introduction a brief review of commutative banach algebras · introduction to c*-algebras 3...

INTRODUCTION TO C*-ALGEBRAS

TRON OMLAND AND NICOLAI STAMMEIER

Abstract. These are lectures notes for an introductory course on C∗-algebras heldin Autumn 2016 at the University of Oslo.

Introduction

Good references for the contents (and way beyond) are [Mur90,Ped79].

1. A brief review of commutative Banach algebras

We will mainly be interested in algebras over the field C, so this will implicitly beassumed, unless stated otherwise. Recall the following facts:

(1) For an element a in a unital algebra A, the spectrum is defined as Spa = {λ ∈C | λ− a is not invertible.}.

(2) A Banach algebra is an algebra A with a submultiplicative norm ‖·‖, i.e. ‖ab‖ ≤‖a‖‖b‖, for which A is complete.

(3) For every a in a unital Banach algebra A, the spectrum Spa is a nonempty,

compact set with spectral radius r(a) := limn→∞ ‖an‖1n ≤ ‖a‖.

Now suppose A is a commutative, unital Banach algebra. Then an ideal I in A ismaximal if and only if A/I ∼= C. This yields a correspondence between maximal idealsand nonzero homomorphisms ϕ : A→ C. For every such homomorphism ϕ and a ∈ A

ϕ(ϕ(a)− a) = 0⇒ ϕ(a) ∈ Sp a⇒ |ϕ(a)| ≤ ‖a‖

holds true. Thus every nonzero homomorphism ϕ : A→ C is continuous, and Spec A :={ϕ : A → C nonzero homomorphism} becomes a compact Hausdorff space under thetopology of pointwise convergence. This leads to χ : A → C(Spec A), a 7→ a witha(ϕ) := ϕ(a), the homomorphism known as the Gelfand transformation.

Remark 1.1. If X is a compact Hausdorff space and A = C(X), then Spec A is homeo-morphic to X with χ being the identity. Hence the range of a equals Sp a for all a ∈ Ahere, and consequently ‖a‖ = r(a).

2. C*-Algebras: Uniqueness of the C*-norm, unitization, and thefunctional calculus for normal elements

Recall that an involution ∗ : A → A, a 7→ a∗ on a normed algebra (A, ‖ · ‖) is anconjugate linear, anti-multiplicative symmetry, i.e. (λa+bc)∗ = λa∗+c∗b∗, and (a∗)∗ = afor λ ∈ C, a, b, c ∈ A. Involutions are norm-decreasing, and they are often isometric.

Date: December 13, 2016.1

2 T. OMLAND AND N. STAMMEIER

Definition 2.1. A C∗-algebra is a Banach algebra (A, ‖ · ‖) with an involution ∗ satis-fying the C∗-norm condition ‖aa∗‖ = ‖a‖2 for all a ∈ A.

Remark 2.2. As ‖aa∗‖ ≤ ‖a‖‖a∗‖ = ‖a‖2, it suffices to request ‖aa∗‖ ≥ ‖a‖2.

Examples 2.3. Here are some basic examples to get started:

(a) The algebra of bounded, linear operators on a Hilbert space H, denoted by L(H),forms a C∗-algebra with the operator norm and involution given by taking theadjoint.

(b) For every C∗-algebra A, every subalgebra B ⊂ A that is closed and invariantunder the involution yields a C∗-algebra.

(c) Given two Banach algebras A,B, their direct sum A ⊕ B becomes a Banachalgebra under component-wise operations and ‖(a, b)‖ := max(‖a‖, ‖b‖). If Aand B are C∗-algebras, then A⊕B becomes a C∗-algebra with (a, b)∗ := (a∗, b∗).

(d) Let A be a C∗-algebra, and M ⊂ A an arbitrary subset. Then there ex-ists a smallest C∗-subalgebra of A containing M which is given by C∗(M) :=⋂B⊂AC∗-algebra:

M⊂BB. Equivalently, C∗(M) can be described as the closure of all

polynomials in M ∪M∗.

If A is a Banach algebra, then it can be embedded into a unital Banach algebra Ain an elementary way: On the C-vector space A := A ⊕ C, define multiplication by(a, λ)(b, µ) := (ab+ µa+ λb, λµ). Then A is an algebra with unit 1 = (0, 1). It becomesa Banach algebra for the norm ‖(a, λ)‖ := ‖a‖+‖λ‖, and A embeds into A isometricallyvia a 7→ (a, 0).

For C∗-algebras, the story is a little more subtle, but has a happy ending.

Proposition 2.4. If a is a normal element in a C∗-algebra A, then r(a) = ‖a‖. There-fore, every unital ∗-algebra A admits at most one norm satisfying the C∗-norm condition.

Proof. Suppose first that a is self-adjoint. Then the C∗-norm condition gives ‖a2‖ =‖a‖2, and thus r(a) = limn→∞ ‖a2n‖2−n = ‖a‖. For a normal, we get

‖a‖2n = ‖aa∗‖2n· 12 = ‖(aa∗)2n‖

12 = ‖a2na∗2

n‖12 = ‖a2n‖,

and thus r(a) = ‖a‖. For the second claim, let ‖ · ‖1 and ‖ · ‖2 be C∗-norms on A. Bythe first part, we get

‖a‖21 = ‖aa∗‖1 = r(a) = ‖aa∗‖2 = ‖a‖2

2 for all a ∈ A.�

A remarkable aspect of Proposition 2.4 is that the spectral radius, originally definedalgebraically without reference to a norm, actually coincides with the unique C∗-normfor every C∗-algebra.

Theorem 2.5. Let A be a C∗-algebra with norm ‖·‖A and A the unital algebra describedabove. Then there exists a unique C∗-norm ‖ · ‖ on A.

Proof. By Proposition 2.4, we know that there can only be one such norm, so we needto prove the existence of this norm. Suppose first that A already has a unit e. As 1− eis a self-adjoint idempotent, i.e. a projection, with (1− e)a = 0 for all a ∈ A, we get a

INTRODUCTION TO C*-ALGEBRAS 3

decomposition A = A⊕C(1−e) ∼= A⊕C as a direct sum of involutive Banach algebras.In that case, we get a C∗-algebra-structure on A via Example 2.3 (c).

Now suppose that A does not have a unit. For x = (a, λ) ∈ A, define a linear operatorLx : A→ A by Lx(b) = xb = ab+ λb. Then Lx is bounded with ‖Lx‖ ≤ ‖a‖A + |λ|, andwe define ‖x‖ := ‖Lx‖ = sup{‖Lx(b)‖ | b ∈ A, ‖b‖ ≤ 1}. To see that this defines a normon A, note that

‖x‖ = ‖Lx‖ = 0⇒ Lx = 0⇒ xb = 0 for all b ∈ A.In this case, we must have λ 6= 0 (for otherwise x = 0). Assume λ 6= 0. Then b = −λ−1abfor all b ∈ A would show that A admits a unit on the left. Taking involution, we wouldalso get a unit on the right. In that case, these two units would necessarily coincide andA would be unital - a contradiction. Hence ‖ · ‖ defines a norm on A.

Next, we show that it satisfies the C∗-norm condition, that is ‖x∗x‖ = ‖x‖2: Letε > 0, and b ∈ A, ‖b‖A ≤ 1 with ‖xb‖ ≥ ‖xb‖ − ε. As ‖b∗‖A ≤ 1, we get

‖x∗x‖ ≥ ‖b∗‖A‖x∗xb‖ ≥ ‖b∗x∗xb‖ = ‖(xb)∗xb‖ = ‖(xb)∗xb‖A = ‖xb‖2 ≥ (‖x‖ − ε)2,

which suffices, see Remark 2.2. Finally, A is complete with respect to ‖ · ‖ as a sum ofa finite dimensional space and a Banach space inside the C∗-algebra of bounded linearoperators on A denoted by L(A). Another way to see this is to note that A is a closed∗-invariant subalgebra of the C∗-algebra L(A). �

Remark 2.6. Let A,B be algebras, and assume that B is unital. Then every homomor-phism ϕ : A→ B admits a unique extension ϕ : A→ B with ϕ(1A) = 1B.

Theorem 2.7. Suppose A is a Banach algebra with isometric involution and B is aC∗-algebra. Then every ∗-homomorphism ϕ : A → B is contractive, i.e ‖ϕ(a)‖ ≤ ‖a‖for all a ∈ A.

Proof. Consider the unique extension ϕ : A→ B. As SpBϕ(a) ⊂ SpAa, we get

‖a‖2 = ‖a∗‖‖a‖ ≥ ‖aa∗‖ ≥ r(aa∗) ≥ r(ϕ(aa∗)) = ‖ϕ(aa∗)‖ = ‖ϕ(a)‖2,

so that ϕ is contractive. �

We now revisit and extend some spectral considerations for elements in general C∗-algebras, starting with a straightforward observation.

Proposition 2.8. Suppose X is a compact Hausdorff space and x0 ∈ X. Then Y :=X \ {x0} is locally compact, C0(Y ) ∼= {f ∈ C(X) | f(x0) = 0}, and C(X) = C0(Y )⊕Cas vector spaces.

Remark 2.9. For a locally compact Hausdorff space Y , define a topology on X :=Y t{∞} (extending the original one on Y ) by declaring N ⊂ X to be a neighbourhoodof ∞ if there exists K ⊂ Y compact such that X \ K ⊂ N . With this topology, Xbecomes a compact Hausdorff space, and Y = X \ {∞} with the subspace topology. Xis called the Alexandroff compactification of Y . In this context, Proposition 2.8 statesthat C0(Y ) ∼= {f ∈ C(X) | f(∞) = 0}.

Example 2.10. The Alexandroff compactification of a nonempty half-open interval in Rgives a space that is homeomorphic to [0, 1], whereas any nonempty open interval yieldsthe torus.


Recall that a character on an algebra A over a field K is an algebra homomorphismϕ : A→ K.

Definition 2.11. For a commutative C∗-algebra A, the spectrum of A is Spec A :={ϕ nonzero character on A}.

Recall that the Gelfand transformation χ : A → C(Spec A) defines an isometric ∗-isomorphism for every unital commutative C∗-algebra A.

Note that Spec A sits inside Spec A, which corresponds to the space of all characterson A. Thus we have Spec A = Spec A \ {0}, and we conclude that Spec A is locallycompact and Hausdorff, see Proposition 2.8.

Proposition 2.12. Let A be a commutative C∗-algebra. Then the restriction of theGelfand transformation for A defines an isometric ∗-isomorphism

A→ C0(Spec A) ∼= {f ∈ C(Spec A) | f(0) = 0}.

Proof. Using the previous results, this is an easy exercise. �

We will see that Proposition 2.12 is particularly interesting in connection with Exam-ple 2.3 (d) applied to M = {a} or M = {a, 1} for a normal element a ∈ A. This yieldscommutative C∗-subalgebras C∗(a) ⊂ A and C∗(a, 1) ⊂ A.

We want to define a notion of a spectrum for elements inside a not necessarily unitalC∗-algebra. To this end, we make the following preliminary definition: For an elementa in a C∗-algebra A, we define Sp′ a := SpA a.

Remark 2.13. If A is a C∗-algebrawith unit e, then we know that A ∼= A ⊕ C(1 − e).Using the canonical embedding A → A, a 7→ (a, 0), we see that λ1 − a = (λe − a, λ) isinvertible in A if and only if λ 6= 0 and λ /∈ Sp a. This means that Sp′ a = Sp a ∪ {0}.

Theorem 2.14. Let a be a normal element in a C∗-algebra A.

(1) If A is unital, then the map Spec C∗(a, 1) → SpA a, ϕ 7→ ϕ(a) is a homeo-morphism and the Gelfand transformation defines an isomorphism C∗(a, 1) ∼=C(Sp a).

(2) The map Spec C∗(a)→ Sp′ a \ {0}, ϕ 7→ ϕ(a) is a homeomorphism and Gelfandtransformation defines an isomorphism C∗(a) ∼= {f ∈ C(Sp′ a) | f(0) = 0} ∼=C0(Sp′ a \ {0}).

Proof. (1) The map is injective: If ϕ and ψ are characters on C∗(a, 1) with ϕ(a) =ψ(a), then they agree on all polynomials in a and a∗. Hence we get ϕ = ψ byExample 2.3 (d). Surjectivity is easy as every λ ∈ Sp a defines a nonzero char-acter ϕ ∈ Spec C∗(a, 1) via ϕ(a) := λ. As Spec C∗(a, 1) carries the topology ofpointwise convergence, the map is also continuous, and hence a homeomorphismbecause domain and range are compact.

(2) According to (1), the map Spec C∗(a)→ Sp′ a is a homeomorphism with 0 7→ 0

and C∗(a) ∼= C(Sp′ a). Therefore, we have Spec C∗(a) = Spec C∗(a) \ {0} ∼=Sp′ a \ {0} and C∗(a) ∼= C0(Sp′ a \ {0}) by Proposition 2.12 and the precedingdiscussion.

�


Remark 2.15. If A is a unital C∗-algebra, then Sp′ a\{0} = Sp a\{0} so that Spec C∗(a)is homeomorphic to Sp a \ {0}.

Definition 2.16. For an element a in a C∗-algebra A, we define its spectrum Sp a as

Sp a =

{SpA a , if A is unital, and

SpA a , otherwise.

Remark 2.17 (The functional calculus for normal elements in C∗-algebra). Let a be anormal element in a C∗-algebra A. Then the following hold by invoking Theorem 2.14:

(i) Every f ∈ C(Sp a) defines an element f(a) ∈ C∗(a, 1). This gives rise to afunctional calculus with the properties that

λf +µg)(a) = λf(a) +µg(a), (fg)(a) = f(a)g(a), f(a) = f(a∗), and f ◦ g(a) = f(g(a)).

(ii) Every element f ∈ C(Sp a) with f(0) = 0 then defines an element f(a) ∈ C∗(a).(iii) Due to the isomorphisms in Theorem 2.14, every element in C(a, 1) and C∗(a)

admits such a description.

Proposition 2.18. Let A be a unital C∗-algebra and a ∈ A normal. Then a is invertiblein A if and only if 1 ∈ C∗(a).

Proof. Recall that a is invertible in A if and only if 0 does not belong to Sp a. If0 /∈ Sp a, then f(x) := x−1 defines an element of C(Sp a) for which 1 = f(a)a ∈ C∗(a)by the properties listed under Remark 2.17 (i). Conversely, let 1 ∈ C∗(a). Under thefunctional calculus, the unit corresponds to the characteristic function χSp a, and henceχSp a(0) = 0 according to Remark 2.17 (ii), which implies 0 /∈ Sp a. �

Proposition 2.19. Let a and b be elements in a unital C∗-algebra A. Then Sp ab∪{0} =Sp ba ∪ {0} holds.

Proof. Suppose λ /∈ Sp ab ∪ {0}. Without loss of generality, we can assume that λ = 1(by replacing a with λ−1x, if necessary). Set u := (1− ab)−1 ∈ A. Then we get

(1− ba)(1 + bua) = 1− ba+ bua− babua = 1− ba+ b(1− ab)ua = 1 and(1 + bua)(1− ba) = 1− ba+ bua− buaba = 1− ba+ bu(1− ab)a = 1.

Thus 1 = λ /∈ Sp ba. �

Remark 2.20. If A is a C∗-algebra with unit 1 and B is a C∗-subalgebra of A with 1 ∈ B,then SpA b = SpB b for every normal element b ∈ B. Thus it suffices to calculate thespectrum of b in C∗(b, 1). This is also true for not necessarily normal elements, but theproof involves spectral measures, see [Mur90, Section 2.5].

3. Positive elements in a C*-algebra

For the whole section, A is assumed to be a C∗-algebra.

Definition 3.1. An element a ∈ A is called positive (denoted a ≥ 0) if a is self-adjointand Sp a ⊂ [0,∞).

Remark 3.2. Due to the functional calculus from Remark 2.17, every positive elementa has a (uniquely determined) square root

√a = a

12 ∈ A.


Lemma 3.3. Let A be unital, a ∈ A self-adjoint, and λ ≥ ‖a‖. Then a is positive ifand only if ‖λ− a‖ ≤ λ.

Proof. This follows easily from the functional calculus in Remark 2.17. �

Proposition 3.4. The sum of two positive elements in A is positive.

Proof. Let a and b be positive elements. For λ := ‖a‖+ ‖b‖, we get

‖λ− (a+ b)‖ ≤ ‖‖a‖ − a‖+ ‖‖b‖ − b‖ ≤ λ

by applying Lemma 3.3 twice. Thus, Lemma 3.3 now implies that a+ b ≥ 0. �

Remark 3.5. Elements in A can be expressed as linear combinations of positive elements:

(i) If a ∈ A is self-adjoint, then there are a+, a− ≥ 0 with a+a− = 0 such thata = a+ − a−. To see this, use the functional calculus from Remark 2.17 (ii) forf+ := max(id, 0) and f− := min(id, 0).

(ii) If a ∈ A is an arbitrary element, we can decompose as a = a1 + ia2 witha1 = Re a = a+a∗

2and a2 = Im a = a−a∗

2i. As a1, a2 are self-adjoint, we end up

witha = a1,+ − a1,− + ia2,+ − ia2,−.

Proposition 3.6. For a ∈ A, the following conditions are equivalent:

(1) a ≥ 0.(2) ∃h ∈ A, h∗ = h with h2 = a.(3) ∃b ∈ A with b∗b = a.

Proof. For (1) implies (2), take h :=√a, see Remark 3.2. (2) implies (3) is obvious, so

it remains to prove that (3) implies (1): As a first step, we show that −c∗c ≥ 0 forcesc = 0 for any c ∈ A. Note that −c∗c ≥ 0 is equivalent to −cc∗ ≥ 0, see Proposition 2.19.If we now invoke Remark 3.5 (ii) to write c = c1 + ic2 with ci self-adjoint, then we getc∗c + cc∗ = 2c2

1 + 2c22. Thus −cc∗ ≥ 0 would imply that c∗c = 2c2

1 + 2c22 + (−cc∗) is

positive by Proposition 3.4. As we also have −c∗c ≥ 0, we get Sp c∗c = {0}, and hence

‖c‖2 = ‖c∗c‖ = r(c∗c) = 0.

Now let a = b∗b = u−v with u, v ≥ 0, uv = 0, see Remark 3.5 (i), and set c := bv. Then−c∗c = −vb∗bv = v3 ≥ 0 as v ≥ 0. According to the first step, this forces c = 0 so thatv3 = 0. Hence v = 0 and thus a ≥ 0. �

Remark 3.7. We shall denote the set of self-adjoint elements by Asa, and the set ofpositive elements by A+. Then Remark 3.5, Proposition 3.4, and Proposition 3.6 showthat A+ is a cone in Asa, i.e.

(i) Asa = A+ − A+,(ii) A+ + A+ ⊂ A+,

(iii) λA+ ⊂ A+ for all λ ∈ [0,∞), and(iv) A+ ∩ − A+ = {0}.

Thus a ≥ b :⇔ a− b ≥ 0 defines a partial order on Asa.

Theorem 3.8. Let A be a C∗-algebra. Then the following statement hold:

(1) A+ = {a∗a | a ∈ A}.


(2) For a, b ∈ Asa, c ∈ A, a ≤ b implies c∗ac ≤ c∗bc.(3) For a, b ∈ A, 0 ≤ a ≤ b implies ‖a‖ ≤ ‖b‖.(4) If A is unital, then 0 ≤ a ≤ b for invertible a and b implies 0 ≤ b−1 ≤ a−1.

Proof. We address the items in the same order:

(1) This has been proven in Proposition 3.6.(2) We have c∗bc − c∗ac = c∗(b − a)c = (

√b− ac)∗

√b− ac ≥ 0 using Remark 3.2

and Proposition 3.6.(3) By passing to the unitization of A if necessary, see Theorem 2.5, we can assume

that A is unital. Then ‖b‖ = inf{λ ∈ (0,∞) | λ ≥ b(≥ a)} ≥ ‖a‖.(4) The assumption 0 ≤ a ≤ b implies that a−1 and b−1 are positive, for instance

by using Definition 3.1 and functional calculus. As√a−1a√a−1 = 1, we can

use (2) for c =√a−1 to get 1 ≤

√a−1b√a−1. Then functional calculus shows

1 ≥ (√a−1b√a−1)−1 =

√ab−1√a. Now we apply (2) for c =

√a−1 again.

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Remark 3.9. Suppose a, b ∈ A satisfy 0 ≤ a ≤ b. Then we have 0 ≤ aα ≤ bα for everyα ∈ (0, 1]. If a and b commute, then 0 ≤ a2 ≤ b2 holds as well. However, this is not truein general. Indeed, take A = M2(C). Then

p =

(1 00 0

)and q =

1

2

(1 11 1

)are projections, i.e. p∗ = p = p2. Then p ≤ p+ q by Proposition 3.4, but

(p+ q)2 − p = pq + qp+ q = 12

[(1 10 0

)+

(1 01 0

)+

(1 11 1

)]= 1

2

(3 22 1

)has eigenvalues 2±

√5

2and thus Sp(p + q)2 − p2 is not contained in [0,∞). So we have

0 ≤ p ≤ p + q, but p2 6≤ (p + q)2. In fact, if 0 ≤ aα ≤ bα for all α ≥ 1, then a and bnecessarily commute.

4. Approximate units, ideals, and quotients

Definition 4.1. Let A be a C∗-algebra. An approximate unit in A is a monotoneincreasing net (uλ)λ∈Λ ⊂ A with 0 ≤ uλ ≤ 1 (in A) and uλa→ a← auλ for all a ∈ A.

Example 4.2. Of course, the unit in a unital C∗-algebra yields an approximate unit, buthere are two natural, non-tautological examples:

(a) For A = C0(R), an approximate unit is given by un := (n + 1− t)χ[−(n+1),−n) +χ[−n,n] + (n+ 1− t)χ(n,n+1].

(b) Let H be a Hilbert space, e.g. H = `2(N), with orthonormal basis (ei)i∈I , andA = K(H), the C∗-algebra of compact operators on H. Note that the family Λof finite subsets F of I form a directed set with respect to inclusion. For F ⊂ Ifinite, let PF denote the orthogonal projection onto the space 〈{ei | i ∈ F}〉 ⊂ H.Then 0 ≤ PF ≤ 1, and F ⊂ G corresponds to 0 ≤ pF ≤ pG. Thus (pF )F∈Λ is amonotone increasing net of positive contractions. Moreover, for every compactoperator k ∈ K(H), we have pFk, kpF → k as the finite subset F gets larger.


The next theorem shows that an approximate unit is not available in contexts thatwe arguably understand rather well, but a general tool that is always at our disposal.

Theorem 4.3. Let A be a C∗-algebra (or a norm-closed ideal in a C∗-algebra). Then(uh)h∈Λ given by

Λ = {h ∈ A+ | ‖h‖ < 1} and uh = h

constitutes an approximate unit for A.

Proof. We need to prove that

a) Λ is direct, andb) ha→ a← ah for all a ∈ A with the limit taken over h ∈ Λ.

For λ ∈ (0,∞) and a, b ∈ A with 0 ≤ a ≤ b, the elements λ−1 + a and λ−1 + b areinvertible (in A), and λ−1 + a ≤ λ−1 + b. Thus Theorem 3.8 (4) gives

(λ−1 + a)−1 ≥ (λ−1 + b)−1.

Therefore we get λ− (λ−1 + a)−1 ≤ λ− (λ−1 + b)−1. Since λ = (1 + λc)(λ−1 + c)−1 forc = a, b, this leads to

λa(λ−1 + a)−1 ≤ λb(λ−1 + b)−1

and then

(4.1) a(λ−1 + a)−1 ≤ b(λ−1 + b)−1.

Let us denote by h′ the positive contraction h(1 + h)−1 for h ∈ Λ. Then functionalcalculus shows that h = h′(1 + h′)−1, so the process is self-inverse. Therefore, (4.1)states that we have g ≤ h if and only if g′ ≤ h′ for g, h ∈ Λ. Now let g, h ∈ Λ. As‖c(1 + c)−1‖ < 1 for any c ∈ A+, we have (g′ + h′)(1 + g′ + h′)−1 ∈ Λ. By (4.1),g′, h′ ≤ g′ + h′ gives g, h ≤ (g′ + h′)(1 + g′ + h′)−1, which proves a).

For b), let a ∈ A and consider b := a∗a. For n ∈ N×, define hn := b( 1n

+ b)−1 ∈ Λ.

Then b− bhn ≤ 1n. So if h ∈ Λ satisfies hn ≤ h, we get

‖b− hb‖2 = ‖b(1− h)2b‖3.8 (2),(3)

≤ ‖b(1− h)b‖3.8 (2),(3)

≤ ‖b(1− hn)b‖≤ 1

n‖b‖ → 0.

Thus we have buh, uhb→ b for all b ∈ A+ (by taking adjoints). For a ∈ A, we now use

‖a− ah‖2 = ‖(1− h)b(1− h)‖ ≤ ‖1− h‖‖b(1− h)‖ → 0.

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Definition 4.4. A C∗-algebra A is called σ-unital if it admits a countable approximateunit.

Proposition 4.5. Every separable C∗-algebra A is σ-unital.

Proof. Let (an)n∈N ⊂ A be dense, and (νλ)λ∈Λ an approximate unit for A, which existsby Theorem 4.3. Choose (λn)n∈N such that

‖νλak − ak‖ ≤1

n≥ ‖akνλ − ak‖ for all k ≤ n and λ ≥ λn,


and set un := νλn . Then (un)n∈N is an approximate unit for A: For a ∈ A and ε > 0,there is k ∈ N such that ‖a− ak‖ < ε. Then we can also choose n ≥ k such that ε > 1

n.

This allows us to compute for m ≥ n:

‖uma− a‖ ≤ ‖uma− umak‖+ ‖umak − ak‖+ ‖ak − a‖ < 3ε.

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Unless stated otherwise, an ideal in a C∗-algebra means a two-sided ideal, and isdenoted by I / A.

Corollary 4.6. Let A be a C∗-algebra. If I / A is closed, then I∗ = I. Moreover, ifJ / I is closed, then J is a closed ideal in A.

Proof. Fix an approximate unit (uλ) for I, which exists by Theorem 4.3. For a ∈ I, wehave a∗uλ ∈ I as I is an ideal in A. Since ∗ is continuous on A, we get a∗ = lim(uλa)∗ =lim a∗uλ ∈ I.

For the second claim, it is clear that J is closed as the C*-norm on I is the restrictionof the one on A, see Proposition 2.4. If b ∈ J, a ∈ A then buλa ∈ J as uλa ∈ I, andbuλa→ ba as buλ → b. Thus ba ∈ J since J is a closed ideal in I. �

It follows from Corollary 4.6 that every closed ideal of a C∗-algebra is again a C∗-algebra. Next, we will show that C∗-algebras are also well-behaved under taking quo-tients with respect to closed ideals. To do so, we need to combine algebraic and analyticconsiderations:

If A is an algebra and I / A, then A/I is an algebra with the natural multiplication.If an algebra A has an involution ∗, and I∗ = I / A, then A/I has an involution givenby a∗ := a∗.

If A is a Banach space and I is a closed subspace, then A/I is a Banach space for‖a‖ = infi∈I ‖a + i‖. Now if A is a Banach algebra and I / A is a closed ideal, thenA/I is a Banach algebra with the quotient norm described above. Indeed, the norm issubmultiplicative because for a, b ∈ A and ε > 0, we can find i, j ∈ I with

‖a‖‖b‖ ∼ε ‖a+ i‖‖b+ j‖ ≥ ‖(a+ i)(b+ j)‖ ≥ ‖ab‖.Theorem 4.7. For every closed ideal I / A in a C∗-algebra A, the quotient A/I is aC∗-algebra.

Proof. By the preceding remarks we know that A/I is an involutive Banach algebra,so we need to show that the quotient norm satisfies the C*-norm condition. For thispurpose, we invoke an approximate unit (uλ) for I, whose existence is guaranteed byTheorem 4.3. We claim that ‖a‖ = lim ‖a − auλ‖ for every a ∈ A. Indeed, for ε > 0,choose b ∈ I with ‖a‖ + ε ≥ ‖a + b‖, and λ0 such that ‖b − buλ‖ < ε for all λ ≥ λ0.Then

‖a‖ ≤ ‖a− auλ‖ ≤ ‖(a+ b)(1− uλ)‖+ ‖b(1− uλ)‖ ≤ ‖a‖+ 2ε

for λ ≥ λ0, so ‖a = lim ‖a− auλ‖. This in turn gives

‖a‖2 = lim ‖a− auλ‖2 = lim ‖(a(1− uλ))∗a(1− uλ)‖≤ lim ‖(1− uλ)‖‖a∗a(1− uλ1)‖≤ lim ‖a∗a(1− uλ1)‖ = ‖a∗a‖

�


Remark 4.8. If ϕ : A → B is a ∗-homomorphism between C∗-algebras A and B, thenSp ϕ(a) ⊂ Sp a for every a ∈ A. Moreover, for every normal a ∈ A and f ∈ C0(Sp a),we have ϕ(f(a)) = f(ϕ(a)).

Corollary 4.9. Let A and B be C∗-algebras, and ϕ : A→ B a ∗-homomorphism.

(1) If ϕ is injective, then it is isometric.(2) The image ϕ(A) is a C∗-algebra, which is isomorphic to A/ kerϕ.

Proof. By Theorem 2.7, we know that ‖ϕ‖ ≤ 1. Now suppose there was a ∈ A suchthat ‖ϕ(a)‖ < ‖a‖. This is equivalent to ‖ϕ(a∗a)‖ < ‖a∗a‖. Then there would exist acontinuous function f on Sp a∗a ⊂ [0, ‖a∗a‖] with the properties

f = 0 on [0, ‖ϕ(a∗a)‖] while f(‖a∗a‖) 6= 0.

It would follow that f(a∗a) 6= 0, but ϕ(f(a∗a)) = f(ϕ(a∗a)) = 0, see Remark 4.8, acontradiction to the hypothesis that ϕ is injective. Thus ϕ is isometric.

The natural map ϕ given by a 7→ ϕ(a) is a well-defined ∗-homomorphism.

Aϕ //

��

B

A/ kerϕ

ϕ

BB

By Theorem 4.7 A/ kerϕ is a C∗-algebra as kerϕ / A is closed. The map ϕ is clearlyinjective, hence also isometric by (1). Therefore, ϕ(A) is complete, and thus a C∗-algebra as a closed involutive subalgebra of B. Surjectivity of ϕ then shows that ϕdefines an isomorphism A/ kerϕ ∼= ϕ(A). �

5. Positive linear functionals

Definition 5.1. Let A be a C∗-algebra. A linear functional ϕ : A→ C is called positive(denoted ϕ ≥ 0) if ϕ(a) ≥ 0 for all a ≥ 0.

Note that A+ = {a ∈ A | a ≥ 0} = {a∗a | a ∈ A}, see Proposition 3.6.Whenever there is no confusion, we will suppress linearity and speak of (positive)

functionals to improve the readability.

Examples 5.2. Here are three canonical examples of positive functionals on C∗-algebras:

(a) For A = C([0, 1]), the functionals ϕλ(f) :=∫

[0,1]f(x) dx and ϕ1/2(f) := f(1

2)

for f ∈ A are positive. More generally, if X is a compact Hausdorff space,then Riesz’s Theorem asserts that the positive functionals on C(X) are in 1 : 1-correspondence with Borel measures on X via µ 7→ ϕµ with ϕµ(f) :=

∫Xf dµ.

(b) For a Hilbert space H and A = L(H), pick ξ ∈ H and define ϕξ(a) := 〈aξ, ξ〉.Then ϕξ is positive because ϕξ(a

∗a) = ‖aξ‖2 ≥ 0.(c) If A = Mn(C) = L(Cn) for some n ∈ N, then the trace τ(a) :=

∑1≤k≤n akk is

a positive functional on A, where a = (ak`)1≤k,`≤n. This follows from (b) andτ =

∑1≤k≤n ϕξk , where ξ1, . . . , ξn are the canonical orthonormal basis for Cn.


Theorem 5.3. If ϕ is a positive functional on a C∗-algebra A, then ϕ is bounded.

Proof. According to Remark 3.5 (ii), every element in A can be displayed as a linearcombination of at most four positive elements. Thus it suffices to show that ϕ is boundedon A1

+ := {a ∈ A+ | ‖a‖ ≤ 1}. If ϕ was not bounded on A1+, there would exist a

sequence (an)n≥1 ⊂ A1+ with ϕ(an) ≥ 2n for all n. Define a :=

∑n≥1 2−nan ∈ A using

completeness of A. Then a ≥∑

1≤n≤N 2−nan ≥ 0 for every N ≥ 1. Hence ϕ(a) would

need to satisfy ϕ(a) ≥ ϕ(∑

1≤n≤N 2−nan) ≥ N for all N ≥ 1, which is impossible. �

Proposition 5.4. If ϕ is a positive functional on a C∗-algebra A, then

(1) ϕ(a∗) = ϕ(a), and(2) |ϕ(a)|2 ≤ ‖ϕ‖ϕ(a∗a)

hold for all a ∈ A.

Proof. Consider 〈·, ·〉ϕ : A × A → C given by 〈a, b〉ϕ := ϕ(b∗a). This defines a positivesemidefinite sesquilinear form on A as 〈a, a〉ϕ = ϕ(a∗a) ≥ 0. Hence, the polarization

identity for such maps gives 〈a, b〉ϕ = 〈b, a〉ϕ. Now take an approximate unit (uλ) in Aand let a ∈ A. Then

ϕ(a)← ϕ(uλa) = 〈a, uλ〉ϕ = 〈uλ, a〉ϕ = ϕ(a∗uλ)→ ϕ(a∗)

because ϕ is bounded, see Theorem 5.3.For (2), an application of the Cauchy-Schwartz inequality gives

|ϕ(a)|2 ← |〈a, uλ〉|2 ≤ |〈uλ, uλ〉|2|〈a, a〉|2 = ϕ(u2λ)ϕ(a∗a) ≤ ‖ϕ‖ϕ(a∗a).

�

The relevance of approximate units in the context of positive functionals goes farbeyond Proposition 5.4, as we shall now see.

Theorem 5.5. For every bounded functional ϕ on a C∗-algebra A, the following areequivalent:

(1) ϕ is positive.(2) ‖ϕ‖ = limϕ(uλ) for every approximate unit (uλ) in A.(3) ‖ϕ‖ = limϕ(uλ) for some approximate unit (uλ) in A.

Proof. Suppose (1) holds. We can assume that ‖ϕ‖ = 1. Let (uλ) be an approximateunit in A. Then (ϕ(uλ)) is a monotone increasing net in C with an upper bound 1, dueto Proposition 5.4. Thus ϕ(uλ) → α for a suitable α ∈ [0, 1]. We need to show thatα = 1. To this end, let a ∈ A with ‖a‖ ≤ 1. Then

|ϕ(a)|2 ← |ϕ(uλa)|2 ≤ ϕ(u2λ)ϕ(a∗a) ≤ ϕ(uλ)ϕ(a∗a) ≤ limϕ(uλ) = α

shows that 1 = ‖ϕ‖ ≤ α.The implication from (2) to (3) is trivial.Now suppose (3) holds, and for convenience assume ‖ϕ‖ = 1. First, we show that

ϕ(Asa) ⊂ R. So let a ∈ Asa and ϕ(a) = α+ iβ, α, β ∈ R. We can assume that β ≤ 0 bya 7→ −a if necessary. For every n ≥ 1, the C∗-norm condition gives

‖a− inuλ‖2 = ‖a2 + n2u2λ − in(auλ − uλa)‖ ≤ 1 + n2 + n‖auλ − uλa‖.


While ‖ϕ‖ = 1 gives

|ϕ(a− inuλ)|2 ≤ ‖(a− inuλ)∗(a− inuλ)‖‖a‖,‖uλ‖≤1

≤ 1 + n2 + n‖auλ − uλa‖ → 1 + n2,

(3) implies|ϕ(a− inuλ)|2 → |ϕ(a)− in|2 = α2 + (β − n)2.

This forces β = 0. Now let a ∈ A1+. Then −1 ≤ −a ≤ uλ − a ≤ uλ ≤ 1 so that

‖uλ − a‖ ≤ 1 and hence

1 ≥ ϕ(uλ − a)→ 1− ϕ(a) =⇒ ϕ(a) ≥ 0.

�

Note that the conditions in Theorem 5.5 are non-void by Theorem 4.3.

Corollary 5.6. If A is a unital C∗-algebra, then a functional ϕ on A is positive if andonly if ϕ(1) = ‖ϕ‖.

Proof. This follows immediately from (1)⇔(3) in Theorem 5.5 applied to the unit ofA. �

Corollary 5.7. If ϕ and ψ are positive functionals on a C∗-algebra A, then ‖ϕ+ψ‖ =‖ϕ‖+ ‖ψ‖.

Proof. This follows immediately from linearity and Theorem 5.5. �

Definition 5.8. A positive functional ϕ on a C∗-algebra A is called a state if ‖ϕ‖ = 1.

Theorem 5.9. For every normal element a in a C∗-algebra A, there exists a state ϕ onA with |ϕ(a)| = ‖a‖.

Proof. Note that |ψ(a)| ≤ ‖a‖ for every state ψ on A. As a is normal, there existsλ ∈ Sp a with |λ| = r(a) = ‖a‖, see Proposition 2.4. This gives rise to a characterϕλ : C(Sp a∪{0})→ C, f 7→ f(λ). As C(Sp a∪{0}) ∼= C∗(a, 1), see Theorem 2.14, thismap defines a state on the subalgebra C∗(a, 1) of A. Then we can use the Hahn-Banachextension theorem to get a linear extension ϕ : A → C with ‖ϕ‖ = ‖ϕλ‖ = 1. ByTheorem 5.5, ϕ ≥ 0 and hence ϕ is a state on A. The desired ϕ is now given by therestriction of ϕ to A ⊂ A. Here is a visualization:

A �� // A

ϕ

))C∗(a, 1)?�

OO

∼= // C(Sp a ∪ {0}) ϕλ

f 7→f(λ)// C

�

Proposition 5.10. Let ϕ be a positive functional on a C∗-algebra A.

(1) For a ∈ A, we have ϕ(a∗a) = 0 if and only if ϕ(ba) = 0 for all b ∈ A.(2) For a, b ∈ A, we have ϕ(b∗a∗ab) ≤ ‖a∗a‖ϕ(b∗b).

Proof. For (1), the Cauchy-Schwartz inequality yields |ϕ(ba)|2 ≤ ϕ(b∗b)ϕ(a∗a) for a, b ∈A. To get (2), we just note that b∗a∗ab ≤ ‖a∗a‖b∗b, see Theorem 3.8 (2). �


Theorem 5.11. Let ϕ be a positive functional on a C∗-subalgebra B of a C∗-algebra A.Then ϕ extends to a positive functional ϕ′ on A with ‖ϕ′‖ = ‖ϕ‖.Proof. First, suppose A = B. Define ϕ′(λ+ b) := λ‖ϕ‖+ ϕ(b) for all b ∈ B and λ ∈ C.Then ‖ϕ′‖ = ‖ϕ‖ < ∞ because for b ∈ B, λ ∈ C, and an approximate unit (uν) in B,we have

|ϕ′(λ+ b)| = |λ‖ϕ‖+ ϕ(b)|= lim |λϕ(uν) + ϕ(buν)|= lim |ϕ((λ+ b)uν)|≤ lim ‖ϕ‖‖λ+ b‖‖uν‖ ≤ ‖ϕ‖‖λ+ b‖.

Thus Corollary 5.6 shows that ϕ′ ≥ 0 as ϕ′(1) = ‖ϕ‖ = ‖ϕ′‖.For the general case, the strategy is displayed by the following diagram:

A� _

��

ϕ′

��

B? _oo� _

��

ϕ

��A

ψ

77B? _oo ψ // C

We use the first part of the proof to extend ϕ to ψ ≥ 0 with ‖ψ‖ = ‖ϕ‖. Applying the

Hahn-Banach theorem yields an extension of ψ to a bounded linear functional ψ on Asatisfying ψ(1) = ‖ψ‖ = ‖ψ‖ = ‖ϕ‖. Due to Corollary 5.6, we can thus conclude that

ψ is positive. Its restriction to A ⊂ A gives the desired positive functional ϕ′. �

6. The GNS-representation

The aim of this section is to prove that every C∗-algebra is isomorphic to a subal-gebra of the bounded linear operators on some Hilbert space. This celebrated result isattributed to Gelfand, Naimark, and Segal.

Definition 6.1. A representation of a C∗-algebra A is a pair (π,H) consisting of aHilbert space H and a ∗-homomorphism π : A → L(H). A representation (π,H) iscalled

(i) nondegenerate if π(A)H = H, and

(ii) cyclic if there is ξ ∈ H such that π(A)ξ = H. In this case, we write (π,H, ξ).

Remark 6.2. We note the following observations:

(i) If (πi, Hi)i∈I is a family of representations of a C∗-algebra A, then their directsum (⊕i∈Iπi,⊕i∈IHi) is a representation of A.

(ii) If (π,H) is a representation of A and K ⊂ H is a closed invariant subspace,that is, π(A)K ⊂ K, then its orthogonal complement K⊥ is also a closed in-variant subspace. Thus, π decomposes into a direct sum π = πK ⊕ πK⊥ of tworepresentations (πK , K) and (πK⊥ , K

⊥).

A natural question is whether representations can be described as directs sums ofrepresentations that form irreducible components. Note that every cyclic representationis necessarily irreducible in this informal sense. For this purpose, we need a notion ofequivalence that identifies representations that are morally the same.


Definition 6.3. Two representations (π,H) and (π′, H ′) of a C∗-algebra A are unitarilyequivalent, denoted π ∼ π′, if there exists a unitary U : H → H ′ such that π′(a) =Uπ(a)U∗ for all a ∈ A.

Proposition 6.4. Let A be a C∗-algebra. Then the following statements hold:

(1) Every representation of A is unitarily equivalent to a direct sum of nondegeneraterepresentations and a 0-representation.

(2) Every nondegenerate representation of A is unitarily equivalent to a direct sumof cyclic representations.

Proof. Let (π,H) be a representation. For part (1), note that K := π(A)H is a closedinvariant subspace of H. Hence π ∼ πK ⊕ πK⊥ , see Remark 6.2 (ii). Then πK isnondegenerate because

πK(A)K = πK(A)πK(A)HA2=A

= πK(A)H = K.

Moreover, πK⊥ = 0 as π(a)h ∈ K for every h ∈ K⊥.For (2), let (ξi)i∈I ⊂ H \ {0} be a maximal family with the property that the closed

invariant subspaces Ki := π(A)ξi are mutually orthogonal. Assume that⊕

i∈I Ki is a

proper subspace of H. Then there exists ξ ∈ (⊕

i∈I Ki)⊥ \ {0}, and K := π(A)ξ is

orthogonal to every Ki as

〈Ki, K〉 = 〈π(A)Ki, ξ〉 = 〈Ki, ξ〉 = 0.

We arrive at a contradiction and thus deduce that H =⊕

i∈I Ki, so that π ∼ ⊕i∈Iπiwith πi = πKi . �

Cyclic representations also have the advantage that unitary equivalence is witnessedby a uniquely determined unitary.

Proposition 6.5. Let (π1, H1, ξ1) and (π2, H2, ξ2) be cyclic representations of a C∗-algebra A. If the positive functionals ϕi on A given by ϕi(a) := 〈πi(a)ξi, ξi〉 coincide,then there exists a unique unitary U : H1 → H2 with Uξ1 = ξ2 and π2(a) = Uπ1(a)U∗.In particular, we have π1 ∼ π2.

Proof. The map V : π1(A)H1 → π2(A)H2 defined by π1(a)ξ1 7→ π2(a)ξ2 is well-definedand isometric because

〈π2(a)ξ2, π2(a)ξ2〉 = 〈π2(a∗a)ξ2, ξ2〉 = ϕ2(a∗a) = ϕ1(a∗a) = 〈π1(a)ξ1, π1(a)ξ1〉.As ξ1 and ξ2 are both cyclic vectors, domain and range of V are dense in H1 and H2,respectively, so V admits a unique extension U : H1 → H2. Then U is necessarily aunitary, and

Uπ1(a)U∗π2(b)ξ2 = Uπ1(ab)ξ1 = π2(a)π2(b)ξ2

shows that Uπ1(a)U∗ = π2(a) for all a ∈ A on H2. �

Note that ξ1 and ξ2 in Proposition 6.5 necessarily need to have the same norm. Byscaling, we can restrict our attention to normalized vectors, i.e. ‖ξi‖ = 1. In this case,the map ϕi is a state on A.

Theorem 6.6 (GNS-representation for a state). If ϕ is a state on a C∗-algebra A,then there exists a cyclic representation (πϕ, Hϕ, ξϕ) with 〈πϕ(a)ξϕ, ξϕ〉 = ϕ(a). Therepresentation (πϕ, Hϕ, ξϕ) is unique up to unitary equivalence.


Proof. Uniqueness up to unitary equivalence will a consequence of Proposition 6.5, oncewe prove the existence of such a cyclic representation. The basic idea is to use ϕ toproduce a Hilbert space out of A. To do so, we start by noting that 〈a, b〉ϕ := ϕ(b∗a)defines a positive semidefinite Hermitian form on A. Let Nϕ := {a ∈ A | 〈a, a〉ϕ = 0}denote the null space. For a, b ∈ Nϕ, we have

|ϕ((a+ b)∗(a+ b))| = |ϕ(a∗b) + ϕ(b∗a)| ≤ |ϕ(a∗b)|+ |ϕ(b∗a)| 5.10 (1)= 0.

Thus Nϕ is a (necessarily closed) subspace of A. Therefore, Kϕ := A/Nϕ is a Hermitianvector space for the scalar product induced by 〈·, ·〉, and its completion Hϕ with respectto the norm induced by this scalar product is a Hilbert space. Let γ : A → Hϕ denotethe map a 7→ a+Nϕ, and define π(a) : γ(A)→ γ(A) by γ(b) 7→ γ(ab) for all a ∈ A. As

‖γ(ab)‖2ϕ = ϕ(b∗a∗ab)

5.4

≤ ‖a∗a‖ϕ(b∗b) = ‖a‖2‖γ(b)‖2ϕ,

for all a, b ∈ A, each map π(a) is well-defined and ‖π(a)‖ ≤ ‖a‖. Since γ(A) is dense inHϕ, we can thus extend π(a) to a bounded linear operator πϕ(a) on Hϕ. This defines a∗-homomorphism πϕ because π is multiplicative on elements in γ(A), and, similarly,

〈π(a)γ(b), γ(c)〉ϕ = ϕ(c∗ab) = ϕ((a∗c)∗b) = 〈γ(b), π(a∗)γ(c)〉ϕimplies πϕ(a)∗ = πϕ(a∗).

The map f : Kϕ → C determined by a + Nϕ 7→ ϕ(a) is continuous because |f(a)| ≤ϕ(a∗a) = ‖a‖2

ϕ for all a ∈ A. Thus Riesz’s theorem asserts that the continuous extensionf ′ of f to Hϕ admits a unique ξϕ ∈ Hϕ such that

ϕ(a) = f(a+Nϕ) = f ′(γ(a)) = 〈γ(a), ξϕ〉ϕ for all a ∈ A.Then

〈γ(b), π(a)ξϕ〉ϕ = 〈γ(a∗b), ξϕ〉ϕ = ϕ(a∗b) = 〈γ(b), γ(a)〉ϕfor a, b ∈ A implies πϕ(a)ξϕ = γ(a). Therefore, ξϕ is a cyclic vector for πϕ, and〈πϕ(a)ξϕ, ξϕ〉ϕ = 〈γ(a), ξϕ〉ϕ = ϕ(a). Note that since ϕ is a state, ξϕ has norm 1. �

Remark 6.7. The unique cyclic ξϕ ∈ Hϕ can be obtained explicitly by observing that itis the norm limit of (γ(uλ)) for any approximate unit (uλ) in A.

Remark 6.8. Theorem 6.6 yields a one-to-one correspondence

{(π,H, ξ) cyclic representation of A}/ ∼←→ {ϕ state on A}.For a state ϕ on A, the triple (πϕ, Hϕ, ξϕ) is also called the GNS-representation for ϕ.

Combining the GNS-representation for states with Theorem 5.9, which guarantees asufficient supply of states on C∗-algebras, we arrive at:

Theorem 6.9 (The universal GNS-representation). Every C∗-algebra A admits a faith-ful representation (π,H). In particular, every C∗-algebra is (isometrically) isomorphicto a subalgebra of bounded linear operators on a Hilbert space.

Proof. Define (π,H) as the direct sum representation over (πϕ, Hϕ)ϕ state on A, see Re-mark 6.2 (i) and Theorem 6.6. For a ∈ A \ {0}, there exists a state ϕ on A withϕ(a∗a) = ‖a∗a‖, see Theorem 5.9. Thus we get

‖π(a)‖2 ≥ ‖πϕ(a)‖2 ≥ ‖〈πϕ(a∗a)ξϕ, ξϕ〉ϕ = ϕ(a∗a) = ‖a∗a‖ = ‖a‖2,


so that π is faithful. As every injective ∗-homomorphism is isometric, see Corollary 4.9,π is isometric. �

The pair (π,H) is commonly referred to as the universal GNS-representation of theC∗-algebra A.

Examples 6.10. Let us look at two explicit examples:

(a) Consider A = C([0, 1]) with the state given by integration with respect to theLebesgue measure λ, i.e. ϕ(f) :=

∫[0,1]

f(t) λ(dt). Then we have 〈f, g〉ϕ =∫[0,1]

g(t)f(t) λ(dt), which coincides with the usual scalar product on L2([0, 1], λ).

As C([0, 1]) is dense in L2([0, 1], λ) with respect to the L2-norm, we get Hϕ∼=

L2([0, 1]). Moreover, πϕ is given by multiplication, and ξϕ = 1.(b) For n ≥ 1, let A = Mn(C) and ϕ((aij)1≤i,j≤n) = a11. Then 〈a, b〉ϕ = ϕ(b∗a) =∑

1≤i≤n bi1ai1 is the euclidean scalar product on Cn (on the first column of the

matrices). Thus Hϕ∼= Cn, γ(a) = (a11, a21, . . . , an1)t, and πϕ is given by left

multiplication of a vector by a matrix.

Remark 6.11. If A is a C∗-algebra, then there exists a unique C∗-norm on Mn(A) :={(aij)1≤i,j≤n | aij ∈ A} for all n ≥ 1. Indeed, the universal GNS-representation fromTheorem 6.9 allows us to view A as a subalgebra of L(H). Therefore Mn(A) is asubalgebra of L(⊕1≤i≤nH), and the latter has a C∗-norm. Uniqueness is an outcome ofProposition 2.4.

7. Irreducible representations

Definition 7.1. A representation (π,H) of a C∗-algebra A with π 6= 0 is irreducible ifthe only π(A)-invariant closed subspaces of H are 0 and H.

We will simply talk about invariant closed subspaces whenever there is no ambiguityconcerning the representation. Moreover, we mostly suppress H by saying that π isirreducible.

Proposition 7.2. For a representation (π,H) of a C∗-algebra A with π 6= 0, the fol-lowing statements are equivalent:

(1) (π,H) is irreducible.(2) The relative commutant π(A)′ := {T ∈ L(H) | Tπ(a) = π(a)T for all a ∈ A}

equals C1.(3) Every nonzero vector in H is cyclic for π.

Proof. We prove (1) ⇔ (2) and (1) ⇔ (3). To show that (1) ⇒ (2), we observe thatπ(A)′ is a C∗-subalgebra of L(H). Let T ∈ π(A)′sa. If T /∈ R1, then Sp T contains atleast two distinct real numbers, say λ1 < λ2. This allows us to choose (nonnegative)bump functions f1, f2 ∈ C(Sp T ) with disjoint support (so that f1f2 = 0) and fi(λi) > 0.

Then Hi := fi(T )H defines two nontrivial, closed subspaces of H that are π(A)-invariantas

π(A)HiT∈π(A)′

= fi(T )π(A)H ⊂ Hi.

Moreover, f1f2 implies that H1 and H2 are orthogonal as

〈H1, H2〉 = 〈f1(T )H, f2(T )H〉 = 〈H, f1(T )f2(T )H〉 f1f2=0= 0.


This contradicts irreducibility so that we must have T ∈ R1, which amounts to π(A)′ =C1. For the converse direction, Let K be a closed invariant subspace of H, and denoteby PK ∈ L(H) the orthogonal projection onto K. As π(a)PKξ ∈ K for all ξ ∈ H byπ(A)-invariance of K, we have PKπ(a)PK = π(a)PK for all a ∈ A. So if a ∈ Asa, then

PKπ(a) = (π(a)PK)∗ = (PKπ(a)PK)∗ = PKπ(a)PK = π(a)PK .

Thus we have PK ∈ π(A)′ = C1, which forces K ∈ {0, H}.For (1) ⇒ (3), let ξ ∈ H \ {0} and consider K := π(A)ξ. This is a closed, invariant

subspace of H. If we had K = 0, then Cξ would be a nonzero, closed, invariant subspaceand (1) would imply H = Cξ. However, then we would have π = 0, which is ruled outby the standing assumptions. Thus K 6= 0 and (1) implies K = H so that ξ is cyclicfor π.

Conversely, every nonzero, closed, invariant subspace K of H contains some nonzerovector ξ, and (3) implies that H = π(A)ξ ⊂ π(A)K ⊂ K. �

Examples 7.3. Let us look at some examples:

(a) If A is a C∗-algebra and π : A → C is a character, then (π,C) is an irreduciblerepresentation.

(b) If A is a commutative C∗-algebra and (π,H) is irreducible, then Proposition 7.2tells us that π(A) ⊂ π(A)′ = C1, so that π(A) = C1. Hence every closed sub-space of H is invariant, and thus H ∼= C so that π : A→ C is in fact a character.So we see that for commutative C∗-algebras, irreducible representations are inone-to-one correspondence with characters.

(c) If (π,H) is a finite-dimensional, nondegenerate representation of a C∗-algebra A,then (π,H) is unitary equivalent to a finite direct sum of irreducible representa-tions (using an induction argument).

We have seen in Theorem 6.6 that every state on a C∗-algebra A gives rise to a cyclicrepresentation. Our next goal is to characterize the irreducible representations amongthese. This requires some preparation.

Lemma 7.4. Let H be a Hilbert space and (·, ·) a continuous sesquilinear form on H,i.e. |(ξ, η)| ≤ C‖ξ‖‖η‖ for all ξ, η ∈ H for some C < ∞. Then there exists T ∈ L(H)such that (ξ, η) = 〈Bξ, η〉 for all ξ, η ∈ H.

Proof. For every fixed η ∈ H, the map ξ 7→ (ξ, η) is a continuous functional on H. Thusthe Riesz representation theorem asserts that there is η ∈ H such that the functionalis given by ξ 7→ 〈ξ, η〉. The map η 7→ η is apparently linear, bounded. Then its adjointT ∈ L(H) satisfies

〈Tξ, η〉 = 〈ξ, η〉 = (ξ, η).

�

Note that an operator T ∈ L(H) on some Hilbert space H is positive if and only if〈Tξ, ξ〉 ≥ 0. The next lemma is crucial.

Lemma 7.5. Let (π,H) be a representation of a C∗-algebra A, ξ ∈ H and ϕ(a) :=〈π(a)ξ, ξ〉 for a ∈ A. Then the following statements hold:

1) For every T ∈ π(A)′, 0 ≤ T ≤ 1, the positive functional ϕT on A given byϕT (a) := 〈π(a)Tξ, Tξ〉 for a ∈ A satisfies ϕT ≤ ϕ, i.e. ϕ− ϕT ≥ 0 on A+.


2) If ξ is cyclic for π, then the mapping T 7→ ϕT from 1) is injective.3) If ψ is a positive functional on A such that ψ ≤ ϕ, then there exists T ∈

π(A)′, 0 ≤ T ≤ 1 with ϕT = ψ.

Proof. For every a ∈ A+, we have

ϕT (a) = 〈π(a)Tξ, Tξ〉= 〈Tπ(

√a)π(√a)Tξ, ξ〉 (T ∗ = T, a ≥ 0)

= 〈π(√a)T 2π(

√a)ξ, ξ〉 (T ∈ π(A)′)

= 〈π(a)ξ, ξ〉 (3.8 (2), (3) and ‖T‖ ≤ 1)= ϕ(a),

which proves 1). For 2), let S and T as in 1) with ϕS = ϕT . Then we get

〈S2π(a)ξ, π(b)ξ〉 = 〈π(b∗a)Sξ, Sξ〉 = ϕS(b∗a) = ϕT (b∗a) = 〈T 2π(a)ξ, π(b)ξ〉,as S, T ∈ π(A)′sa. If ξ is cyclic for π, this implies S2 = T 2, and hence S = T byuniqueness of the square root for positive operators, see Remark 3.2.

Part 3) will be more involved than 1) and 2), and uses Lemma 7.4. Given a positive

fucntional ψ on A with ψ ≤ ϕ, we define a sesquilinear form (·, ·)ψ on K := π(A)ξ by(π(a)ξ, π(b)ξ)ψ := ψ(b∗a) for a, b ∈ A. Note that (·, ·)ψ is well-defined and ‖(·, ·)ψ‖ ≤ 1because

|(π(a)ξ, π(b)ξ)ψ| = |ψ(b∗a)| ≤ ψ(b∗b)12ψ(a∗a)

12

ψ≤ϕ≤ ϕ(b∗b)

12ϕ(a∗a)

12 = ‖π(b)ξ‖ · ‖π(a)ξ‖.

Thus Lemma 7.4 yields T0 ∈ L(K) with

ψ(b∗a) = (π(a)ξ, π(b)ξ)ψ = 〈T0π(a)ξ, π(b)ξ〉 for all a, b ∈ A.Note that T0 is a positive contraction as

〈T0π(a)ξ, π(a)ξ〉 = ψ(a∗a) ≥ 0 for all a ∈ A and ‖T0‖ ≤ ‖(·, ·)ψ‖ ≤ 1.

We can therefore consider an operator T ∈ L(H) that agrees with√T0 on K, and

vanishes on K⊥. We claim that ψ = ϕT . Clearly, 0 ≤ T ≤ 1, so we need to show that

a) T ∈ π(A)′, andb) ψ = ϕT .

Note that K = π(A)ξ and its orthogonal complement K⊥ are π-invariant subspaces ofH. Thus Tπ(a) = 0 = π(a)T on K⊥. So, it suffices to show that π(a)T0π(b)ξ = T0π(ab)ξfor all a, b ∈ A. This follows from

〈T0π(ab)ξ, π(c)ξ〉 = ψ(c∗ab) = ψ((a∗c)∗b) = 〈T0π(b)ξ, π(a∗c)ξ〉 = 〈π(a)T0π(b)ξ, π(c)ξ〉,so T ∈ π(A)′. But then

ψ(b∗a) = 〈T0π(a)ξ, π(b)ξ〉 = 〈T 2π(a)ξ, π(b)ξ〉 a)= 〈π(a)Tξ, π(b)Tξ〉 = ϕT (b∗a).

If we use an approximate unit (uλ) in A in place of b, which exists by Theorem 4.3,we conclude that ψ = ϕT , because both functionals are positive, hence continuous, seeTheorem 5.3. �

Definition 7.6. A state ϕ on a C∗-algebra A is called pure if 0 ≤ ψ ≤ ϕ implies thatψ = λϕ for some λ ∈ [0, 1].


Example 7.7. If A is a commutative C∗-algebra,i.e. A = C0(X) for some locally compactHausdorff space X, then the pure states on A are the Dirac measures on X.

Theorem 7.8. A state ϕ on a C∗-algebra A is pure if and only if its GNS-representation(πϕ, Hϕ, ξϕ) is irreducible.

Proof. Suppose (πϕ, Hϕ, ξϕ) is irreducible, and let 0 ≤ ψ ≤ ϕ. Due to Lemma 7.5 3),there exists T ∈ πϕ(A)′, 0 ≤ T ≤ 1 such that ψ = ϕT . As πϕ is irreducible, Propo-sition 7.2 implies πϕ(A)′ = C1, so that T = λ1 with λ ∈ [0, 1]. In this case, we getψ = ϕT = λ2ϕ, so ϕ is pure.

Conversely, assume that ϕ is pure, and let K be a closed πϕ-invariant subspace of Hϕ.Then the orthogonal projection P onto K is a positive contraction in π(A)′. Therefore,Lemma 7.5 1) implies that the positive functional ϕP satisfies 0 ≤ ϕP ≤ ϕ. Since ϕ ispure, there is λ ∈ [0, 1] with ϕP = λϕ. We use this to obtain

〈Pπϕ(a)ξϕ, πϕ(b)ξϕ〉 = 〈πϕ(b∗a)Pξϕ, P ξϕ〉 (P is a projection in π(A)′.)= ϕP (b∗a)= λϕ(b∗a)= 〈πϕ(a)ξϕ, λπϕ(b)ξϕ〉.

for arbitrary a, b ∈ A. Since ξϕ is cyclic for πϕ, this shows that Pη = λη for all η ∈ Hϕ.Since P is the orthogonal projection onto K, we conclude that P ∈ {0, 1} correspondingto K = 0 and K = Hϕ, respectively. Thus πϕ is irreducible. �

We now show that every positive functional on a C∗-algebra with norm at most 1is a convex combination of pure states. This essentially reduces the study of positivefunctionals to the study of pure states. In combination with Theorem 7.8, this providesus with an ample supply of irreducible representations for arbitrary C∗-algebras.

Definition 7.9. Let V be a vector space.

(i) A subset C of V is convex if for all x, y ∈ C and t ∈ [0, 1], the element tx+(1−t)ybelongs to C.

(ii) A subset M of a convex set C is extremal if it has the following property: Ifx, y ∈ C and t ∈ (0, 1) satisfy tx+ (1− t)y ∈M , then x, y ∈M .

(iii) A point x in a convex set C is called an extreme point if {x} is extremal.

Proposition 7.10. Let A be a C∗-algebra and B := {ϕ : A → C | ϕ ≥ 0, ‖ϕ‖ ≤ 1}.Then the following statements hold:

(1) B is convex and compact with respect to the weak topology.(2) The extreme points of B are the pure states and 0.

Proof. For (1), note that the set of continuous functionals ϕ : A → C with ‖ϕ‖ ≤ 1 iscompact with respect to the weak topology. B forms a closed subset of this former setas ϕ ≥ 0 is a closed condition. Convexity is clear.

For (2), we first note that 0 is an extreme point as 0 = ϕ+ ψ for positive functionalsϕ, ψ implies ϕ = 0 = ψ. Next, let ϕ be a pure state, and suppose we have ϕ =tψ1 + (1− t)ψ2 for some ψi ≥ 0 with ‖ψi‖ ≤ 1 and t ∈ (0, 1). Since ϕ is pure, there areλ1, λ2 ∈ [0, 1] such that ψi = λiϕ for i = 1, 2. Thus

ϕ = tψ1 + (1− t)ψ2 = (tλ1 + (1− t)λ2)ϕ


implies tλ1 + (1 − t)λ2 = 1, and therefore λi = 1, i.e. ψi = ϕ. Conversely, let ϕ ∈ Bbe an extreme point with ϕ 6= 0. Then we necessarily have ‖ϕ‖ = 1. Suppose there is0 ≤ ψ ≤ ϕ. Then ϕ = ψ + ψ′ for ψ′ := ϕ− ψ ∈ B. Consider

ϕ = ‖ψ‖ ψ

‖ψ‖+ ‖ψ′‖ ψ′

‖ψ′‖.

As ϕ is an extreme point and ‖ψ‖ + ‖ψ′‖ = ‖ψ + ψ′‖ = ‖ϕ‖ = 1 by Corollary 5.7, weget ψ = ‖ψ‖ϕ, so ϕ is pure. �

Let us recall the celebrated theorem of Krein and Milman:

Theorem 7.11 (Krein–Milman). Let V be a locally convex Hausdorff space. Then everynonempty, compact, convex set C ⊂ V has an extreme point.

Definition 7.12. Let M be a subset of a vector space V . The convex hull of M , denotedby conv(M), is the smallest convex subset of V containing M .

Remark 7.13. The convex hull can be described explicitly by conv(M) = {∑

1≤k≤n ckxk |n ∈ N, xk ∈M, and

∑1≤k≤n ck = 1}.

Corollary 7.14. Let V be a locally convex R-vector space and C ⊂ V convex andcompact. Then C = conv(Ext C), where Ext C is the set of extreme points in C.

Proof. Let us set A := conv(Ext C) for convenience. As C is convex and closed, A is asubset of C because Ext C ⊂ C. So suppose that there exists c ∈ C \A. We can assumethat 0 ∈ C so that 0 ∈ Ext C ⊂ A. Then there is a convex neighbourhood of 0 U ′ suchthat c+U ′∩A = ∅. Then U := 1

2U ′ is a convex neighbourhood of 0 with c+U∩A+U = ∅.

Consider the Minkowski functional pB(v) := inf{t > 0 | v ∈ t(B)} for v ∈ V fora nonempty convex B ⊂ C containing 0. For the convex sets A + U and U , we havepA+U ≤ pU . As pB is always sublinear and pU is continuous because U is a neighbourhoodof 0, the map pA+U is continuous and thus bounded on C since the latter is compact.By the Hahn-Banach theorem for the locally convex case, there exists a continuous mapf : V → R with f(c) = pA+U(c) > 1 and f(b) ≤ pA+U(v) for all v ∈ V . Since pA+U

is bounded on C, so is f . Therefore, the set K := {c′ ∈ C | f(c′) = supb∈C f(b)} isnon-empty and compact. It is also extremal in C, so Theorem 7.11 implies that Kcontains an extreme point c′ ∈ Ext C ⊂ A. However, f(c′) = supb∈C f(b) ≥ f(c) > 1contradicts f |A ≤ 1. Thus we conclude that A = C. �

Corollary 7.14 has a great application to the study of representations of C∗-algebras:

Corollary 7.15. Let A be a C∗-algebra and a ∈ A\{0}. Then there exists an irreduciblerepresentation (π,H) such that ‖π(a)‖ = ‖a‖.Proof. With the notation of Proposition 7.10, consider K := {ϕ ∈ B | ϕ(a∗a) = ‖a‖2}.Then K is non-empty and extremal, and compact as a closed subset of B, see Propo-sition 7.10 (1). By Theorem 7.11, K contains an element ϕ ∈ Ext B. Accordingto Proposition 7.10 (2), ϕ is a pure state. Thus its associated GNS-representation(πϕ, Hϕ, ξϕ) is irreducible by Theorem 7.8. In addition,

‖πϕ(a)ξϕ‖2 = ϕ(a∗a) = ‖a‖2

implies ‖π(a)‖ = ‖a‖ as ‖ξϕ‖ = 1. �


8. von Neumann algebras

In the previous Section, we saw that every C∗-algebra A can be thought of as aconcrete norm-closed, ∗-invariant subalgebra of L(H) for some Hilbert space, see The-orem 6.9. But there are more interesting topologies on L(H) that are weaker thanthe norm topology, namely the strong operator topology (SOT) and the weak operatortopology (WOT).

Norm: It suffices to consider sequences instead of nets. A sequence (an) converges to aif ‖an−a‖ → 0. Multiplication, addition, and similar operations are continuous,the involution is isometric.

SOT: A net (aλ) converges to a if aλξ → aξ for all ξ ∈ H. This can be described bythe collection of seminorms pξ, ξ ∈ H, where pξ(a) := ‖aξ‖. The multiplicationL(H)× L(H)→ L(H), (a, b) 7→ ab is not continuous. A counterexample can begiven using the unilateral shift on `2(N). However, for fixed b ∈ L(H), left andright multiplication by b, i.e. a 7→ ba and a 7→ ab, are continuous. Moreover,multiplication is continuous when restricted to bounded domains in L(H). Thatis to say, if aλ → a and bµ → b strongly with ‖aλ‖, ‖bµ‖ ≤ C for some C < ∞,then aλbµ → ab as

‖aλbµξ − abξ‖ ≤ C‖bµξ − bξ‖+ ‖aλbξ − abξ‖ → 0.

Also, the involution is not strongly continuous: Let H = `2(N) with ONB (ξn)n∈Nand define operators an ∈ L(H) by anη := 〈η, ξn〉ξ1. Then (an) tends to 0strongly, but a∗nξ1 = ξn for all n ∈ N.

WOT: A net (aλ) converges to a if 〈aλξ, η〉 → 〈aξ, η〉 for all ξ, η ∈ H. As for SOT,left and right multiplication by a fixed element in L(H) is weakly continuous,but multiplication is not jointly continuous. Indeed, let V denote the unilateralshift on `2(N), i.e. V ξn = ξn+1 for the canonical ONB (ξn). Then V n → 0 andV ∗n → 0 weakly, but V ∗nV n = 1 for all n ∈ N. However, the involution is weaklycontinuous because 〈a∗λξ, η〉 = 〈aλη, ξ〉.

Remark 8.1. For every Hilbert space H, the unit ball L(H)1 = {a ∈ L(H) | ‖a‖ ≤ 1}is weakly compact. To see this, note that C1 is the closed unit disc, which is compact.Due to Tychonov’s theorem, CI

1 is compact for every index set I. Now the embedding

a 7→ 〈aξ, η〉 of L(H)1 into C{(ξ,η)∈H×H|‖ξ‖,‖η‖≤1}1 is a homeomorphism onto its range,

which is closed, hence compact.If H is separable, then bounded nets can be replaced by sequences when concerned

with questions around SOT and WOT.

Remark 8.2. Note that convergence with respect to the norm topology implies conver-gence in SOT, which in turn implies convergence in WOT. Hence, every weakly closed(open) set is strongly closed (open), and every strongly closed (open) set is norm-closed(open).

Even though it already appeared in Proposition 7.2, let us now define a commutantfor subsets in L(H) properly.

Definition 8.3. For a Hilbert space H and X ⊂ L(H), the commutant of X in L(H)is X ′ := {T ∈ L(H) | Tx = xT for all x ∈ X}.


Remark 8.4. The commutant X ′ is always weakly closed as multiplication by a fixedelement is weakly continuous. In addition, we have (X∗)′ = (X ′)∗. So if X∗ = X, thenX ′ is a C∗-algebra, see Remark 8.2.

Observe that X ⊂ X ′′ and X ′ ⊂ X ′′′. In principle, if X∗ = X, then these could bethe starting bits of two increasing chains of C∗-algebras (ignoring the fact that X maynot be a C∗-algebra). But the next lemma shows that these sequences stabilize afterthe first step already, if X is suitably chosen.

Lemma 8.5. If A ⊂ L(H) is a ∗-subalgebra with 1 ∈ A, then A is strongly dense in A′′.

Proof. Let b ∈ A′′ and ε > 0. We need to show that for all n ∈ N and ξ1, . . . , ξn ∈ H,there exists a ∈ A such that pξk(b− a) = ‖(b− a)ξk‖ < ε for 1 ≤ k ≤ n.

For n = 1, let ξ ∈ H and K := Aξ. Then K is a closed A-invariant subspace of Hthat contains ξ (as 1 ∈ A). Thus the orthogonal projection PK belongs to A′, and hencecommutes with b so that bξ ∈ K. Therefore, there exists a ∈ A with ‖bξ − aξ‖ < ε.

Now let n > 1 and ξ1, . . . , ξn ∈ H. Consider the map ϕ : L(H)→Mn(L(H)) given byT 7→ diag(T, . . . , T ). This is an isometric homomorphism of ∗-algebras, so ϕ(A)∗ = ϕ(A)is a ∗-algebra with 1Mn(L(H)) ∈ ϕ(A). As

ϕ(A)′ = {(Tij)1≤i,j≤n | Tij ∈ A′ for all i, j},we know that b ∈ A′′ implies ϕ(b) ∈ ϕ(A)′′. Now, applying the case n = 1 to ϕ(A),ξ := (ξ1, . . . , ξn)t and ϕ(b) yields an element a ∈ A with

ε > ‖ϕ(b− a)ξ‖ = max1≤k≤n ‖(b− a)ξk‖.�

Theorem 8.6. If A ⊂ L(H) is a ∗-subalgebra with 1 ∈ A, then the following areequivalent:

(1) A = A′′.(2) A is weakly closed.(3) A is strongly closed.

Proof. Clearly, we have (1) ⇒ (2) ⇒ (3), and by Lemma 8.5, (3) forces A = ASOT

=A′′. �

Definition 8.7. A von Neumann algebra is a weakly closed ∗-subalgebra A ⊂ L(H)that contains the identity.

We will now collect a number of relatively straightforward, but nonetheless usefulobservations. For the rest of the section, let M be a von Neumann algebra.

Remark 8.8. For every normal element a ∈M , we can perform functional calculus withBorel functions on Sp a inside M . The reason is that the algebra of bounded Borelfunctions Bb(Sp a) is the smallest algebra of functions on Sp a that contains C(Sp a)and is closed under pointwise limits of sequences.

Remark 8.9. If K is a closed subspace of a Hilbert space H and a ∈ L(H), thenaK ⊂ K if and only if aPK = PkaPK for the orthogonal projection PK onto K. ThusPK commutes with a if and only if K is invariant under a and a∗. This allows us todraw the following consequence:


For a ∈ M , let K := aH and L := ker a. Then PK and PL belong to M as well.Indeed, K and L are invariant under b and b∗ for every b ∈ M ′ = (M ′)∗ becauseab = ba. Thus PK , PL ∈M ′′ which equals M , by Theorem 8.6.

Remark 8.10. Every element in M can be approximated in norm by linear combinationsof projections. Since M is a ∗-algebra, it suffices to show this for a ∈ Msa. Via theGelfand transformation, a corresponds to the identity function on Sp a. This functioncan be approximated by linear combinations of characteristic functions coming fromfiner and finer finite partitions of Sp a into Borel sets. By Remark 8.8, each suchcharacteristic function give rise to projections with the desired property.

Remark 8.11. Recall from a previous exercise that in every unital C∗-algebra, everyelement is a linear combination of four unitaries. Thus M = {a ∈ L(H) | uau∗ =a for all u ∈ U(M ′)}, that is, M is the set of all elements in L(H) that are fixed underconjugation by all unitaries from the von Neumann algebra M ′.

Remark 8.12. For a ∈M , let a = v|a| be its polar decomposition in L(H), and recall thatv is a partial isometry and |a| =

√a∗a. Recall that it is unique if we request v| ker a = 0,

which we shall do. Then |a| belongs to M by functional calculus, but Remark 8.11 alsoimplies a = uau∗ = uvu∗|a| for every u ∈ U(M ′). Notice that uvu∗ is a partial isometrythat vanishes on ker a. So by uniqueness of the polar decomposition, we get uvu∗ = v,and hence v ∈M by Remark 8.11.

Definition 8.13. For X ⊂ L(H), the von Neumann algebra generated by X W ∗(X) isthe smallest von Neumann subalgebra of L(H) that contains X.

Proposition 8.14. For every X ⊂ L(H), the von Neumann algebra W ∗(X) is given by(X ∪X∗)′′.

Proof. By Remark 8.4, (X∪X∗)′′ is a von Neumann algebra, and contains X. So supposeM is a von Neumann algebra with X ⊂M . Then

X ∪X∗ ⊂M =⇒ (X ∪X∗)′ ⊃M ′ =⇒ (X ∪X∗)′′ ⊂M ′′ = M,

so that W ∗(X) = (X ∪X∗)′′. �

Definition 8.15. For a von Neumann algebra M , the set Z(M) := M ∩M ′ is calledthe center of M .

Note that Z(M ′) = Z(M).

Proposition 8.16. For every von Neumann algebra M , its center Z(M) is a von Neu-mann algebra, and Z(M)′ = W ∗(M ∪M ′).

Proof. It is clear that Z(M) is a von Neumann algebra as unitality, WOT, and ∗-subalgebra structures are well-behaved with respect to finite intersections. The secondclaim follows from

(M ∪M ′)′ = M ′ ∩M ′′ = Z(M)

as this leads toW ∗(M ∪M ′) = (M ∪M ′)′′ = Z(M)′.

�


Example 8.17. For M = L(H), we get M ′ = C1, so that Z(M) = Z(M ′) = C1.

There are more von Neumann algebras with trivial center, which motivates the nextdefinition.

Definition 8.18. A von Neumann algebra M is called a factor if Z(M) = C1.

Proposition 8.19. For a von Neumann algebra M , the following are equivalent:

(1) M is a factor.(2) M ′ is a factor.(3) W ∗(M ∪M ′) = L(H).

Proof. As Z(M) = Z(M ′), (1) and (2) are equivalent. Using Proposition 8.16, we haveZ(M)′ = W ∗(M ∪M ′) and Z(M) = Z(M)′′, so the equivalence between (1) and (3)follows from Example 8.17. �

9. Tensor products

In this section we discuss the notion of a tensor product of Hilbert spaces, which leadsto an elementary class of factors.

Let V and W be vector spaces over some field K. We want to define an object V ⊗Wsuch that all bases A for V and B for W yield a basis A × B for V ⊗W . If we tookthis as a definition for V ⊗W , we would have to show that it does not depend on theparticular choice of bases. Therefore, we make the following alternative definition.

Definition 9.1. For vector spaces V and W over some field K, the (algebraic) tensorproduct V ⊗ W is defined to be the quotient space of the K-vector space with basisgiven by the symbols ξ ⊗ η, ξ ∈ V, η ∈ W , by the linear subspace generated by

(a) (ξ1 + ξ2)⊗ η − (ξ1 ⊗ η + ξ2 ⊗ η),(b) ξ ⊗ (η1 + η2)− (ξ ⊗ η1 + ξ ⊗ η2),(c) λ(ξ ⊗ η)− (λξ)⊗ η, and λ(ξ ⊗ η)− ξ ⊗ λη

for all λ ∈ K, ξ, ξ1, ξ2 ∈ V, η, η1, η2 ∈ W .

Remark 9.2. The algebraic tensor product V ⊗ W is characterized by the followinguniversal property: The assignment (ξ, η) 7→ ξ ⊗ η defines a bilinear map α : V ×W →V ⊗W , and for every bilinear map β : V ×W → Z there exists a unique linear mapβ′ : V ⊗W → Z such that β = β′ ◦ α. This results in a bijection between bilinear mapson V ×W and linear maps on V ⊗W .

Moreover, one can show that the Cartesian product A × B of bases A for V and Bfor W defines a basis for V ⊗W , as desired.

Example 9.3. The tensor product Km⊗Kn is isomorphic to Kmn, where m,n ∈ N. Nowlet m = n and consider the bilinear map 〈·, ·〉 : Kn×Kn → K, (ξ, η) 7→

∑ni=1 ξiηi. Then

ξ ⊗ η 7→ |ξ〉〈η|, where |ξ〉〈η|(ζ) := ξ〈η, ζ〉 for ξ, η, ζ ∈ Kn defines an element in Mn(K).As the image of {ξ ⊗ η | ξ, η ∈ B} for a basis B of Kn is linearly independent, the mapdefines an isomorphism Kn ⊗ Kn ∼= Mn(K). Under this isomorphism, the universalproperty of Kn ⊗Kn from Remark 9.2 applied to 〈·, ·〉 : Kn ×Kn → K yields the traceon Mn(K).


Remark 9.4. Let H be a C-vector space. We define a new C-vector space H that isgiven by H as a set and additive group, but with scalar multiplication λ · ξ := λξfor λ ∈ C, ξ ∈ H. The identity defines a linear map between the spaces H and Hunder which antilinear maps ϕ : H → X, i.e. additive maps with ϕ(λξ) = λϕ(ξ) forλ ∈ C, ξ ∈ H, correspond to linear maps H → X. This is actually a universal propertycharacterizing H. In particular, we get a bijection between antilinear maps H → X andlinear maps H → X (for every X).

We also observe that H1 ⊗H2∼= H1 ⊗H2 due to their universal properties.

Proposition 9.5. Suppose H1 and H2 are Hilbert spaces (over C). Then

〈·, ·〉 : H1 ⊗H2 ×H1 ⊗H2 → C given by (ξ1 ⊗ η1, ξ2 ⊗ η2) 7→ 〈ξ1, η1〉H1〈ξ2, η2〉H2

defines an inner product on H1 ⊗H2.

Proof. Recall that the inner product on Hi is a (particular) sesquilinear form 〈·, ·〉 : Hi×Hi → C. Using Remark 9.4, we can pass to a bilinear map Hi ×H i → C. Thus

H1 ×H2 ×H1 ×H2 → C given by (ξ1, ξ2, η1, η2) 7→ 〈ξ1, η1〉H1〈ξ2, η2〉H2

is a 4-linear map corresponding to a bilinear mapH1⊗H2×H1⊗H2 → C, see Remark 9.2.By the last line of Remark 9.4, we thus see that H1⊗H2×H1 ⊗H2 → C, (ξ1⊗ η1, ξ2⊗η2) 7→ 〈ξ1, η1〉H1〈ξ2, η2〉H2 is a bilinear map. Hence we end up with a sesquilinear map

〈·, ·〉 : H1 ⊗H2 ×H1 ⊗H2 → C given by (ξ1 ⊗ η1, ξ2 ⊗ η2) 7→ 〈ξ1, η1〉H1〈ξ2, η2〉H2 .

It remains to show that 〈·, ·〉 is positive definite. To this end, let ξ ∈ H1 ⊗ H2. Then

ξ =∑n

i=1 ξ(i)1 ⊗ ξ

(i)2 for some n ∈ N and suitable ξ

(i)j ∈ Hj, j = 1, 2. Without loss of

generality, we can assume that (ξ(i)2 )1≤i≤n is an orthonormal family in H2, as the finite

linear combination for ξ can be adjusted accordingly by choosing an orthonormal basis

for the subspace of H2 spanned by {ξ(1)2 , . . . , ξ

(n)2 }. Then

〈ξ, ξ〉 =∑

1≤i,j≤n〈ξ(i)

1 ⊗ ξ(i)2 , ξ

(j)1 ⊗ ξ

(j)2 〉 =

∑1≤i≤n

‖ξ(i)1 ‖2

H1

shows that 〈·, ·〉 is indeed positive definite. �

Definition 9.6. For two complex Hilbert spaces H1 and H2, the Hilbert space tensorproduct H1⊗H2 is defined as the completion of H1⊗H2 with respect to the norm inducedby the inner product 〈·, ·〉 from Proposition 9.5.

Proposition 9.7. If (ξi)i∈I and (ηj)j∈J are orthonormal bases for (complex) Hilbertspaces H1 and H2, respectively, then (ξi⊗ηj)(i,j)∈I×J is an orthonormal basis for H1⊗H2.

Proof. It is clear that (ξi ⊗ ηj)(i,j)∈I×J is an orthonormal family in H1⊗H2, so we needto show that its span is dense. First, let α ∈ H1, so α =

∑i∈I ciξi for suitable ci ∈ C.

Then∑

i∈F ciξi ⊗ ηj → α ⊗ ηj, where F ranges over finite subsets of I (directed withrespect to inclusion). Hence α ⊗ ηj ∈ span{ξi ⊗ ηj | (i, j) ∈ I × J} for every j ∈ J .Displaying β ∈ H2 similarly as β =

∑j∈J djηj with suitable dj ∈ C, we get

α⊗ β = limF⊂J finite

∑j∈F

α⊗ djηj ∈ span{ξi ⊗ ηj | (i, j) ∈ I × J}


for all α ∈ H1, β ∈ H2, so H1 ⊗H2 ⊂ span{ξi ⊗ ηj | (i, j) ∈ I × J}. As H1⊗H2 is thecompletion of the algebraic tensor product H1⊗H2, the subspace span{ξi⊗ ηj | (i, j) ∈I × J} is all of H1⊗H2. �

The following is immediate from Proposition 9.7.

Corollary 9.8. For all index sets I and J , the Hilbert space `2(I)⊗`2(J) is isomorphicto `2(I × J).

Proposition 9.7 also implies that L2(Rm)⊗L2(Rn) is isomorphic to L2(Rm+n) for allm,n ∈ N: The map

L2(Rm)× L2(Rn) → L2(Rm+n)(f, g) 7→ [(x, y) 7→ f(x)g(y)]

is bilinear and thus induces a linear map ϕ : L2(Rm) ⊗ L2(Rn) → L2(Rm+n). The mapϕ is isometric because the inner product on L2(Rm) ⊗ L2(Rn) from Proposition 9.5 isequal to the pullback of the inner product on L2(Rm+n) under ϕ. However, proving thatthe unique extension of ϕ to an isometric linear map ϕ′ : L2(Rm)⊗L2(Rn)→ L2(Rm+n)is a surjective map is more difficult.

Proposition 9.9. Let Hi be a Hilbert space and xi ∈ L(Hi) for i = 1, 2. Then thereexists x1 ⊗ x2 ∈ L(H1⊗H2) uniquely determined by the property x1 ⊗ x2(ξ1 ⊗ ξ2) =x1ξ1 ⊗ x2ξ2 for all ξi ∈ Hi. The operator x1 ⊗ x2 has the following properties:

(i) ‖x1 ⊗ x2‖ = ‖x1‖ · ‖x2‖.(ii) (x1 + y1)⊗ x2 = x1 ⊗ x2 + y1 ⊗ x2 and x1 ⊗ (x2 + y2) = x1 ⊗ x2 + x1 ⊗ y2.

(iii) (x1 ⊗ x2)(y1 ⊗ y2) = x1y1 ⊗ x2y2.(iv) (x1 ⊗ x2)∗ = x∗1 ⊗ x∗2.

Proof. Note that if T ∈ L(H1⊗H2) has the property that T (ξ1 ⊗ ξ2) = x1ξ1 ⊗ x2ξ2 forall ξi ∈ Hi, then T agrees with x1⊗x2 on H1⊗H2, and hence T = x1⊗x2 by denseness.

Suppose first that x2 = 1. A generic element of H1⊗H2 is of the form∑n

i=1 ξ(i)1 ⊗ ξ

(i)2

with n ∈ N. As in the proof of Proposition 9.7, we can arrange for (ξ(i)2 )1≤i≤n to be

orthonormal. Then we get

‖(x1 ⊗ 1)(n∑i=1

ξ(i)1 ⊗ ξ

(i)2 )‖2 =

n∑i=1

‖x1ξ(i)1 ‖2 ≤ ‖x1‖2

n∑i=1

‖ξ(i)1 ‖2 = ‖x1‖2‖

n∑i=1

ξ(i)1 ⊗ ξ

(i)2 ‖.

Therefore, the linear map x1 ⊗ 1: H1 ⊗ H2 admits a unique extension to an elementof L(H1⊗H2) (again denoted by x1 ⊗ 1) with ‖x1 ⊗ 1‖ ≤ ‖x1‖. The case of 1 ⊗ x2 isanalogous, and we define x1 ⊗ x2 := (x1 ⊗ 1)(1⊗ x2). Clearly, ‖x1 ⊗ x2‖ ≤ ‖x1‖ · ‖x2‖by construction, but ‖(x1⊗ x2)(ξ1⊗ ξ2)‖ = ‖x1ξ1‖ · ‖x2ξ2‖ yields the reverse inequality.This shows (i). Using denseness of H1 ⊗H2 ⊂ H1⊗H2 and linearity for the vectors, itsuffices to check (ii)–(iv) on elementary tensors ξ1 ⊗ ξ2. But for these the claims aretrivial by the defining property of x1 ⊗ x2. �

Remark 9.10. If (ξi)i∈I and (ηj)j∈J are orthonormal bases for Hilbert spaces H1 and H2,respectively, then we know from Proposition 9.7 that (ξi⊗ ηj)(i,j)∈I×J is an orthonormal

basis for H1⊗H2. This leads to

H1⊗H2∼=⊕i∈I

Cξi⊗H2∼=⊕j∈J

H1⊗Cηj,


where we note that there is no need to take the completion for each of summandsappearing in the direct sum decompositions. But we use the symbol ⊗ to stress that weare considering Hilbert spaces. For j ∈ J , the map

uj : H1 → H1⊗Cηjξ 7→ ξ ⊗ ηj

is an isometric isomorphism with u∗juj = 1H1 and uju∗j = PH1⊗Cηj . We observe that for

i, j ∈ J , we have uiu∗j(ξ ⊗ ηk) = δj,k ξ ⊗ ηi for all k ∈ J . Thus eij := uiu

∗j = 1⊗ e′ij for

e′ij ∈ L(H2) determined by e′ij(ηk) = δj,k ηi for k ∈ J . In particular, (ejj)j∈J is a familyof mutually orthogonal projections (whose sum converges strongly to the identity onH1⊗H2).

Given x ∈ L(H1⊗H2), we use the maps uj, j ∈ J from Remark 9.10 to define xij :=u∗ixuj for i, j ∈ J .

Proposition 9.11. Let H1 and H2 be Hilbert spaces. For x ∈ L(H1⊗H2), the followingconditions are equivalent:

(1) x = x1 ⊗ 1 for some x1 ∈ L(H1).(2) x ∈ (1⊗ L(H2))′, where 1⊗ L(H2) = {1⊗ x2 | x2 ∈ L(H2)}.(3) xij = δi,j x1 for some x1 ∈ L(H1).

Proof. Clearly, (1) implies (2). Suppose (2) holds and let i, j ∈ J, i 6= j. Then we get

xij = u∗ixuj = u∗i eiixejjuj(2)= u∗ixeiiejjuj

9.10= 0.

Moreover, we have

xii = u∗ixui = u∗iuiu∗j uju

∗i︸︷︷︸

=eji=1⊗e′ji

xui(2)= u∗iuiu

∗jxuju

∗iui = xjj,

which together with the first computation gives (3). Given (3), we use the fact that∑j∈F ejj → 1H1⊗H2

strongly as F ranges over finite subsets of J to compute

x(ξ ⊗ ηi) = xui(ξ) =∑j∈J

uj u∗jxui︸︷︷︸=xji

(ξ)(3)= uix1(ξ) = x1(ξ)⊗ ηi.

According to Proposition 9.9, this shows x = x1 ⊗ 1, so we arrive at (1). �

Proposition 9.12. Let H1 and H2 be Hilbert spaces. Then the following statementshold:

(1) (L(H1)⊗ 1)′ equals 1⊗L(H2) and (1⊗L(H2))′ = L(H1)⊗ 1. In particular, bothalgebras are factors.

(2) L(H1)→ L(H1)⊗ 1, x 7→ x⊗ 1 is a ∗-isomorphism.(3) L(H2)→ 1⊗ L(H2), y 7→ 1⊗ y is a ∗-isomorphism.

Proof. Part (1) is essentially due to Proposition 9.11 as Z(L(H1)⊗1) = L(H1)⊗1∩1⊗L(H2) = C1H1⊗H2

. For parts (2) and (3), it is clear that the map defines a surjective∗-homomorphism by Proposition 9.9. But injectivity follows from x ⊗ 1 = 0 ⇒ x = 0,and similarly for 1⊗ y. �


10. Murray-von Neumann (sub-)equivalence for factors

In this section, we introduce the notion of Murray-von Neumann (sub-)equivalence ofprojections and prove that projections in factors are comparable. We assume from nowon that H is a separable Hilbert space and that M ⊂ L(H) is a von Neumann algebra.

Definition 10.1. Two projections e, f ∈ M are Murray-von Neumann equivalent, de-noted e ∼ f , if there exists v ∈M such that e = v∗v and vv∗ = f .

It is sometimes convenient to write e ∼v f if v ∈ M satisfies the condition fromDefintion 10.2. We note that any such v is necessarily a partial isometry with sourceprojection e and range projection f . The relation ∼ is an equivalence relation. Indeed,relfexivity and symmetry are apparent. For transitivity, let e ∼v f ∼w g. Then wvsatisfies

(wv)∗wv = v∗fv = v∗vv∗v = e and wv(wv)∗ = wfw∗ = ww∗ww∗ = g.

Next, we introduce a weakening of Murray-von Neumann equivalence.

Definition 10.2. A projection e ∈M is Murray-von Neumann subequivalent to anotherprojection f ∈ M , denoted e ≺ f , if there exists a partial isometry v ∈ M such thate = v∗v and vv∗ ≤ f .

Note that ≺ is also transitive. The proof is analogous to the one for Murray-vonNeumann equivalence.

Remark 10.3. Note that for projections e, f ∈M the following conditions are equivalent:

(a) eH ⊂ fH.(b) fe = e.(c) e ≤ f , i.e. f − e ≥ 0.

Example 10.4. Let M = L(H) and e = PK , f = PL for closed subspaces K,L ⊂ H.Then e ≺ f holds if and only if there is a closed subspace K ⊂ L such that K ∼= K,which is equivalent to dimK ≤ dimL.

Proposition 10.5. If (ei)i∈I , (fi)i∈I ⊂ M are two families of mutually orthogonal pro-jections. If ei ∼ fi for all i ∈ I, then

∑i∈I ei ∼

∑i∈I fi. Similarly, if ei ≺ fi for all

i ∈ I, then∑

i∈I ei ≺∑

i∈I fi.

Proof. By assumption, we have ei ∼vi fi for suitable partial isometries vi ∈ M forall i ∈ I. Then w ∈ L(H) defined by w|eiH := vi and w|(⊕i∈I eiH)⊥ = 0 satisfies∑

i∈I ei = w∗w and ww∗ =∑

i∈I fi. We need to show that w belongs to M , for instanceby showing that it commutes with M ′. Every element in M ′ is a linear combination ofunitaries, so it suffices to prove this for all unitaries u ∈ M ′, see Remark 8.11. Thisfollows from

uw|eiH = uvi|eiHvi∈M= viu|eiH = wu|eiH and uw|(⊕i∈I eiH)⊥ = 0 = wu|(⊕i∈I eiH)⊥

where we used that eiH and (⊕

i∈I eiH)⊥ are M ′-invariant subspaces.If the families only satisfy ei ≺ fi for all i ∈ I, say via partial isometries vi, then

the previous argument applied to (ei)i∈I and (viv∗i )i∈I yields a partial isometry w with∑

i∈I ei ∼w∑

i∈I viv∗i ≤

∑i∈I fi, which proves the claim. �


Proposition 10.6. Two projections e, f ∈ M satisfy e ∼ f if and only if e ≺ f ande � f .

Proof. It is clear that e ∼ f implies e ≺ f and e � f . The proof for the conversedirection is analogous to the proof of the Schroder-Bernstein theorem from set theory:Let e, f ∈ M be projections with e ≺ f and e � f . By replacing f by the rangeprojection from f ≺ e we can assume f ≤ e. Due to e ≺ f , there is a partial isometryv ∈ M such that e = v∗v and vv∗ ≤ f . Then e2n := vnev∗n and e2n+1 := vnfv∗n forn ≥ 0 defines a family (en)n∈N of projections in M with

e2n ≥ e2n+1, e2n ∼ e2n+2, and e2n+1 ∼ e2n+3 for all n ≥ 0.

Observe that e2n − e2n+1 ∼ e2n+2 − e2n+3 via vn := v(e2n − e2n+1) as

(v(e2n − e2n+1))∗v(e2n − e2n+1) = (e2n − e2n+1)e(e2n − e2n+1) = e2n − e2n+1

and v(e2n − e2n+1)(v(e2n − e2n+1))∗ = e2n+2 − e2n+3. Noting that e =∑

n≥0(en − en+1)and f =

∑n≥1(en− en+1) (both as strong limits), we get a partial isometry w ∈M with

e ∼w f by setting w|(1−e)H = 0, w|(e2n−e2n+1)H = vn, and w|(e2n+1−e2n+2)H = id. �

For an element a in a von Neumann algebra M , we denote by supp a the orthogonalprojection onto (ker a)⊥. According to Remark 8.9, supp a also belongs to M .

Proposition 10.7. For every element a in a von Neumann algebra M , supp a∗ ∼ supp aholds.

Proof. If a = v|a| is the polar decomposition for a, then v ∈ M by Remark 8.12, andwe have supp a = v∗v and supp a∗ = vv∗. �

Lemma 10.8. Let M ⊂ L(H) be a factor. If e, f ∈M \ {0} are projections, then thereexist projections e1, f1 ∈M \ {0} such that e ≥ e1 ∼ f1 ≤ f .

Proof. Since e 6= 0, K := MeH is a non-trivial, closed M -invariant subspace of H.Thus we have 0 6= PK ∈ M ′, and also PK ∈ M ′′. Due to M being a factor, we getPK ∈ M ′ ∩ M ′′ = Z(M) = C1. Since PK is a nonzero projection, this amounts toPK = 1. Therefore, there exists z ∈ M such that fze 6= 0 because f 6= 0. Nowe ≥ supp fze ∼ supp ez∗f ≤ f by Proposition 10.7, which proves the claim as supp fzeand supp(fze)∗ = supp ez∗f vanish if and only if fze = 0. �

Theorem 10.9. If e and f are projections in a factor M , then e ≺ f or e � f holds.

Proof. Consider families of pairs of non-zero projections (ei, fi)i∈I ⊂ M ×M such that(ei)i∈I and (fi)i∈I are families of mutually orthogonal projections with e ≥ ei ∼ fi forall i ∈ I. Such families are partially ordered by inclusion, and every totally orderedcollection of such families admits a maximal element. By Lemma 10.8, there exist suchfamilies, at least with |I| = 1. So by Zorn’s lemma, there exists a maximal family(ei, fi)i∈I . Then e′ :=

∑i∈I ei and f ′ :=

∑i∈I fi are projections in M (as strong limits

of the respective partial sums) satisfying e ≥ e′ ∼ f ′ ≤ f by Proposition 10.5. Assumee−e′ 6= 0 6= f−f ′. Then Lemma 10.8 implies that there are projections e′′, f ′′ ∈M \{0}with e − e′ ≥ e′′ ∼ f ′′ ≤ f − f ′. As this would contradict maximality of (ei, fi)i∈I , wemust have e = e′ or f = f ′, and hence e ≺ f or e � f , respectively. �


Theorem 10.10. Let M be a factor. If e and f are projections in M with e 6= 0, thenthere exists a family (ei)i∈I ⊂ M of mutually orthogonal projections and a projectionr ∈M with e ∼ ei ⊥ r for all i ∈ I such that

f =∑i∈Iei + r and e 6∼ r ≺ e.

If I is infinite, then one can arrange for r = 0.

Proof. Similar to the proof of Theorem 10.9, Zorn’s lemma yields a maximal family ofprojections (ei)i∈I ⊂M with respect to the conditions that eiej = 0 for i 6= j and ei ∼ efor all i, j ∈ I. Then r := f −

∑i∈I ei is a projection in M . According to Theorem 10.9,

we have r ≺ e or r � e. But then maximality of (ei)i∈I forces e 6∼ r ≺ e.Now let I be infinite, and pick k ∈ I. Then∑

i∈Iei =

∑i∈I\{k}

ei + ekek∼e�r�

∑i∈I\{k}

ei + r10.5 for I\{k}&I∼ f,

yields f ≺∑i∈Iei, and hence f ∼

∑i∈Iei by Proposition 10.6. If v ∈M is a partial isometry

with f = v∗v and vv∗ =∑i∈Iei, then ei ≤ f leads to

f = v∗vv∗v =∑i∈Iv∗eiv with v∗eiv ∼ e, v∗eivv

∗ejv = δi,jv∗eiv for all i, j ∈ I.

�

11. Discrete factors

In this short section we characterize discrete factors M ⊂ L(H), which are factorsthat are isomorphic to L(H) (as C∗-algebras for some Hilbert space H.

Definition 11.1. Let M be a factor. A nonzero projection e ∈ M is minimal, if theonly projections in M below e are 0 and e.

Proposition 11.2. If e and f are minimal projections in a factor M , then e ∼ f .

Proof. By definition, e 6= 0 6= f . Due to Theorem 10.9 we have e ≺ f or e � f , buteither option gives e ∼ f by minimality of f or e, respectively. �

Proposition 11.3. A nonzero projection e in a factor M is minimal if and only ifeMe = Ce.

Proof. If eMe = Ce, then 0 ≤ f ≤ e for a projection f ∈ M implies f = efe ∈Ce, which amounts to f ∈ {0, e} since it is a projection. For the converse, we notethat eMe ⊂ L(eH) is a von Neumann algebra. Therefore, eMe contains all spectralprojections of its self-adjoint elements, see Remark 8.10. Since e is minimal in M , thisforces eMe = Ce. �

Theorem 11.4. For a factor M ⊂ L(H), the following conditions are equivalent:

(1) M is discrete, i.e. M ∼= L(H1) for some Hilbert space H1.(2) Every nonzero projection in M dominates a minimal projection in M .(3) M has a minimal projection.(4) M ′ is discrete, i.e. M ′ ∼= L(H2) for some Hilbert space H2.


(5) Every nonzero projection in M ′ dominates a minimal projection in M ′.(6) M ′ has a minimal projection.

Proof. It is apparent that (1)⇒ (2)⇒ (3) and (4)⇒ (5)⇒ (6). We will show that (3)implies (4), which will then also prove (6)⇒ (1).

For (3) implies (4), we let e be a minimal projection in M . We claim that

(a) M ′ →M ′e, a 7→ ea(= ae = eae) is an injective ∗-homomorphism, and(b) M ′e ⊂ L(eH) is a von Neumann algebra with (M ′e)′ ∩ eL(H)e = eMe = Ce.

For (a), the closed subspace M ′aH for every a ∈ M ′ \ {0} is M - and M ′-invariant. AsM is a factor, a 6= 0 gives PK = 1, that is, K = H. Thus e 6= 0 yields eK 6= 0, and thusea 6= 0.

For (b), we note that M ′e ⊂ L(eH) is a von Neumann algebra as the property of beingSOT-closed is preserved under compression by any projection in L(H). The second partof the equality is due to Proposition 11.3. For the first one, let a ∈ M ′, b ∈ M . Then(ebe)ae = eabe = ae(ebe) shows eMe ⊂ (M ′e)′ ∩ eL(H)e. The reverse inclusion isobtained from

(M ′e)′ ∩ eL(H)e = {b ∈ L(H) | b = be = eb, aeb = bae for all a ∈M ′}ae=ea

= {b ∈ L(H) | b = be = eb, ab = ba for all a ∈M ′}= {b ∈ L(H) | b = be = eb, b ∈M ′′}

M ′′=M= {b ∈M | b = be = eb}= eMe.

We thus conclude by (b) that for H2 := eH, we have M ′ ∼= M ′e = (M ′e)′′ ∩ L(H2) =(Ce)′ ∩ L(H2) = L(H2), which completes (4).

Now suppose the equivalent conditions (1)–(6) hold, and let e be a minimal projectionin M . Then 1 =

∑i∈I ei for some index set I with projections ei ∈ M such that

e ∼ ei ⊥ ej for all i, j ∈ I, i 6= j by Theorem 10.10. Note that the remainder r vanishesas e is minimal, and that I 6= ∅. Choose k ∈ I to define H2 := ekH and H1 := `2(I).

�

Remark 11.5. The conditions (1)–(6) from Theorem 11.4 are also equivalent to:

(7) There are Hilbert spaces H1 and H2 such that there is an isomorphism H ∼=H1⊗H2 that takes M to L(H1)⊗ 1 and M ′ to 1⊗ L(H2).

However, we do not provide a proof.

12. Group C∗-algebras

Recall that a topological group is a group together with a topology such that the mapG × G → G given by (g, h) 7→ gh, and the map G → G given by g 7→ g−1, are bothcontinuous (the former with respect to the product topology).

In operator algebras one generally considers locally compact Hausdorff groups.However, in this course, we will only consider discrete groups (although some ex-

ceptions could briefly be mentioned). Recall that G is discrete if all of its subsets areopen. In particular, this means that every map from G into any space is automaticallycontinuous.


Remark 12.1. If G is a locally compact Hausdorff group and the cardinality of G iscountable, then G must be discrete.

Examples 12.2. Some locally compact Hausdorff groups:

(i) Finite groups are discrete (but not so interesting from the viewpoint of operatoralgebras, as they only give rise to finite-dimensional C∗-algebras).

(ii) Zn is discrete and abelian (Z is probably the most important group).(iii) Fn is discrete, but not abelian for n ≥ 2 (also a very important group for operator

algebras).(iv) T, the circle group under multiplication and topology inherited from the complex

plane; this group is non-discrete, compact, and abelian.(v) R, the real numbers under addition and the usual topology; this group is non-

discrete, non-compact, and abelian,(vi) For any Hilbert space H, define the unitary group U(H) as the set {U ∈ B(H) :

UU∗ = U∗U = I}; then U(H) is a group under multiplication (and we usuallygive it the strong operator topology).

(vii) the above are just the most standard examples – even for countable discretegroups, the class of examples is vast!

Let G be a discrete group. For any g ∈ G, define a function δg : G→ C by

δg(h) =

{1 if g = h,

0 else.

Define the group ring C[G] as the vector space of functions G→ C with finite support.Since G is discrete, this coincides with Cc(G), the vector space of functions G→ C withcompact support. For a function ϕ ∈ C[G], define

‖ϕ‖1 =∑g∈G

|ϕ(g)|,

and denote the completion of C[G] with respect to ‖·‖1 by `1(G).

Proposition 12.3. `1(G) is a Banach ∗-algebra when equipped with convolution, invo-lution, and norm given by

(ϕ ∗ ψ)(s) =∑g∈G

ϕ(g)ψ(g−1s),

ϕ∗(s) = ϕ(s−1),

‖ϕ‖1 =∑g∈G

|ϕ(g)|.

Proof. Exercise; for example, you need to show that

‖ϕ ∗ ψ‖1 ≤ ‖ϕ‖1 · ‖ψ‖1.

�

Definition 12.4. A unitary representation of G on a Hilbert space H is a homomor-phism π : G→ U(H).


This induces a (nondegenerate) representation π : `1(G)→ B(H) given by

π(ϕ) =∑g∈G

ϕ(g)π(g).

Check that ‖π(ϕ)‖ ≤ ‖ϕ‖1. In fact, every nondegenerate representation of `1(G) arisesthis way.

Definition 12.5. For any G, we define the Hilbert space `2(G) as the space of allfunctions ξ : G→ C such that ∑

g∈G

|ξ(g)|2 <∞,

together with inner product

〈ξ, ζ〉 =∑g∈G

ξ(g)ζ(g).

We also notet that {δh}g∈G forms an orthonormal basis for `2(G).

Definition 12.6. For any G, we define the unitary representation λ : G→ U(`2(G)) by

(λ(g)ξ)(s) = (δg ∗ ξ)(s) = ξ(g−1s).

Then λ is called the left regular representation of G.Its extension to `1(G) satisfies λ(ϕ)ξ = ϕ ∗ ξ.

Definition 12.7. The reduced group C∗-algebra of G is denoted C∗r (G) (or sometimesC∗λ(G)), and is the norm-closure of λ(`1(G)) inside B(`2(G)).

It coincides with the C∗-subalgebra of B(`2(G)) generated by the set {λ(g)}g∈G.

Definition 12.8. The group von Neumann algebra of G is denoted W ∗(G) or sometimesL(G), and is the closure in the weak operator topology of λ(`1(G)) inside B(`2(G)). Thatis, W ∗(G) = C∗r (G)′′.

It coincides with the von Neumann subalgebra of B(`2(G)) generated by the set{λ(g)}g∈G.

For every ϕ ∈ `1(G), define

‖ϕ‖max = sup{‖π(ϕ)‖ : π is a representation of `1(G)}.Since ‖π(ϕ)‖ ≤ ‖ϕ‖1 for all ϕ ∈ `1(G) and all representations π, the subset of R on theright-hand side is bounded for every ϕ ∈ `1(G). Moreover, it is nonempty (contains theleft regular representation). Hence, the supremum always exists. Moreover, ‖·‖max is aC∗-norm on `1(G).

Definition 12.9. The full group C∗-algebra of G is denoted C∗(G) and it is the com-pletion of `1(G) with respect to the ‖·‖max-norm.

Next, for a discrete group G and an element g ∈ G, we define its conjugacy classCg = {hgh−1 : h ∈ G}. We say that G is ICC if Cg is infinite for every g ∈ G, g 6= e.Note that if G is abelian, then Cg = {g} for all g ∈ G, so ICC groups are in some sensefar away from being abelian. For example, the free group on two generators F2 is ICC.

Theorem 12.10. The group von Neumann algebra W ∗(G) is a factor if and only if Gis ICC.


Proof. Coming soon. �

Remark 12.11. If G is nontrivial, countable, and ICC, then W ∗(G) is a II1 factor.Indeed, W ∗(G) has a tracial state, namely the vector state associated with δe, so

W ∗(G) is finite. Moreover, since G is countably infinite, W ∗(G) is infinite-dimensionaland separable, so W ∗(G) is a II1 factor.

Example 12.12. W ∗(Z2 ∗ Z2) is not a factor, but W ∗(Z2 ∗ Z3) is a factor (exercise).

Remark 12.13. For every G, the left regular representation λ extends to a surjective∗-homomorphism C∗(G)→ C∗r (G).

Definition 12.14. A group G is called amenable if λ is faithful on C∗(G).

For example, all finite and abelian groups are amenable, while F2 is not amenable.

Remark 12.15. For every G, we can define the unitary representation ι : G → C byι(g) = 1 for all g ∈ G. This gives rise to a surjective ∗-homomorphism C∗(G)→ C, andker ι is an ideal of C∗(G) of codimension one. Thus C∗(G) is never simple, unless G istrivial.

On the other hand, C∗r (G) can be simple, which is the case, for example for G = F2.Determining when C∗r (G) is in general hard, and is an active research area.

Proposition 12.16. C∗r (Z) ∼= C(T).

Proof. Coming soon. �

Remark 12.17. For every locally compact Hausdorff abelian group G, one can define

its Pontryagin dual G = Hom(G,T) with pointwise operations and the compact-open

topology. Then C∗r (G) ∼= C0(G) by essentially the same argument as above. In par-

ticular, Z ∼= T via z 7→ (n 7→ zn). Moreover, note that for all G we have G ∼= G via

x 7→ (χ 7→ χ(x)) for x ∈ G and χ ∈ G.

13. Tensor products

Given Hilbert spaces H1, H2, there is a unique inner product on H1 ⊗H2 such that

〈ξ1 ⊗ ξ2, η1 ⊗ η2〉 = 〈ξ1, η1〉 · 〈ξ2, η2〉.Let H1⊗H2 denote its completion (see Section 9 and [Mur90, p. 184–186]).

Given x1 ∈ B(H1) and x2 ∈ B(H2), there exists unique element in B(H1⊗H2),denoted x1 ⊗ x2 such that

(x1 ⊗ x2)(ξ1 ⊗ ξ2) = x1ξ1 ⊗ x2ξ2

for all ξi ∈ H1. Moreover, ‖x1 ⊗ x2‖ = ‖x1‖ · ‖x1‖ (see Section 9 and [Mur90,Lemma 6.3.2]).

Let A1 and A2 be ∗-algebras. There exists unique multiplication and involution onA1 ⊗ A2 such that (see [Mur90, p. 188])

(a1 ⊗ a2)(b1 ⊗ b2) = a1a2 ⊗ b1b2 (a1 ⊗ a2)∗ = a∗1 ⊗ a∗2.Let ϕ : A1 → C and ψ : A2 → C be ∗-homomorphisms such that ϕ(a1)ψ(a2) =

ψ(a2)ϕ(a1) for all a1 ∈ A1 and a2 ∈ A2. Then there is a unique ∗-homomorphism


π : A1 ⊗ A2 → C such that π(a1 ⊗ a2) = ϕ(a1)ψ(a2). Indeed, the map (a1, a2) 7→ϕ(a1)ψ(a2) is bilinear, so it induces a map A1 ⊗ A2 → C (see [Mur90, p. 189]).

Henceforth A1 ⊗ A2 will always denote the algebraic tensor product, even when A1

and A2 are C∗-algebras.

Theorem 13.1. Suppose that ϕ : A1 → B(H1) and ψ : A2 → B(H2) are representa-tions of C∗-algebras A1 and A2. There exists a unique ∗-homomorphism π : A1 ⊗ A2 →B(H1⊗H2) such that π(a1 ⊗ a2) = ϕ(a1)⊗ ψ(a2). If ϕ and ψ are both injective, then πis injective. We write π = ϕ⊗ ψ.

Proof. See [Mur90, Theorem 6.3.3]. �

Definition 13.2. Let A1 and A2 be C∗-algebras, and let ϕ and ψ denote the universalGNS-representations of A1 and A2, respectively, and set π = ϕ⊗ψ. For x ∈ A1⊗A2, set‖x‖∗ = ‖π(x)‖. This norm is called the spatial C∗-norm on A1⊗A2. The completion ofA1⊗A2 with respect to ‖·‖∗ is denoted A1⊗∗A2, and called the spatial tensor product.

We note that ‖a1 ⊗ a2‖∗ = ‖a1‖ · ‖a2‖.

Remark 13.3. In general, there are more than one C∗-norm on A1⊗A2. If γ is a C∗-normon A1 ⊗ A2, we denote the completion with respect to γ by A1 ⊗γ A2.

Theorem 13.4. Let A1 and A2 be nonzero C∗-algebras, and let γ be a C∗-norm onA1 ⊗γ A2. Let π : A1 ⊗γ A2 → B(H) be a nondegenerate representation. There existunique ϕ : A1 → B(H) and ψ : A2 → B(H) such that π(a1 ⊗ a2) = ϕ(a1)ψ(a2) =ψ(a2)ϕ(a1). We write ϕ = π|A1 and ψ = π|A2.

Proof. See [Mur90, Theorem 6.3.5]. �

Corollary 13.5. Let A1 and A2 be C∗-algebras and γ a C∗-seminorm on A1⊗A2. Thenγ(a1 ⊗ a2) ≤ ‖a1‖ · ‖a2‖.

Proof. See [Mur90, Corollary 6.3.6]. �

Let Γ be the set of C∗-norms and for each x ∈ A1 ⊗ A2, set

‖x‖max = supγ∈Γ

γ(c).

Define the maximal tensor product A1⊗maxA2 as the completion of A1⊗A2 with respectto ‖·‖max.

Definition 13.6. A C∗-algebra A is called nuclear if for all C∗-algebras B, there is aunique C∗-norm on A⊗B.

Remark 13.7. The spatial norm is the smallest C∗-norm on A ⊗ B, so A is nuclear ifand only if A⊗∗ B = A⊗max B for all C∗-algebras B.

Moreover, since the spatial norm is minimal, if γ is any C∗-norm on A1 ⊗ A2, then

‖a1‖ · ‖a1‖ = ‖a1 ⊗ a2‖∗ ≤ γ(a1 ⊗ a2)

for all a1 ∈ A1 and a2 ∈ A2, that is, combined with the above corollary, we have thatγ(a1 ⊗ a2) = ‖a1‖ · ‖a2‖ for all a1 ∈ A1 and a2 ∈ A2.

Examples 13.8. Some tensor products of familiar algebras:


(i) Every finite-dimensional C∗-algebra is nuclear and Mn(C)⊗γ A = Mn(A) for allγ.

(ii) Every commutative C∗-algebra is nuclear and C0(X)⊗γ A = C0(X,A) for all γ.(iii) C∗(G) is nuclear if and only if C∗r (G) is nuclear if and only if G is amenable.(iv) The compact operators K(H) on a Hilbert space H is a nuclear C∗-algebra.

14. C∗-dynamical systems and crossed products

Let A be a C∗-algebra and define the automorphism group of A as the set

Aut(A) = {ϕ : A→ A | ϕ is a ∗-isomorphism},

which is a group under composition (and is often given the point-norm topology). Anaction of a discrete group G on a C∗-algebra A is a homomorphism α : G → Aut(A),i.e., αg is an automorphism of A and αgh = αg ◦ αh for all g, h ∈ G.

Definition 14.1. A C∗-dynamical system is a triple (A,G, α), where A is a C∗-algebra,G is a discrete group, and α is an action of G on A.

Definition 14.2. Let (A,G, α) be a C∗-dynamical system. A covariant representationof (A,G, α) on a Hilbert space H is a pair (π, U), where π : A→ B(H) is a representationof A, U : G→ U(H) is a unitary representation of G, and

U(s)π(a)U(s)∗ = π(αs(a))

for all s ∈ G and a ∈ A.

Define `1(G,A) as the space of all functions ϕ : G→ A such that∑

g∈G‖ϕ(g)‖ <∞.This is a Banach ∗-algebra with convolution and involution given by

(ϕ ∗ ψ)(t) =∑s∈G

ϕ(s)αs(ψ(s−1t)),

ϕ∗(t) = αt(ϕ(t−1)∗),

and norm ‖ϕ‖1 =∑

g∈G‖ϕ(g)‖. (Checking that this is indeed a Banach ∗-algebra is a

technical exercise with summation formulas.)Let (π, U) be a covariant representation of a C∗-dynamical system (A,G, α) on H.

Define its integrated form π × U as the representation of `1(G,A) on H given by

(π × U)(ϕ) =∑t∈G

π(ϕ(t))U(t).

Note that π × U is norm-decreasing (check!).Let (A,G, α) be a C∗-dynamical system. Let π : A → B(H) be a faithful (nonde-

generate) representation (e.g. the universal GNS-representation). Define a covariantrepresentation (π, λ) of (A,G, α) on `2(G,H) by

(π(a)ξ)(s) = π(αs−1(a))ξ(s),

(λ(t)ξ)(s) = ξ(t−1s).

For ϕ ∈ `1(G,A), define ‖ϕ‖r = ‖(π × λ)(ϕ)‖. This norm is independent of the choiceof faithful π.


Definition 14.3. Let (A,G, α) be a C∗-dynamical system. The reduced crossed prod-uct, denoted Aoα,r G is the completion of `1(G,A) with respect to ‖·‖r.

For ϕ ∈ `1(G,A), define

‖ϕ‖max = sup{‖(π × U)(ϕ)‖ : (π, U) is a covariant representation}.The right hand side is always bounded and nonempty, so the supremum exists.

Definition 14.4. Let (A,G, α) be a C∗-dynamical system. The full crossed product,denoted Aoα G is the completion of `1(G,A) with respect to ‖·‖max.

If A is represented on H, one may think of AoαG as the C∗-subalgebra of B(`2(G,H))

generated by all the products pi(a)λ(g) for a ∈ A, g ∈ G.There is always a surjection Aoα G → Aoα,r G. If G is amenable, then the map is

also injective. But in contrast to the group C∗-algebra case it is also possible that thefull and reduced crossed products are isomorphic when G is nonamenable.

Example 14.5. If α is trivial, i.e., αg is the identity for all g ∈ G, then A oα,r G ∼=A ⊗∗ C∗r (G) and A oα G ∼= A ⊗max C

∗(G). In some sense, one can think of crossedproducts as “skew” tensor products

Example 14.6. If A = C, then α must be the identity, so C oα,r G ∼= C∗r (G) andCoα G ∼= C∗(G).

Proposition 14.7. Define the left translation action lt of G on C0(G) by (ltsf)(t) =f(s−1t). Let G be a finite group with |G| = n. Then C0(G) olt G ∼= Mn(C).

Proof. Later. �

In fact, for any group G, one has that C0(G) olt G ∼= K(`2(G)).

Example 14.8. Let X be a locally compact Hausdorff space and ϕ : X → X a home-omorphism. Then there is an action of Z on C0(X) given by αn(f) = ϕ ◦ f−n, and(C0(X),Z, α) is a C∗-dynamical system.

Example 14.9. Let θ ∈ R and let αθ be the action of Z on C(T) given by (αθnf)(z) =f(e−2πinθz). Then (C(T),Z, αθ) is a C∗-dynamical system and its crossed product, de-noted Aθ, is called the rotation algebra. If θ /∈ Z, it is also often called a noncommutativetwo-torus.

Define M : C(T) : B(L2(T)) by (M(h)f)(z) = h(z)f(z) and U : Z → B(L2(T)) by(Unf)(z) = f(e−2πinθz). Then (M,U) is a covariant representation of (C(T),Z, αθ) onL2(T).

Theorem 14.10. Let θ ∈ R \Q, consider the C∗-dynamical system (C(T),Z, αθ), andits crossed product algebra Aθ.

(i) Aθ is generated by two unitaries u, v satisfying uv = e2πiθvu.(ii) If H is any Hilbert space and U, V are unitaries in B(H) such that UV = e2πiθV U ,

then there exists a faithful representation L : Aθ → B(H) such that L(u) = U andL(v) = V .

(iii) Aθ is simple.

Proof. Later. �

Remark 14.11. Aθ ∼= Aθ′ if and only if θ + θ′ ∈ Z or θ − θ′ ∈ Z.


15. Universal C∗-algebras by generators and relations

Remark 15.1. In this course, we restrict to unital algebras and algebraic relations.

Let G = {xi}i∈I be a set of generators. By a set of relations R for G, we mean aset of polynomials in {xi, x∗i }i∈I (with complex coefficients). A representation of (G,R)is a set of operators {Ti, T ∗i }i∈I ⊆ B(H) for some Hilbert space H, satisfying all therelations in R. Let A denote the (unital, complex) free ∗-algebra on G (i.e., consistingof all polynomials in {xi, x∗i }i∈I). For every x ∈ A, set

ρ(x) = sup{‖π(x)‖ : π is a representation of (G,R)}.

If this supremum exists for every x ∈ A, then it defines a C∗-seminorm on A. We thendivide out by all elements of norm zero to get a pre-C∗-algebra, and then complete itto get a C∗-algebra (see Remark 15.4). This resulting C∗-algebra is denoted C∗(G,R)and called the universal C∗-algebra of (G,R).

Remark 15.2. Note that:

• Universal C∗-algebras do not always exist.• If it exists, it will in many cases just be C.• Every unital C∗-algebra is isomorphic to C∗(G,R) for some (G,R).• Universal C∗-algebras are especially interesting when they have a simple descrip-

tion.

Suppose that (G,R) are generators and relations such that C∗(G,R) exists, and sup-pose that {Ti, T ∗i }i∈I are operators on a Hilbert space H satisfying the relations from R.Then there exists a ∗-homomorphism π : C∗(G,R)→ B(H) such that π(xi) = Ti for alli ∈ I. Moreover, C∗(G,R) satisfies the following universal property: whenever B is anyother C∗-algebra generated by set {x′i}i∈I satisfying the relations from R, then thereexists a unique surjective ∗-homomorphism π : C∗(G,R) → B such that π(xi) = x′i forall i ∈ I.

Examples 15.3. Consider the following examples:

(i) C∗(Z) ∼= C∗r (Z) ∼= C(T) is the universal C∗-algebra C∗(G,R) with G = {u} andR = {uu∗ = u∗u = 1}.

(ii) Let H be a discrete group. Then the full group C∗-algebra C∗(H) is the universalC∗-algebra C∗(G,R) with G = {Uh : h ∈ H} and

R = {UgUh = Ugh, U∗h = Uh−1 , U(e) = 1, UhU

∗h = U∗hUh = 1 for all g, h ∈ H}.

(It is possible to describe the above with fewer relations.)(iii) Let θ ∈ R, and let Aθ denote the rotation algebra (the (non-)commutative two-

torus). Then Aθ is the universal C∗-algebra C∗(G,R) with G = {u, v} andR = {uu∗ = u∗u = vv∗ = v∗v = 1, uv = e2πiθvu}. Note that Aθ is commutativeif and only if θ ∈ Z.

(iv) The Toeplitz algebra T is the universal C∗-algebra C∗(G,R) with G = {v} andR = {v∗v = 1}.


(v) For n ≥ 2, the Cuntz algebras On is the universal C∗-algebra C∗(G,R) withG = {si}ni=1

R = {s∗i si = 1 for all i andn∑i=1

sis∗i = 1}.

(vi) Let G = {v, w} and R = {vv∗ = w,w2 = v, w∗w = 1}. Then C∗(G,R) = C1.(vii) Let G = {v} and R = ∅. Then the universal C∗-algebra of (G,R) does not

exist.(viii) Crossed product by Z.(ix) Tensor products.

Remark 15.4. Let A be a ∗-algebra. We say that a seminorm ρ on A is a C∗-seminormif it satisfies

ρ(xy) ≤ ρ(x)ρ(y) ρ(x) = ρ(x∗) ρ(x∗x) = ρ(x)2.

Let N := p−1(0). Then N is a self-adjoint ideal of A, so the quotient A/N is a pre-C∗-algebra (i.e., a ∗-algebra with a C∗-norm). The usual Banach space completion of A/Nis then a C∗-algebra, sometimes called the enveloping algebra of (A, ρ).

16. Direct limits of C∗-algebras

Let (An)∞n=1 be a sequence of ∗-algebras, and (ϕn)∞n=1 a sequence of ∗-homomorphismsϕn : An → An+1. For n ≤ m, define ϕn,m : An → Am by composition, and set ϕn,n+1 =ϕn and ϕn,n = id. Thus, if n ≤ m ≤ k, we have ϕn,k = ϕm,kϕn,m.

The product∏∞

n=1An is a ∗-algebra under pointwise operations. Define A ⊆∏∞

n=1 Anas all elements x = (xn)∞n=1 ∈ A for which there exists an N such that xn+1 = ϕn(xn)for all n ≥ N . Then one can check that A is a ∗-subalgebra of

∏∞n=1 An.

Moreover, for x, y ∈ A, we write x ∼ y if there exists some N such that xn = yn forall n ≥ N , i.e., if their sequences eventually become equal. Then ∼ is an equivalencerelation on A and A/ ∼ is a ∗-algebra, which is the algebraic direct limit of the system(An, ϕn)∞n=1.

Now, let us turn to C∗-algebras. Let (An, ϕn)∞n=1 be a system of C∗-algebras and∗-homomorphisms. As above, we define the ∗-subalgebra A of

∏∞n=1 An. Let x ∈ A, and

let N be such that xn+1 = ϕn(xn) for all n ≥ N . Since each ϕn is a ∗-homomorphism,it is norm-decreasing for all n, so we have that

‖xn+1‖ = ‖ϕ(xn)‖ ≤ ‖xn‖

for all n ≥ N . That is, (‖xn‖)∞n=1 is eventually decreasing and bounded below, so itconverges. We set

ρ(x) = limn→∞‖xn‖,

and then x 7→ ρ(x) is a C∗-seminorm on A.

Definition 16.1. The direct limit (sometimes called the inductive limit) of the system(An, ϕn)∞n=1 is the enveloping algebra (see Remark 15.4) of (A, ρ), where A ⊆

∏∞n=1 An

is defined as above. We denote the direct limit by lim−→(An, ϕn) or just lim−→An if the mapsare understood.


Further, define maps Φn : An → A by

(Φn(x))k =

{0 if k < n,

ϕn,k if k ≥ n,

and let Φn : An → lim−→An be the composition, i.e., An → A→ A/N → A/N = lim−→An.

Then Φn(An) ⊆ Φn+1(An+1), and

∞⋃n=1

Φn(An)

is dense in lim−→An. Henceforth, we omit the tilde, and write Φn for the map An → lim−→An.We have that Φn = Φn+1 ◦ ϕn for all n

Theorem 16.2. Given (An, ϕn)∞n=1, lim−→(An, ϕn), and (Φn)∞n=1 as above.(i) Let x ∈ An, y ∈ Am such that Φn(x) = Φm(y). For all ε > 0, there exists k ≥ n,m

such that ‖ϕn,k(x)− ϕm,k(y)‖ < ε.(ii) If A′ and (Φ′n : An → A′)∞n=1 are such that Φ′n = Φ′n+1 ◦ ϕn for all n, then there

exists a unique ∗-homomorphism Φ: lim−→(An, ϕn)→ A′ such that Φ′n = Φ ◦Φn for all n.

Proof. See [Mur90, Theorem 6.1.1 and 6.1.2]. �

Example 16.3. If A is a C∗-algebra and A1 ⊆ A2 ⊆ · · ·An ⊆ An+1 · · · is an increasingsequence of C∗-subalgebras, such that the union

⋃∞n=1An is dense in A, then lim−→(An, ιn),

where ιn is the inclusion map.If in addition all the An’s are finite-dimensional, then A is called an approximately

finite-dimensional algebra, or more commonly, just an AF-algebra (these are discussedin the next section).

Theorem 16.4. Suppose that A is an increasing union of simple C∗-subalgebras An.Then A is simple.

Proof. See [Mur90, Theorem 6.1.3 and 6.1.4], but the proof given in class is maybe alittle bit more direct. �

17. Approximately finite-dimensional C∗-algebras

First, we discuss finite-dimensional C∗-algebras.

Lemma 17.1. Let π is an irreducible representation of a (nonzero) C∗-algebra A on afinite-dimensional Hilbert space H. Then π(A) = MdimH(C).

Proof. Let n = dimH. Since π is irreducible, we have π(A)′ = C1, and thus π(A)′′ =C1′ = B(H) ∼= B(Cn) = Mn(C). On a finite-dimensional Hilbert space, the strong andoperator topologies all coincide, so by invoking the bicommutant theorem, we get

π(A) = π(A)op

= π(A)SOT

= π(A)′′ = Mn(C). �

Proposition 17.2. Let A be a (nonzero) finite-dimensional C∗-algebra. Then A issimple if and only if A ∼= Mn(C) for some n ≥ 1.


Proof. If A ∼= Mn(C) = B(Cn) = K(Cn), then A is simple, since the compact operatorson any Hilbert space is a simple C∗-algebra.

If A is simple, then pick an irreducible representation π of A on a Hilbert space H.Since π is irreducible, it is cyclic, so there exists ξ ∈ H such that π(A)ξ = H. Thus,since A is finite-dimensional, we must have that H is finite-dimensional. It followsfrom Lemma 17.1 that π(A) = Mn(C) for some n ≥ 1. Moreover, as A is assumed tobe simple, then kerπ = {0}, so π is also faithful, and hence π gives an isomorphismA→Mn(C). �

Theorem 17.3. Every (nonzero) finite-dimensional C∗-algebra is a direct sum of matrixalgebras.

Proof. There exist irreducible representations π1, π2, . . . , πN of A such that π = π1 ⊕π2 ⊕ · · · ⊕ πN is faithful. Indeed, ... (to be continued) �

Corollary 17.4. Every (nonzero) finite-dimensional C∗-algebra is unital (whether {0}is unital or not is a somewhat open to opinion).

Definition 17.5. AF-algebras

Remark 17.6. Let A be an AF-algebra, and J an ideal of A. Then both J and A/J areAF-algebras. AF-algebras are always nuclear.

Remark 17.7. AF-algebras has a lot of projections. For example, a commutative C∗-algebra C0(X) is an AF-algebra if and only if X is totally disconnected.

Example 17.8. Let H be an infinite-dimensional separable Hilbert space. Then thecompact operators K(H) is a two-sided closed ideal of B(H). Let F (H) be the finite-rank operators of B(H). Then F (H) is a two-sided selfadjoint (nonclosed) ideal ofB(H). If J is any nonzero, proper, two-sided, selfadjoint, (not necessarily closed) idealof B(H), then F (H) ⊆ J ⊆ K(H). Thus, K(H) is the only nonzero, proper two-sidedclosed ideal of B(H).

Let Pn be a sequence of rank n operators converging strongly to the identity in B(H).Set An = PnK(H)Pn. Then An ' Mn(C), and K(H) is the direct limit of the system(Mn(C), ϕn), where ϕn maps Mn(C) into the upper left corner of Mn+1(C). Thus K(H)is an AF-algebra.

Moreover, K(H) is simple (to be continued...).

Definition 17.9. A C∗-algebra A is called a uniformly hyperfinite algebra (UHF-algebra), if it is unital and has an increasing sequence of simple finite-dimensionalC∗-subalgebras containing the unit of A, such that its union is dense in A.

Let (kn)∞n=1 be a sequence of natural numbers ≥ 2 such that kn divides kn+1 for alln. Let An = Mkn , and let ϕn : An → An+1 be the embedding kn+1/kn copies of Mkn

into the diagonal of Mkn+1 . Let A = lim−→An. Then A is an UHF-algebra, and everyUHF-algebra is isomorphic to one of this form.

Remark 17.10. Supernatural numbers form a complete invariant for the UHF-algebras.They also form a complete invariant for the Bunce-Deddens algebras. The Bunce-Deddens algebras also have a direct limit description of algebras Mn(C(T)).


References

[Mur90] Gerard J. Murphy, C∗-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990.[Ped79] Gert K. Pedersen, C∗-algebras and their automorphism groups, London Mathematical Society

Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.

Department of Mathematics, University of Oslo, NorwayE-mail address: [email protected], [email protected]

introduction a brief review of commutative banach algebras · introduction to c*-algebras 3...

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