƒ ) ⊆[−12,

12]. Then

‖(λ()− λ(0))ƒ‖2L2= 2

so‖λ()− λ(0)‖B(H) ≥

p

2 ∀ 6= 0.

Therefore λ() 6→ λ(0) as → 0 in norm.

Definition III.1.7. The strong operator topology (SOT) on B(H) is thecoarsest topology such that the maps B(H) → H; T 7→ T are contin-

coarser =moins fineuous for every ∈ H. In other words, the net (T) ⊂ B(H) converges

strongly to T ∈ B(H) iff T→ T for all ∈ H.

A base of open neighbourhoods of 0 is the following: for any 1, . . . , n ∈H, ε > 0

U(0;1, . . . , n;ε) = T ∈ B(H) | ‖T‖ < ε ∀ = 1 dots, n.

(One could take just ε = 1 here without changing anything.)

Remark III.1.8. There is also a weak operator topology (WOT), whichis the coarsest topology s.t. the maps B(H) → C; T 7→ ⟨, T⟩ arecontinuous for every , ∈ H.

For linear subspaces of B(H), closure in the WOT and SOT are equiv-alent. A ∗-subalgebra A of B(H) which is WOT or SOT closed is called

64

a (concrete) von Neumann algebra. The abelian von Neumann alge-bras (acting on separable H) are all of the form L∞(X;μ) for someprobability measure space X, and in general von Neumann algebrasbehave like “noncommutative measure spaces” in the same way thatC∗-algebras behave like “noncommutative” topological spaces. Inparticular, there is a Borel functional calculus

L∞(Sp(T);μ)→ A; ƒ → ƒ (T)

for any T ∈ A, where μ is some Borel measure on the Sp(T).

We won’t discuss von Neumann algebras any further in this course.

III.1.4 Unitary representations of topological groups

Definition III.1.9. By a unitary representation of a locally compactHausdorff topological group G we shall mean a map π : G → U(H)which is continuous for the strong operator topology.

Example III.1.10. The left regular representation

λ : G→ L2(G); [λ()ƒ](y) = ƒ (−1y).

is a strongly continuous unitary representation. Unitarity is immedi-ate from the left-translation invariance of the Haar measure. Strongcontinuity is a consequence of the Dominated Convergence Theorem.

Lemma III.1.11. Let G be a l.c. group. There is a bijective corre-spondence between:

• unitary representations of G, and

• non-degenerate ∗-representations of L1(G).

Specifically, the correspondence sends a unitary representation π :G→ U(H) to the integrated representation

π : L1(G)→ B(H); π(ƒ ) :=∫

Gƒ ()π()d. (III.1.1)

Proof. Full proof for G discrete:

Let π be a unitary representation of G. The integrated representationis

π(ƒ ) =∑

∈Gƒ ()π().

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This series is norm-convergent since ‖π()‖ = 1 and ƒ ∈ ℓ1(G), andπ(ƒ ) ≤ ‖ƒ‖1 so π is bounded. Note that π([]) = π(). Therefore

π([])π([y]) = π([y])

and by linearity, π(ƒ )π(g) = π(ƒg) for all ƒ , g ∈ ℓ1. By unitarity

π(ƒ )∗ =∑

∈Gƒ ()π(−1)d = π(ƒ∗).

It is non-degenerate since π([e]) = .

Conversely, if π : ℓ1(G) → B(H) is a ∗-representation then we candefine a representation π : G→ U(H) by π() := π([]).

Sketch proof for G locally compact: (Details in Davidson [Dav96,p.183] or Dixmier [Dix96, §13.3])

The integral (III.1.1) is bounded since

‖π(ƒ )‖ =

∫

Gƒ ()π()d

≤∫

G|ƒ ()|‖π()‖d = ‖ƒ‖1.

Thus ‖π‖ ≤ 1. The fact that π is a ∗-representation is a direct check(although one needs to use the modular function Δ when checkingthe ∗-invariance). For non-degeneracy, we need to use the approxi-mate unit of norm 1 functions supported in decreasing neighbour-hoods of e.

Conversely, if π : L1(G) → B(H) is a ∗-representation then we candefine a unitary representation of G as follows. Let ∈ H be suchthat = π(ƒ ) for some ƒ ∈ L1(G) and ∈ H (such are dense in Hby non-degeneracy). Define

π() = π()π(ƒ ) := π(ƒ (−1 · )).

One can check, that

π() = SOT− limπ((−1 · )),

which implies in particular that π() is well-defined.

With the limit definition above, one gets π()π(y) = π(y) and π(e) =. One also gets ‖π()‖ ≤ 1 and ‖π()−1‖ = ‖p(−1)‖ ≤ 1 so π() isunitary and π(−1) = π()∗.

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Example III.1.12. The left regular representation

λ : G→ U(L2(G)); (λ()ƒ )(y) = ƒ (−1y)

integrates to the representation

λ : L1(G)→ B(L2(G)); λ(ƒ )g = ƒ ∗ g

for ƒ ∈ L1(G), g ∈ L2(G). In particular, we get ∀ƒ ∈ L1(G), g ∈ L2(G)

‖ƒ ∗ g‖2 ≤ ‖ƒ‖1‖g‖2.

One can also check this directly.

III.1.5 The C∗-enveloping algebra of a Banach ∗-algebra

We want to make a C∗-algebra out of the L1-algebra. Let us start byconsidering the general problem of embedding a Banach ∗-algebrain a C∗-algebra.

The easiest way to turn a Banach algebra into a C∗-algebra is torepresent it on a Hilbert space. Given a ∗-representation of norm 1

π : A→ B(H)

the closure π(A)‖ · ‖

is a C∗-algebra.

NB: This will not be a embedding of A unless π is faithful.

We would like to define a “universal representation”

=⊕

π : A→ B(⊕

Hπ)

which is the direct sum of all ∗-representations of norm 1. This hasannoying set-theoretic problems (the class of all representations istoo big to be a set), so we need to restrict the class of representationsin the direct sum.

We could take all irreducible representations (up to unitary equiva-lence). But this is a little hard to work with because of the fact thatnot every infinite dimensional ∗-representation decomposes into ir-reducibles.

In the end, it is technically easiest to use the cyclic representations.These have the nice property that every ∗-representation decom-poses into a direct sum of cyclic representations (by Zorn’s Lemma).

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On the other hand, the class of all cyclic representations (up to uni-tary equivalence) is not too large thanks to the GNS construction.

Let A be a Banach ∗-algebra. We can still talk of states: A state isa linear functional σ : A → C of norm 1 with σ(∗) = 1 for all ∈ A.Then we can define the inner product

⟨, b⟩σ = σ(b∗)

just as for a C∗-algebra. The GNS completion gives a representa-tion πσ : A → B(Hσ). Moreover, any cyclic representation of A is iso-morphic to the GNS representation of the vector state for the cyclicvector.

Definition III.1.13. The C∗-enveloping algebra of a Banach ∗-algebraA is the norm closure of (A), where

=⊕

σ∈S(A)πσ

is the direct sum of all GNS representations of A. It is denoted C∗(A).

Lemma III.1.14. The enveloping C∗-algebra has the following uni-versal property: any ∗-homomorphism of a Banach ∗-algebra A intoa C∗-algebra B factors through C∗(A):

A→ C∗(A)→ B.

Proof. It suffices to take B = B(H) since any C∗-algebra is isomor-phic to a subalgebra of some B(H). But now we can decompose theresulting representation of A on H into a direct sum of cyclic subrep-resentations, and these all factor through C∗(A).

Remark III.1.15. What we cannot show for a Banach ∗-algebra is thatthere are a lot of states. For instance we cannot ensure there areenough states to make a faithful ∗-representation A ,→ B(H) for someH. Constructing states needed the commutative Gelfand Theorem,and the characterization of states by σ(1) = ‖σ‖ = 1. Both of theseonly work for C∗-algebras

In fact, for a general Banach ∗-algebra, there are not always enoughstates. That means that A→ C∗(A) is not always injective.

But we will see that for the L1-algebras of groups this problem doesn’toccur.

III.1.6 The maximal group C∗-algebra

Let G be a l.c. group.

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Definition III.1.16. The maximal group C∗-algebra C∗(G) is the C∗-enveloping algebra of L1(G).

Lemma III.1.17. The map L1(G) → C∗(G) is injective, with denseimage.

Proof. By Lemma III.1.14, it suffices to have one faithful ∗-representationπ : L1(G) ,→ B(H) on a Hilbert space. The regular representationλ : L1(G)→ B(L2(H)) is faithful (exercise).

Density of the image is automatic, since C∗(L1(G)) is defined as theclosure of the image of L1(G) in some representation.

Proposition III.1.18. There is a bijective correspondence between

• Unitary representations of G, and

• ∗-representations of C∗(G).

Proof. It suffices to show that there is a bijective correspondence be-tween ∗-representations of L1(G) and ∗-representations of C∗(G).This is immediate from Lemma III.1.14.

III.1.7 The reduced group C∗-algebra

Definition III.1.19. The reduced C∗-algebra of G is the closure ofthe regular representation:

C∗r (G) := λ(G)‖ · ‖⊆ B(L2(G)).

From the bijective correspondence of Proposition III.1.18, the regularrepresentation gives a map C∗(G) C∗r (G). It is sometimes but notalways an isomorphism.

Remark III.1.20. The groups for which it is an isomorphism are calledamenable groups. We will not define amenability in this course.

amenable =moyennableAbelian groups are amenable. The free group F2 is not.

III.2 C∗-algebras of abelian groups

III.2.1 The Pontryagin dual G

In this section we consider locally compact abelian groups G. Notethat abelian groups are always unimodular (Δ ≡ 1), since left- andright-invariant Haar measure are equal.

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Lemma III.2.1. If G is abelian, then C∗(G) is an abelian C∗-algebraand

C∗(G) ∼= G, where G = Hom(G;T).

Here, Hom(G;T) denotes the set of continuous group homomorphismsfrom G to the circle. Its topology is induced from the weak-∗ topol-ogy on L1(G). That is, a net of continuous homomorphisms φ : G→ T

converges to φ : G→ T iff∫

Gφ()ƒ ()d→

∫

Gφ()ƒ ()d ∀ƒ ∈ L1(G). (III.2.1)

If G is discrete, this is equivalent to pointwise convergence of φ → φ.

Proof. For any , b ∈ L1(G) we have

ƒ ∗ g() =∫

y∈Gƒ (y)g(y−1)dy

=∫

z∈Gƒ (z−1)g(z)dz (using y = z−1)

= g∗ ƒ ().

So L1(G) is commutative, hence also C∗(G) by density.

By the commutative Gelfand Theorem

C∗(G) = C0(X)

where X is the space of characters of C∗(G), i.e. ∗-representationsof dimension 1. These are in bijective correspondence with one-dimensional unitary representations of G, so X ∼= G.

Under the Gelfand transform the topology on X ∼= G is the weak-∗topology, i.e.

φ → φ

⇐⇒ φ(ƒ )→ φ(ƒ ) ∀ƒ ∈ C∗(G)

⇐⇒ φ(ƒ )→ φ(ƒ ) ∀ƒ ∈ L1(G),

since L1(G) is dense in C∗(G). This is exactly Equation (III.2.1) (seeEquation III.1.1).

If G is discrete then pointwise convergence of φ is equivalent to(III.2.1) where ƒ is any delta function. But the delta functions havedense linear span in L1(G).

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Definition III.2.2. The set G is also an abelian group with product,unit, and inverse as follows: for φ,ψ ∈ G,

φψ() = φ()ψ() ∀ ∈ G,

1() = 1 ∀ ∈ G,

φ−1() = φ()−1 ∀ ∈ G.

Moreover, these operations are continuous w.r.t. the weak-∗ topologyon L1(G) (exercise). The l.c. group G is called the Pontryagin dual ofG.

One can also prove that ˆG ∼= G for any l.c. abelian group G, but wewon’t do that here.

Definition III.2.3. The Gelfand homomorphism

F : C∗(G)→ C0(G)

is called the Fourier transform. In particular, the restriction of theFourier transform to L1(G) is given by

(F ƒ )(φ) =∫

∈Gƒ ()φ()d.

Remark III.2.4. It is immediate that F(ƒ ∗ g) = F(ƒ ).F(g) for all ƒ , g ∈L1(G).

Example III.2.5. 1. G = Z.

A homomorphism φ : Z → T is completely determined by theimage of its generator: φ(1). Therefore Z ∼= T via φ 7→ φ(1). Theinverse map is

T 3 eθ 7→ [φθ : n→ enθ] ∈ Hom(Z;T)

The associated Fourier transform C∗(Z)→ C0(T) sends ƒ ∈ ℓ1(Z)to the corresponding Fourier series in C(T).

More generally Zn ∼= Tn.

2. G = R.

Some facts which I will add as exercises: Any continuous grouphomomorphism φ : R → T factors through a group homomor-phism

Rφ→ R

exp( · )→ T.

Moreover, any continuous homomorphism φ : R→ R is automat-ically linear, i.e. φ : → ξ for some θ ∈ R.

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Admitting this, we get R ∼= R via φ 7→ ξ. The inverse map is

R∼=→ R; ξ 7→ eξ•

The associated Fourier transform is the usual Fourier transform(up to factors of 2π).

More generally, Rn ∼= Rn via

Rn → R

n; ξ 7→ e⟨ξ,•⟩

3. G = T.

The homomorphism eξ · : R → T descends to a homomorphismT→ T iff ξ ∈ 2πZ. Therefore T ∼= Z via

R→ R; n 7→ e2πn•

The associated Fourier transform sends ƒ ∈ L1(T) to its Fouriercoefficients in c0(Z).

Remark III.2.6. G is compact iff G is discrete, since

C∗(G) unital ⇐⇒ L1(G) unital ⇐⇒ G discrete.

That deals with C∗(G). What about C∗r (G)?

Let us temporarily write Gr for the Gelfand dual of C∗r (G) which is also

commutative. (We will soon show Gr = G.) The surjection C∗r (G) C∗(G) implies that Gr is a subset of G.

Theorem III.2.7. If G is abelian then C∗r (G) = C∗(G).

Proof. We will show that Gr = G. That is, every unitary characterφ ∈ Hom(G,T) extends continuously to a character of C∗r (G) via the

usual integration formula on L1(G):

φ(ƒ ) =∫

∈Gƒ ()φ()d.

So we need to prove that

|φ(ƒ )| ≤ C‖ƒ‖C∗r (G) := C‖λ(ƒ )‖B(L2(G)) ∀ƒ ∈ L1(G).

Note first that φ ∈ L∞(G) so φƒ ∈ L1(G). We claim that, for any ƒ ∈L1(G) we have

‖λ(φƒ )‖ = ‖λ(ƒ )‖

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where λ is the left regular representation of G. To see this, we calcu-late ∀g ∈ L2(G)),

(λ(φƒ )g)() =∫

y∈Gφ(y)ƒ (y)g(y−1)d

= φ()∫

y∈Gƒ (y)φ(−1y)g(y−1)d

= (Mφλ(ƒ )Mφ−1g)(),

where Mφ is (as usual) the operator of pointwise multiplication by φon L2(G). But Mφ is unitary since im(φ) ⊆ T.

Now fix ψ ∈ G some unitary character which extends continuously toC∗r (G) (at least one such character must exist, since Gr is not empty).

Put φ′ = ψ−1φ. Then

φ(ƒ ) =∫

∈Gφ()ƒ () =

∫

∈Gψ()φ′()ƒ () = ψ(φ′ƒ ).

So|φ(ƒ )| = |ψ(φ′ƒ )| ≤ ‖λ(φ′ƒ )‖ = ‖λ(ƒ )‖.

III.2.2 Plancherel’s Theorem

Theorem III.2.8 (Plancherel’s Theorem). The Fourier transform ex-tends to an isometric isomorphism

F : L2(G)∼=→ L2(G),

for appropriate choice of Haar measure.

This statement is not quite precise.

If G is a discrete group, then ℓ1(G) ⊆ ℓ2(G). In this case, the isometryℓ2(G)→ L2(G) really is an extension of the Fourier transform ℓ1(G)→C(G) of the previous paragraph. The appropriate Haar measures arecounting measure on G and the Haar measure of total mass 1 on G.

If G is not discrete, then we must start with the restriction of theFourier transform to L1(G) ∩ L2(G) → C0(G). There is no canonicalnormalization of the Haar measures in general, which is why all theproblems about factors of 2π in the usual Fourier transform.

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Proof (for G discrete). Let [e] ∈ ℓ2(G) be the delta function at theidentity. Note that the Fourier transform of [e] is the unit 1 ∈ C(G):

F([e])(φ) = φ(e) = 1 ∀φ ∈ G.

The corresponding vector state for the regular representation of C∗(G)is

τ(ƒ ) = ⟨ƒ ∗ [e], [e]⟩

= ƒ (e) (when ƒ ∈ L1(G)).

It is also a state for C(G), so by the Riesz representation theoremthere is a positive Borel probability measure ν on G such that

τ(ƒ ) =∫

φ∈GF(ƒ )(φ)dν ∀ƒ ∈ C∗(G).

If ψ ∈ Hom(G,T) and ƒ ∈ L1(G) then

(F(ƒψ))(φ) =∫

∈Gƒ ()ψ()φ()d = (F(ƒ ))(ψϕ).

Therefore∫

φ∈GF ƒ (ψφ)dν(φ) =

∫

φ∈GF(ƒψ)(φ)dν(φ)

= τ(ƒψ) = ƒ (e)ψ(e) = ƒ (e) =∫

φ∈GF ƒ (φ)dν(φ).

Therefore ν is a left-invariant measure, i.e. a Haar measure.

Both of these Hilbert spaces are cyclic representations of C∗(G) ∼=C(G).

The first is the regular representation of C∗(G), with cyclic vector [e].

The second is the multiplication representation of C(G) with cyclicvector 1.

The corresponding vector states are equal:

⟨(F ƒ )1,1⟩ =∫

φ∈GF ƒ (φ)dν(φ) = τ(ƒ ).

The uniqueness part of the GNS theorem says that there is a unitaryisomorphism

U : ℓ2(G)→ L2(G)

with U : [e]→ 1 such that Uλ(ƒ )U−1 = MF(ƒ ). Then

Uλ(ƒ )[e] = MF(ƒ )1 =⇒ Uƒ = F(ƒ )

where ƒ denotes the function ƒ ∈ ℓ2(G).

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For the general proof, one needs to work with approximate unitsagain.

III.3 The C∗-algebras of the free group

In this section, we study the maximal and reduced C∗-algebras of avery non-abelian group: the free group.

III.3.1 The free group on two generators

Definition III.3.1. Fix two symbols (called “letters”) and b. A wordwill mean a finite string of the symbols ±1, b±1, e.g. bb−1bb−1b−1b−1,which we simplify as b2−1bb−3. The empty word (of length zero)is also allowed, and is denoted e.

A word is reduced if it contains no substring of the form −1, bb−1,1, b−1b.

The free group on two generators F2 is the set of all reduced words.Multiplication is given by concatenation of strings, and reduction, e.g.

(b2b−1).(b−2b2) = b2−1b2.

The inverse is given by reversing a word and inverting each letter:

(b2b−1)−1 = b−1b−2−1.

The unit is the empty word e.

The free group is the universal group on two generators, in the sensethat if G is any group and g, h ∈ G are any two elements, then thereis a homomorphism

φ : F2 → G

such that φ() = g and φ(b) = h. We simply send a word in , b to thecorresponding word in g, h.

It is non-abelian (since b 6= b) and torsion-free: it contains no ele-ments of finite order.

III.3.2 The Cayley graph

Definition III.3.2. The Cayley graph of a group G with respect to agenerating set S is the graph whose vertex set V and edge set E are

vertex =sommet, edge= arête75

V = G,

E = (g, h) ∈ G×G | h = gs±1 for some generator s ∈ S.

The generator s±1 = g−1h will be called the edge label of the edge(g, h)

Example III.3.3. 1. The Cayley graph of Z2 (w.r.t. the canonicalgenerators (1,0) and (0,1)) is a grid.

grid = grille

2. The Cayley graph of F2 (w.r.t. the defining generators , b) is aninfinite tree of valence 4.

(See pictures in class.)

The Cayley graph of F2 is a tree, i.e. it contains no loops. Between ev-ery pair of vertices, there is a unique shortest path, called a geodesic.The edge labels of the geodesic are given by the reduced word g−1h.

III.3.3 The canonical trace

Let G be a discrete group. The Dirac function at the identity [e] ∈ℓ2(G) is a unit vector. It defines a vector state for the regular repre-sentation:

τ(ƒ ) := ⟨λ(ƒ )[e], [e]⟩ (ƒ ∈ C∗r (G)).

If ƒ ∈ ℓ1(G) then τ(ƒ ) = ⟨ƒ , [e]⟩ = ƒ (e) is just evaluation at the identity.In particular,

τ([]) =

(

1, if = e

0, otherwise.

Definition III.3.4. A trace on a C∗-algebra A is a linear functionaltr : A→ C such that

tr(b) = tr(b) ∀nb ∈ A.

Proposition III.3.5. τ is a trace on C∗r (A). It is also faithful.

It is called the canonical trace on C∗r (G).

Proof. First we prove that τ(ƒ ) = ⟨λ(ƒ )[], []⟩ for any ∈ G. It suf-fices to prove this for ƒ = [y] another delta function, since these spana dense subspace of ℓ1(G); We calculate

⟨λ([y])[], []⟩ = ⟨[y], []⟩ = δy, = δy,e = τ([y]).

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Therefore, for any ƒ ∈ ℓ1(G), any ∈ G

τ(ƒ[]) = ⟨λ(ƒ[])[e], [e]⟩= ⟨λ([]ƒ[])[e], λ([])[e]⟩ because [] is unitary

= ⟨λ([]ƒ )[], []⟩= τ([]ƒ ).

Again, by density, we get τ(ƒg) = τ(gƒ ) for all ƒ , g ∈ ℓ1(G).

For faithfulness, observe that [e] is a cyclic vector for the regularrepresentation, since

λ([])[e] = [] ∀ ∈ G

and these span a dense subspace of ℓ2(G). Therefore, the GNS repre-sentation associated to τ is isomorphic to the regular representation,which is faithful.

III.3.4 The canonical trace for F2

In the case of the free group, there is another formula for the canoni-cal trace.

In this section, I will just write instead of λ() whenever ∈ F2. Inparticular, and b are the two unitary generators in C∗r (G).

Proposition III.3.6. For any ƒ ∈ C∗r (F2) we have

limm,n→∞

1

mn

m∑

=1

n∑

j=1

bjƒb−j− = τ(ƒ )1.

Here is the idea of the proof. As usual, it suffices to prove the caseƒ = [] for some ∈ G. Then

bjƒb−j− = bjb−j−.

These are all unitaries in C∗r (G).

If = e then bjb−j− = e for every , j, so the averages of theproposition are all equal to [e] = 1.

If 6= e then the words bjb−j− are (almost) all different. We willshow that these averages tend to 0.

We need some lemmas about operators with orthogonal ranges

77

Lemma III.3.7. Let A,B ∈ B(H) with m(A) ⊥ m(B). Then ‖A+ B‖2 ≤‖A‖2 + ‖B‖2.

Proof.

‖A+ B‖2 = sp‖‖≤1

⟨A + B,A + B⟩

= sp‖‖≤1

(⟨A,A⟩+ ⟨B,B⟩) ≤ ‖A‖2 + ‖B‖2.

Lemma III.3.8. Let H = H1 ⊕ H2 be an orthogonal decompositionof a Hilbert space. Let T ∈ B(H) map H2 into H1. Let U1, . . . , Un beunitaries such that U∗j U maps H1 into H2 for all 6= j. Then

1n

∑n=1UTU

∗

≤ 2pn‖T‖.

Proof. First, suppose that T maps all of H into H1.

Then,

∑n=1UTU

∗

2=

T +∑

6=1U∗1UTU∗ U1

2

≤ ‖T‖2 +

∑

6=1U∗1UTU∗ U1

2

≤ ‖T‖2 +

∑

6=1UTU∗

2.

By induction,

∑n=1UTU

∗

2≤∑n=1 ‖T‖

2 = n‖T‖2. (III.3.1)

Now, suppose that T only maps H2 into H1.

Write T = TP1 + TP2 where P is the orthogonal projection onto H.Note that TP2 maps all of H into H1. Also P1T = (TP1)∗ maps all ofH into H1. So each of TP1 and TP2 satisfy the norm estimate (III.3.1).We get

∑n=1UTU

∗

≤ 2pn‖T‖2.

The result follows.

Proof of proposition III.3.6. Put H = L2(F2). Let

Sn : B(H)→ B(H); Sn(T) =1

n

n∑

=1

bjTb−j

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denote the averaging of the conjugates by b, b2, . . . , bn. We considerSn() where ∈ G.

If = bk for some k then Sn() = . Otherwise, I claim that

‖Sn(λ())‖ ≤2pn. (III.3.2)

Note that if the word 6= b then = bk′b for some k, ∈ Z where ′

starts and ends with ±1. Then

‖Sn(λ())‖ = ‖λ(bk)Sn(λ(′))λ(b)‖ = ‖Sn(λ(′))‖.

So it suffices to prove the estimate (III.3.2) for words which beginand end with ±1.

Let us decompose H = H1 ⊕H2 where

H1 = spn[y] | y begins with ±1,

H2 = spn[y] | y begins with b±1 or y = e.

If ∈ G begins and ends with ±1 then λ() maps H2 to H1. Theoperators Uj = j are such that U∗j U = −j = λ(b−j) which maps H1to H2 for every 6= j. The previous lemma immediately gives (III.3.2).

Now let

S′m : B(H)→ B(H); Sm(T) =1

m

m∑

=1

T−

be the average of conjugates by . We get:

if 6= bk, ‖S′m(Sn())‖ ≤ ‖Sn()‖ ≤2pn,

if = bk (k 6= 0), ‖S′m(Sn())‖ = ‖S′m()‖ ≤

2pm

,

if = e, S′m(Sn(e)) = e = 1

Therefore,

limm,n→∞

S′m(Sn()) =

(

e, if = e

0, otherwise.

This completes the proof.

III.3.5 Simplicity of C∗r (F2)

Theorem III.3.9. C∗r (F2) is simple, i.e. it has no non-trivial ideals.

Remark III.3.10. It is equivalent to say C∗r (F2) has no non-trivial closedideals, since maximal proper ideals are always closed.

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Proof. Let J / C∗r (F2) be a C∗-ideal. Let ∈ J, 6= 0. Then τ(∗) 6= 0since τ is faithful. But S′m(Sn(

∗)) ∈ J for all m,n, so τ(∗)1 ∈ J.Therefore 1 ∈ J, i.e. J = C∗r (F2).

Corollary III.3.11. C∗r (F2) is not isomomorphic to C∗(F2).

Proof. Recall that any unitary representation of F2 extends to a rep-resentation of the maximal C∗-algebra. For instance, the trivial rep-resentation π : F2 → C extends to a ∗-homomorphism π : C∗(F2)→ C.The kernel is a non-trivial ideal, so C∗(F2) is not simple.

Remark III.3.12. In fact, C∗(F2) has many ideals. For instance, letG be any finite group with two generators, and π : G → U(n) anyirreducible unitary representation of G (which must be finite dimen-sional). Then there is a finite dimensional irreducible representation

π : F2 Gπ→ U(n)

by the universal property of the free group, and hence a finite dimen-sional irreducible representation of C∗(F2). None of these represen-tations factor through the reduced C∗-algebra.

The simplicity of C∗r (F2) suggests that there is not much hope of do-ing “Fourier theory” on the free group. Ideals in a C∗-algebra playthe role of open sets in topological spaces: recall that the ideals ofC0(X) are all of the form C0(Y) where Y is an open subset of X. Thus,roughly speaking, the topology on the reduced dual ˆ(F2)r has no non-trivial open or closed subsets.

The situation is very different for real reductive groups, like G =SL(n,C), where Gr is a very nice (almost Hausdorff) locally compactspace. Harish-Chandra described Gr for these groups and showedthere is a Plancherel measure on Gr which allows a generalization ofthe formula L2(G) ∼= L2(G) for abelian groups. These are the groupswhich interest physicists, for instance, so we’re kind of lucky thatthings work so nicely there.

III.3.6 The Kadison-Kaplansky conjecture

Definition III.3.13. A discrete group is called torsion-free if it has nonon-trivial finite subgroups.

Equivalently, G is torsion-free iff it has no elements of finite order,other than the identity.

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If G has a finite subgroup H, then the "average over H"

pH :=1

|H|

∑

∈H[]

is a projection in C∗r (G) (direct check). The Kadison-Kaplansky Con-jecture says that torsion is the only way to get projections in C∗r (G):

Conjecture III.3.14. If G is a torsion-free discrete group then C∗r (G)has no non-trivial projections.

Example III.3.15. G = Z is torsion-free. In this case C∗r (Z)∼= C(T).

A projection in C(T) is a 0,1-valued continuous function on T. ButT is connected, so the only projections are the constant functions 0and 1.

In this sense, the Kadison-Kaplansky Conjecture states that the re-duced dual of a torsion-free group is connected.

The Conjecture has been proven for a very large class of groups, butnot all groups. We will prove it for the free group. The method of proofis perhaps as interesting as the result. It uses K-homology, which isa homology theory for C∗-algebras.

Remark III.3.16. The existence of a simple unital C∗-algebra with noprojections was not known for a long time. The first examples wereconstructed using AF-algebras. But C∗r (F2) is a very pretty example.

III.3.7 Fredholm modules

Here is the fundamental concept which will let us prove that C∗r (F2)is projectionless.

Definition III.3.17. A Fredholm module for a C∗-algebra A is a pairof Hilbert space representations

π : A→ B(H), ( = 0,1)

with a unitary operator U : H0 → H1 that essentially intertwines themin the sense that

Uπ0()− π1()U is a compact operator ∀ ∈ A. (III.3.3)

It is notationally convenient to combine H0 and H1 into a single Z/2Z-gradedHilbert space

H := H0 ⊕H1.

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We keep track of the subspaces H0 and H1 by introducing the gradingoperator

γ =

1 00 −1

∈ B(H)

so that H0 and H1 are the ±1-eigenspaces of γ. A pair (H,γ) (where γis a self-adjoint operator with γ2 = 1) is called a Z/2Z-graded Hilbertspace.

We then put

π() =

π0() 00 π1()

and F =

0 U∗

U 0

in B(H). We then get the following equivalent definition:

Definition III.3.18. A Fredholm module for a C∗-algebra A is M =(H,γ, π, F), where (H,γ) is a Z/2Z-graded Hilbert space, π : A→ B(H)is a ∗-representation, and F ∈ B(H) is a bounded self-adjoint operatorsatisfying the following conditions:

• π() commutes with γ for all ∈ A,

• F anticommutes with γ, i.e. γF = −Fγ,

• F2 = 1

• [F, π()] ∈ K(H) for all ∈ A.

Example III.3.19. Let A = C∗r (Z)∼= C(T). Let π0 = π1 = λ be the

regular representation on H0 = H1 = ℓ2(Z) ∼= L2(T). Let U : ℓ2(Z) →ℓ2(Z) be the operator of pointwise multiplication by

χ ∈ ℓ∞(Z); χ(n) =

(

1, if n ≥ 0,−1, if n < 0.

Then

H,π, F =

0 U∗

U 0

is a Fredholm module for C∗r (Z). It turns out that it’s not a very inter-esting one. We’ll see a more interesting one soon.

Lemma III.3.20. Let

H,π, F =

0 U∗

U 0

be a Fredholm module

for A. For any projection p ∈ A, the operator Up := π1(p)Uπ0(p) :π0(p)H0 → π1(p)H1 is Fredholm.

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Proof. Using the essential intertwiner condition,

U∗p Up = π0(p)U∗π1(p)Uπ0(p)

≡ π0(p)U∗Uπ0(p)π0(p) mod K= π0(p),

which is the identity on π0(p)H0.

Definition III.3.21. We define

ndexM : Proj(A)→ Z,

ndexM(p) := ndex(π1(p)Uπ0(p) : π0(p)H0 → π1(p)H1)

where Proj(A) is the set of projections in A.

Remark III.3.22. By the stability of the Fredholm index, ndexM is lo-cally constant on Proj(A).

In summary, Fredholm modules can be used to detect projections.

III.3.8 Trace class operators

Definition III.3.23. Let H be a Hilbert space with orthonormal basis(e). The trace norm or L1-norm of an operator T ∈ B(H) is

‖T‖L1 :=∑

⟨|T |e, e⟩.

An operator T ∈ B(H) is called trace class if ‖t‖L1 < ∞. The set oftrace class operators is denoted L1(H).

Compare the Hilbert-Schmidt operators T ∈ HS(H) = L2(H), which arethose operators s.t.

‖T‖L2 :=∑

⟨|T |2e, e⟩ =∑

⟨Te, Te⟩ <∞.

Some facts (analogous to those for Hilbert-Schmidt operators):

1. Trace-class operators are compact.

2. If T ∈ L1(H) then the sum of “diagonal matrix entries”

Tr(T) :=∑

⟨Te, e⟩

is finite and independent of the choice of orthonormal basis (e).

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3. If P ∈ sL1(H) is a projection then it is finite dimensional andTrce(P) = rnk(P).

4. If T ∈ L1(H) and S ∈ B(H) then ST, TS ∈ L1(H) and

Tr(ST) = Tr(TS).

Moreover,‖ST‖L1 ≤ ‖S‖B(H)‖T‖L1(H).

The reason why we care about trace class operators here is that theywill give us another way to calculate the index of Fredholm operators.

Proposition III.3.24. Let T ∈ B(H) be Fredholm. If S is an inversefor T modulo trace-class operators, then

ndex(T) = Tr(1− ST)− Tr(1− TS).

Sketch of proof. One has to show that the right-hand side is inde-pendent of the choice of S. Given this, we saw in the section onFredholm operators that one can choose S such that 1− ST = Pker(T)and 1 − TS = Pim(T)⊥ , for which the right-hand side is dim(ker(T)) −dim(coker(T)).

III.3.9 Summability of Fredholm modules

Definition III.3.25. The domain of summability of a Fredholm mod-ule M = (H,π, F) is the set

A := ∈ A | [F, π()] ∈ L1(H).

A Fredhlom module M is called summable if its domain of summabilityis dense in A.

Example III.3.26. The domain of summability of the Fredholm mod-ule of Example III.3.19 includes all finite sums

=∑

n∈Zn[n] ∈ C[Z].

since the commutator [F, ] is finite rank (so trace class). Therefore,this Fredholm module is summable.

In fact, the domain of summability A is always a ∗-subalgebra of A(though usually not closed). It is even a Banach ∗-algebra with thenorm

‖‖A := ‖‖A + ‖[F, π()]‖L1 .

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(exercise).

The point of summability is that it gives us a new way of calculatingthe M-index of a projection in A.

Lemma III.3.27. Let M be a Fredholm module for A. The linear func-tional τM defined on the domain of summability A by

τM() :=12 Tr(γF[F, ])

is a trace on the domain of summability A.

(I am suppressing π from the notation here.)

Proof. Recall that the commutator is a derivation in both variables,e.g. [, yz] = [, y]z + y[, z].

For any , b ∈ A,

τM(b) =12 Tr(γF[F, b] + γF[F, ]b)

= 12 Tr(γF[F, b]− γ[F, ]Fb) (because [F2, ] = [1, ] = 0)

= 12 Tr(γF[F, b] + γFb[F, ]) (because Fγ = −γF).

This is clearly symmetric in , b.

Lemma III.3.28. If p ∈ A is a projection in the domain of summabil-ity, then

τM(p) = ndexM(p).

In particular, τM(p) is an integer.

Proof. We have

τM(p) = τM(p2)

= Tr(γFπ(p)[F, π(p)]) (see previous proof)

= Tr(γFπ(p)2[F, π(p)])

= −Tr(γπ(p)[F, π(p)]Fπ(p))= −Tr(γ(π(p)Fπ(p)Fπ(p)− π(p)))= Tr(γ(π(p)− π(p)Fπ(p)Fπ(p))).

A calculation gives

π(p)Fπ(p)Fπ(p) =

π0(p)U∗π1(p)Uπ0(p) 00 π1(p)Uπ0(p)U∗π1(p)

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So

τM(p) = Tr(π0(p)− U∗p Up)− Tr(π1(p)− UpU∗p )

= ndex[Up : π0(p)H→ π1(p)H] (by Proposition III.3.24)

= ndexM(p).

III.3.10 A Fredholm module for C∗r (F2)

We now apply the above machinery to C∗r (F2). The Fredholm modulewill be defined geometrically using the Cayley graph (V, E) of F2. Letus fix a root vertex, say e. Let us also add one more element to E,which we will denote by ξe.

Put H0 = L2(V) and H1 = L2(E ∪ ξe). We will denote the basisvectors by [] (for ∈ V) and [ξ] (for ξ ∈ E) respectively

Note that H0 = L2(F2), and we equip it with the regular representationπ0 = λ.

On the other hand, we can separate the edges into horizontal andvertical edges, and each will be in bijection with F2 (by taking left-hand vertices and bottom vertices, respectively). Thus, H1 ∼= L2(F2)⊕L2(F2)⊕C. We equip H1 with two copies of the regular representation,with the zero representation on spn[ξe]. That is, π1 = λ⊕ λ⊕ 0.

In other words, π0 and π1 are the representations induced from theobvious translation actions of F2 on V and E respectively.

Now fix a root vertex of the tree, say e. If ∈ V, we let ξ denote thefirst edge along the geodesic path from back towards e. If = e,we get ξe. Note that the map 7→ ξ is a bijection V → E ∪ ξe.Thus, the operator

U : H0 → H1; [] 7→ [ξ]

is a unitary.

Let us show that it is an essential intertwiner. Consider

π1()Uπ0(−1).

This sends [] to the first edge along the geodesic path from to .This is different from the edge towards e only if lies on the geodesicbetween e and . That is π1()Uπ0(−1)[] = U[] for all but a finitenumber of vertices . Thus,

π1()U− Uπ0() is finite rank ∀ ∈ F2.

By density, we obtain a summable Fredholm module M = (H,π, F).

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Proposition III.3.29. The trace τM = τ (the canonical trace of C∗r (F2))on the domain of summability.

Proof. It suffices to check this on the group elements [] ∈ C∗r (F2).

If = e, then [e] = 1 is a projection. We have π0(e) = d whileπ1(e) is the projection onto L2(E) = [ξe]⊥. One gets that τM([e]) =ndexM([e]) = 1.

If 6= e then

τM([]) = Trce(γF[F, π()]) = Trce(γ(π()− Fπ()F).

But translation by fixes no vertices or edges, so the operators π(),Fπ()F have no nonzero diagonal entries. Thus, the trace is zero.

III.3.11 Kadison-Kaplansky for F2

Now, the fruit of all our labour.

Theorem III.3.30. C∗r (F2) has no non-trivial projections.

Proof. Let p ∈ C∗r (F2) be a projection. Let A be the domain of summa-bility of the above Fredholm module M.

If p ∈ A, we getτ(p) = τM(p) = ndexM(p) ∈ Z.

But p is positive of norm 1, so τ(p) ∈ [0,1]. If τ(p) = 0 then p = 0since τ is faithful. If τ(p) = 1 then τ(1− p) = 0 so p = 1.

Finally, a standard argument shows that the projections in A aredense in the projections in A. Doing this requires the holomorphicfunctional calculus on the dense Banach ∗-subalgebra A. Here is theargument.

Let p ∈ A be a projection, and ε > 0. We assume ε > 12 .

Choose any q ∈ A with ‖p − q‖ < ε/2. Since Sp(p) = 0,1 weget Sp(q) ⊆ B(0;ε/2) ∪ B(1, ε/2). The function h which is constant0 on Re(z) < 1

2 and constant 1 on Re(z) > 12 is holomorphic

on a neighbourhood of Sp(q). So p′ = h(q) ∈ A and has Sp(p′) =h(Sp(q)) = 0,1, i.e. p′ is a projection in A. Moreover, ‖h − z‖∞ <ε/2 on Sp(q) (where z is the identity function on C as usual) so‖p′ − q‖ ≤ ε/2. Therefore ‖p′ − p‖ < ε.

Therefore, by continuity of τ, we get τ(p) = 0 or 1 for all projectionsp ∈ C∗r (F2), and the result follows.

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Bibliography

[Dav96] Kenneth R. Davidson. C∗-algebras by example, volume 6 ofFields Institute Monographs. American Mathematical Soci-ety, Providence, RI, 1996.

[Dix96] Jacques Dixmier. Les C∗-algèbres et leurs représentations.Les Grands Classiques Gauthier-Villars. [Gauthier-VillarsGreat Classics]. Éditions Jacques Gabay, Paris, 1996. Reprintof the second (1969) edition.

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