Crash course: the hyperfinite II1 factor and the standard form

Hiroshi Ando

October 9, 2014

Abstract

(Masterclass “Topological quantum field theories, quantum groups and 3-manifold invariants”)This is a follow-up notes for the introductory talks (45min × 2) on the hyperfinite II1 factor andthe standard form prepared for Prof. Ryszard Nest’s talks Oct.9-10. There may be typos orerrors.

1 Talk 1 (Oct. 6 2014, 10:30-11:15)

We define the notion of II1 factors and see one example, namely the hyperfinite II1 factor. We usethe symbol H for complex Hilbert space (all Hilbert spaces in this notes are separable, and infinite-dimensional). The inner product ⟨ · , · ⟩ : H × H → C is assumed to be linear in the first variableand conjugate-linear in the second variable. We use Greek letters ξ, η, · · · for vectors in H and Italicletters x, y, · · · for bounded linear operators on H.

A linear opeartor x : H → H is bounded, if it is continuous in Hilbert space norm. In this case,there exists c > 0 such that ∥xξ∥ ≤ c∥ξ∥ (ξ ∈ H). The infimum of such c is called the (operator)norm of x, denoted as ∥x∥:

∥x∥ = supξ∈H, ∥ξ∥≤1

∥xξ∥.

Definition 1.1. We denote by B(H) the set of all bounded linear maps from H to itself.

B(H) equipped with the operator norm is a Banach space. For each x ∈ B(H) there exists aunique x∗ ∈ B(H) satisfying

⟨xξ, η⟩ = ⟨ξ, x∗η⟩ (ξ, η ∈ H).

x∗ is called the adjoint of x. The map x 7→ x∗ has the following properties (x, y ∈ B(H), λ, µ ∈ C):

(1) (λx+ µy)∗ = λx∗ + µy∗.

(2) (x∗)∗ = x.

(3) (xy)∗ = y∗x∗.

(4) ∥x∗x∥ = ∥x∥2.

B(H) has the *-algebra structure with respect to the operator sum, multiplication, etc. A vectorsubspace A of B(H) is called a *-subalgebra if A is closed under operator product and *. For simplicitywe only consider *-subalgebras of B(H) containing the unit 1 = 1H .

Definition 1.2. A *-subalgebra A of B(H) with unit 1H is called a C∗-algebra, if A is closed inoperator norm topology. A is called a von Neumann algbera on H, if A is closed with respect to thestrong operator topology (SOT).Here, a net {xi}i∈I ⊂ B(H) converges to x ∈ B(H) with respect to SOT, if limi ∥xiξ − xξ∥ = 0 forevery ξ ∈ H.

Below we always assume that all C∗-algebras on H contain the unit 1 = 1H . Next lemma is onlyrelevant for Corollary 1.5, which is not important either for understanding the rest of the topics. Sothey can safely be ignored.

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Definition 1.3. Let S ⊂ B(H). The commutant S ′ of S is the set S ′ := {a ∈ B(H); ax = xa, ∀x ∈S}.

If a subset S ⊂ B(H) is self-adjoint (i.e. x ∈ S ⇒ x∗ ∈ S), then S ′ is a von Neumann algebra.The next theorem shows that von Neumann algbera is characterized by its double commutant:

Theorem 1.4 (von Neumann’s double commutant theorem). Let M be a *-subagebra of B(H) con-

taining unit 1H . Then MSOT

= M ′′(= (M ′)′).

Proof. MSOT ⊂ M ′′ : Let x ∈ M

SOT. Then there exists a net (xi)i∈I ⊂ M converging to x strongly.

Then for every y′ ∈ M ′ and ξ ∈ H, we have

xy′ξ = limi→∞

xiy′ξ = lim

i→∞y′xiξ = y′xξ.

Since ξ is arbitrary, xy′ = y′x holds. Therefore x ∈ (M ′)′.

MSOT ⊃ M ′′ : Let x ∈ M ′′. We show that x is in the strong closure of M . This, by definition of SOT,

amounts to proving the following: For every ε > 0, n ∈ N and ξ1, · · · , ξn ∈ H, there exists x0 ∈ Msuch that

∥xξi − x0ξi∥ < ε, 1 ≤ i ≤ n. (1)

Fix ε > 0 and ξ1, · · · , ξn ∈ H. Let Hn = H ⊕H ⊕· · ·⊕H (n copies) and define π : B(H) → B(Hn) by

π(a)(η1, · · · , ηn) = (aη1, · · · , aηn) (a ∈ B(H)), η1, · · · , ηn ∈ H.

That is, π(a) = diag(a, a, · · · , a) regarded as an n × n matrices with entries in B(H). It is straight-forward to check that the commutant of π(M) is the set of all n× n matrices with entries in M ′:

π(M)′ ={[x′

ij ];x′ij ∈ M ′, 1 ≤ i, j ≤ n

}⊂ B(Hn) ∼= Mn(B(H)).

Let E := π(M)(ξ1, · · · , ξn) ⊂ Hn, and let e be the projection ofHn onto E. Then E is π(M)-invariant.Therefore E⊥ is also π(M)-invariant. This shows that e ∈ π(M)′. Thus e = [eij ], where eij ∈ M ′. Itis then clear that since x ∈ M ′′, π(x) = diag(x, x, · · · , x) commutes with e:

π(x)e = [xeij ] = [eijx] = eπ(x).

It then follows that (since 1 ∈ M , (ξ1, · · · , ξn) ∈ E holds)

π(x)(ξ1, · · · , ξn) = π(x)e(ξ1, · · · , ξn) = eπ(x)(ξ1, · · · , ξn) ∈ E.

Therefore there exists x0 ∈ M such that

∥(π(x)− π(x0))(ξ1, · · · , ξn)∥2 =n∑

i=1

∥xξi − x0ξi∥2 < ε2.

This implies (1). Therefore x ∈ MSOT

.

Corollary 1.5. Let M ⊂ B(H) be a von Neumann algebra, and let x = x∗ ∈ M be its self-adjointelement. Let x =

∫sp(x)

λde(λ) be the spectral decomposition of x. Then e(J) ∈ M for every Borel

subset J ⊂ R.

Proof. Denote by Proj(H) the set of all orthogonal projections on H. Let B(R) be the σ-algebra ofall Borel subsets of R. Fix u ∈ U(M ′) (U(M ′) is the group of unitary elements in M ′). Define a newmap e : B(R) → Proj(H) by

e(J) = ue(J)u∗, J ∈ B(R).It is straightforward to see that e(·) is a projection valued measure, and e(·) is the spectral resolutionof x := uxu∗. Since x ∈ M,u ∈ M ′, we have

uxu∗ =

∫sp(x)

λde(λ) = x.

This shows, by the uniqueness of spetral decomposition, that e = e. That is, ue(J)u∗ = e(J) for everyu ∈ U(M ′). It is also not difficult to show that M ′ is spanned by U(M ′) (cf. proof sketched in Lemma1.12), the unitary elements of M ′. Therefore e(J) ∈ (M ′)′ = M by double commutant theorem.

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Definition 1.6. For a self-adjoint subset S of B(H) we call S ′′ the von Neumann algebra generatedby S.

We now introduce several operator topologies on B(H). Denote by ℓ2(H) the set of all sequences{ξn}∞n=1 ⊂ H such that

∑∞n=1 ∥ξn∥2 < ∞ holds.

Definition 1.7. On B(H), we define the following topologies:

(1) The strong operator topology (SOT for short) is the locally convex topology determined bythe seminorm family {pξ}ξ∈H\{0}, where

pξ(x) := ∥xξ∥, x ∈ B(H).

(2) The weak operator topology (WOT for short) is the locally convex topology determined bythe seminorm family {pξ,η}ξ,η∈H\{0}, where

pξ,η(x) := |⟨xξ, η⟩|, x ∈ B(H).

(3) The strong* operator topology (S*OT for short) is the locally convex topology determined bythe seminorm family {pξ}ξ∈H\{0}, where

pξ(x) := ∥xξ∥+ ∥x∗ξ∥, x ∈ B(H).

(4) The σ-strong topology (σSOT for short) is the locally convex topology determined by theseminorm family {p{ξn}}{ξn}∈ℓ2(H)\{0}, where

p{ξn}(x) :=

{ ∞∑n=1

∥xξn∥2} 1

2

, x ∈ B(H).

(5) The σ-weak topolgy (σWOT for short) is the locally convex topology determined by theseminorm family {p{ξn},{ηn}}{ξn},{ηn}∈ℓ2(H)\{0}, where

p{ξn},{ηn}(x) :=

∣∣∣∣∣∞∑

n=1

⟨xξn, ηn⟩

∣∣∣∣∣ , x ∈ B(H).

(6) The σ-strong* topology (σS*OT for short) is the locally convex topology determined by theseminorm family {p{ξn}}{ξn}∈ℓ2(H)\{0}, where

p{ξn}(x) :=

{ ∞∑n=1

(∥xξn∥2 + ∥x∗ξn∥2

)} 12

, x ∈ B(H).

The product (x, y) 7→ xy is strongly continuous on the unit ball B(H)1 := {a ∈ B(H); ∥a∥ ≤ 1},and on the unit ball, σ-strong topology coincides with strong topology.

Definition 1.8. Let M,N be von Neumann algebras. A linear map φ : M → N is called normal ifit is σ-weakly continuous.

The notion of positivity is important for operator algebras. We recall the following fact fromspectral theory.

Definition 1.9. For x ∈ B(H), we define the spectrum of x, denoted sp(x) by

sp(x) := {λ ∈ C; (x− λ1) is not invertible in B(H)}.

The spectrum sp(x) is always a nonempty compact subset of C. And if x is normal (i.e., xx∗ = x∗x),then x is self-adjoint (i.e., x = x∗) if and only if sp(x) ⊂ R (this equivalence fails for non-normal x:consider nilpotent matrices!).

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Proposition 1.10. For a = a∗ ∈ B(H), the following conditions are equivalent:

(1) sp(a) ⊂ [0,∞).

(2) a = b∗b for some b ∈ B(H).

(3) a = h2 for a unique self-adjoint h ∈ B(H) with sp(h) ⊂ [0,∞).

(4) ⟨aξ, ξ⟩ ≥ 0 for every ξ ∈ H.

If A ⊂ B(H) is a C∗-algebra and a ∈ A then b (resp. h) in (2) (resp. in (3)) also belongs to A.

Definition 1.11. If one (hence all) of the conditions (1)-(4) in Proposition 1.10 are satisfied for

a = a∗ ∈ B(H), we say that a is positive, and denote a ≥ 0. The h in (3) is written as a12 . Also, for

self-adjoint operators a, b ∈ B(H) we write a ≤ b if b− a ≥ 0.

Lemma 1.12. Let A ⊂ B(H) be a C∗-algebra with unit 1H . Then A is spanned by the unitaryelements U(A) = {u ∈ A;u∗u = 1 = uu∗}.

Sketch of Proof. It suffices to show that every element a ∈ A with norm ∥a∥ ≤ 1 is spanned byunitaries in A. Since a = a1 + ia2, where a1 = a+a∗

2 , a2 = a−a∗

2i satisfy ai = a∗i , ∥ai∥ ≤ 1(i = 1, 2),we may from the beginning assume that a = a∗. In this case, −1 ≤ a ≤ 1 (in operator order) andtherefore 1 − a2 ≥ 0 (see Definition 1.11). In this case one can define the square root inside A:√1− a2 ∈ A. Then

a =a+ i

√1− a2

2+

a− i√1− a2

2,

and it is not difficult to see that a± i√1− a2 are unitaries in A.

Proposition 1.13. Let a, b ∈ B(H). The following operator inequalities hold.

(1) b∗a∗ab ≤ ∥a∥2b∗b.

(2) (a+ b)∗(a+ b) ≤ 2(a∗a+ b∗b).

Proof. (1) Let ξ ∈ H. Then

⟨b∗a∗abξ, ξ⟩ = ⟨abξ, abξ⟩ = ∥abξ∥2 ≤ ∥a∥2∥bξ∥2 = ⟨∥a∥2b∗bξ, ξ⟩.

Since ξ ∈ H is arbitrary, we get (1).(2) 2(a∗a+ b∗b)− (a+ b)∗(a+ b) = a∗a+ b∗b− a∗b− b∗a = (a− b)∗(a− b) ≥ 0.

Definition 1.14. Let 1 ∈ A ⊂ B(H) be a C∗-algebra. A linear functional φ : A → C is called

(1) positive if it satisfies φ(a∗a) ≥ 0 for every a ∈ A.

(2) if moreover φ satisfies φ(1) = 1, it is called a state.

(3) A positive linear functional φ is called faithful if φ(a∗a) = 0 implies a = 0.

(4) A positive linear functional φ is called a trace if φ(ab) = φ(ba) for a, b ∈ A.

Remark 1.15. If φ is a state on A, then its norm is attained at 1:

∥φ∥ def= sup{|φ(a)|; a ∈ A, ∥a∥ ≤ 1} = φ(1) = 1.

We use next useful lemma later.

Lemma 1.16 (Cauchy-Schwartz inequality). Let A be a C∗-algebra with unit on a Hilbert space H.Let φ be a state on A. Then for a, b ∈ A, the following inequality hold:

|φ(b∗a)| ≤ φ(b∗b)12φ(a∗a)

12 .

Moreover, we have ∥a+ b∥φ ≤ ∥a∥φ + ∥b∥φ for every a, b ∈ A. Here, ∥a∥φ := φ(a∗a)12 .

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Proof. Step 1 φ(a∗) = φ(a) (complex conjugate) for every a ∈ A.First we see that if a = a∗ ∈ A, then φ(a) ∈ R. This is because of the ineuqlity −∥a∥1 ≤ a ≤ ∥a∥1,we have φ(∥a∥1 − a) ≥ 0 and φ(a + ∥a∥1) ≥ 0. In particular, ∥a∥ ≥ φ(a) and φ(a) ∈ R. Then forgeneral a ∈ A, we have the decompositon a = x+ iy where x = a+a∗

2 , y = a−a∗

2i ∈ A are self-adjoint.Since φ(x), φ(y) ∈ R, we know that

φ(a∗) = φ(x− iy) = φ(x)− iφ(y) = φ(x) + iφ(y) = φ(a).

Step 2 We show the claim. It suffices to consider the case φ(b∗b) > 0. Since the transformatiob 7→ eiθb (θ ∈ R) does not change φ(b∗b), we may assume that φ(b∗a) ∈ R after this transformation.Now consider

(a+ tb)∗(a+ tb) = a∗a+ t(a∗b+ b∗a) + t2b∗b ≥ 0

for t ∈ R. Applying φ to the above expansion we get that

φ(a∗a) + 2tφ(b∗a) + t2φ(b∗b) ≥ 0, t ∈ R (2)

Here we used the fact (Step 1) that φ(a∗b) = φ((b∗a)∗) = φ(b∗a) = φ(b∗a) ∈ R. Since (2) is quadraticin t, (2) holds for every t ∈ R if and only if its discriminant is ≤ 0, whence

φ(b∗a)2 − φ(a∗a)φ(b∗b) ≤ 0,

which shows (2).We show the triangle inequality ∥a+ b∥φ ≤ ∥a∥φ + ∥b∥φ.

Let a, b ∈ A. Then

(∥a∥φ + ∥b∥φ)2 − ∥a+ b∥2φ = (∥a∥2φ + 2∥a∥φ∥b∥φ + ∥b∥2φ)− (∥a∥2φ + φ(a∗b+ b∗a) + ∥b∥φ + ∥b∥2φ)

= 2(φ(a∗a)12φ(b∗b)

12 − Reφ(a∗b))

≥ 0,

by the Cauchy-Schwartz inequality (and φ(x∗) = φ(x)) that we have just proved.

Now we give an (unofficial, but equivalent to the official) definition of II1 factors. The officialdefinition requires the analysis of projection lattice in von Neumann algebras (see e.g., [Su86]).

Definition 1.17. A von Neumann algebra M ⊂ B(H) is called a II1 factor, if

(factor) The center Z(M) of M , Z(M) := M ′ ∩M is C1.

(∞-dim) M is infinite-dimensional (as a vector space over C).

(tracial) M admits a faithful normal tracial state τ .

We will not prove, but the following holds.

Proposition 1.18. In a II1 factor, there is only one normal faithful tracial state.

Next theorem satets that we can construct a *-homomorphism of a C∗-algbera out of states suchthat the resulting *-homomorphism is unique in a strong sense.

Theorem 1.19 (Gelfand-Naimark-Segal (GNS) construction). Let A be a C∗-algebra with unit 1, andlet φ be a state on A. Then there exists a triple (Hφ, πφ, ξφ) where Hφ is a Hilbert space, ξφ ∈ Hφ isa unit vector and πφ : A → B(Hφ) is a *-homomorphism such that

(1) φ(a) = ⟨πφ(a)ξφ, ξφ⟩ (a ∈ A).

(2) πφ(A)ξφ = {πφ(a)ξφ; a ∈ A} is a dense subspace of Hφ.

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The triple is unique in the sense that if (H, ξ, π) is another triple satisfying (1) and (2), then thereexists a unitary V : Hφ → H such that

V πφ(a)V∗ = π(a), a ∈ A,

and V ξφ = ξ.

Proof. Regard A as a pre-inner product space equipped with the following (possibly degenerate)inner-produt

⟨a, b⟩φ := φ(b∗a), a, b ∈ A.

It is clear that ⟨ · , · ⟩φ is sesqui-linear and positive (by the positivity of φ). Let Nφ = {a ∈ A; ∥a∥φ =

⟨a, a⟩12φ = 0}. Then Nφ is a vector subspace of A. It is clear that x ∈ Nφ, λ ∈ C implies λx ∈ Nφ.

If x, y ∈ Nφ, then by Lemma 1.16, ∥x + y∥φ ≤ ∥x∥φ + ∥y∥φ = 0, whence x + y ∈ Nφ. Therefore⟨ · , · ⟩φ induces a positive definite inner-product on the quotient vector space A/Nφ. Let Hφ bethe completion of A/Nφ and denote by a the canonical image of a ∈ A in Hφ. By construction,

A = {a; a ∈ A} is a dense subspace of Hφ. Define ξφ = 1 ∈ Hφ. Then ∥ξφ∥2 = ⟨1, 1⟩φ = φ(1) = 1. So

ξφ is a unit vector. For each a ∈ A, define πφ(a) as a linear map A → A by

πφ(a)b = ab, b ∈ A.

Since b∗a∗ab ≤ ∥a∥2b∗b by Lemma 1, we see that πφ(a) is bounded on the dense subspace A of Hφ:

∥ab∥2 = φ(b∗a∗ab) ≤ ∥a∥2φ(b∗b) = ∥a∥2∥b∥2.

Therefore πφ(a) is uniquely extended to a bounded linear map πφ(a) ∈ B(Hφ). It is straightforwardto see that A ∋ a 7→ πφ(a) ∈ B(Hφ) is a *-homomorphism. For example, if a, b, c ∈ A. Then

πφ(a)πφ(b)c = abc = πφ(ab)c.

Therefore πφ(a)πφ(b) and πφ(ab) agree on the dense subspace A of Hφ, whence by boundedness,πφ(a)πφ(b) = πφ(ab) as operator in B(Hφ). Next, if a ∈ A, then

⟨πφ(a)ξφ, ξφ⟩ = ⟨a, 1⟩φ = φ(a).

It is clear that πφ(A)ξφ = A, which is dense in Hφ.

Finally, suppose (H,π, ξ) is another triple satisfying (1) (2). Then let V : πφ(A)ξφ = A → π(A)ξby

V πφ(a)ξφ := π(a)ξ.

This is well-defined, because ∥πφ(a)ξφ∥2 = φ(a∗a) = ⟨π(a∗a)ξ, ξ⟩ = ∥π(a)ξ∥2, and V is in fact

isometric on A. Therefore V is extended to an isometry fromHφ intoH. If we also defineW : H → Hφ

as the unique isometric extension of the map π(A)ξ → πφ(A)ξφ given by Wπ(a)ξ = πφ(a)ξφ, then

it is straightforward to see that VW = 1H ,WV = 1Hφ (again use densities of A, π(A)ξ in Hφ, H,respectively). Therefore V is onto, and it is clear that V ξφ = ξ by construction. Now let a, b, c ∈ A.Then

⟨V πφ(a)πφ(b)ξφ, π(c)ξ⟩ = ⟨V πφ(ab)ξφ, π(c)ξ⟩ = ⟨π(ab)ξ, π(c)ξ⟩,⟨π(a)V πφ(b)ξφ, π(c)ξ⟩ = ⟨π(a)π(b)ξ, π(c)ξ⟩ = ⟨π(ab)ξ, π(c)ξ⟩.

Therefore since both V πφ(a), π(a)V are bounded operators, the density of πφ(A)ξφ (resp. π(A)ξ) inHφ (resp. in H) shows that V πφ(a) = π(a)V . This finishes the proof.

Now we give an example of a hyperfinite II1 factor.

Definition 1.20. A von Neumann algebraM is called hyperfinite if there exists an increasing sequenceM1 ⊂ M2 ⊂ · · · of finite-dimensional *-subalgebra of M with unit 1, such that its union is SOT-dense

in M : M =∪

n∈N MnSOT

.

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Example 1.21 (Hyperfinite II1 factor). Consider an increasing sequence of matrix algebras

M2(C) ↪→ M2(C)⊗2 ↪→ M2(C)⊗3 ↪→ · · · ,

where the inclusion M2(C)⊗n ↪→ M2(C)⊗(n+1) is given by x 7→ x ⊗ 1 for each n ∈ N. Let M2∞ bethe norm-completion of the above inductive limit

∪n∈N M2(C)⊗n. Consider for each n ∈ N a tracial

state τn = ( 12Tr)⊗n on M2(C)⊗n ∼= M2n(C). Namely τn is uniquely determined (by the universality

of tensor product)

τn(x1 ⊗ · · · ⊗ xn) =1

2Tr(x1) · · ·

1

2Tr(xn), x1, x2, · · · , xn ∈ M2(C).

Since τn(y) = τn+1(y ⊗ 1) (y ∈ M2(C)⊗n), and since states are norm-continuous, {τn}∞n=1 defines apositive linear functional (of norm 1) on

∪n∈N M2(C)⊗n which is extended to a unique state τ on the

(abstract) C∗-algebra M2∞ =∪

N M2(C)⊗n∥·∥

. Let (Hτ , ξτ , πτ ) be the GNS representation associatedwith τ . We then define

R := πτ (M2∞)′′

τ is extended to a normal tracial state by

τ(x) = ⟨xξτ , ξτ ⟩, x ∈ πτ (M2∞)′′ = R.

It is clear from the construction that R is hyperfinite. Below we give a proof that R is indeed a II1factor.

We show that R is a hyperfinite II1 factor. It is clear that R is hyperfinite, infinite-dimensionaland has a normal tracial state τ . We show that

(a) τ is faithful on R.

(b) R is a factor.

For (a), note that we do know that τ is faithful on a strongly dense *-subalgebra∪

N M2(C)⊗n.However, this does not automatically mean that τ is faithful on R (and in fact could fail if we considernon-tracial states). We deduce (b) from

(b’) R has unique normal tracial state.

Lemma 1.22. Let M be a von Neumann algebra on a Hilbert space H. Assume that there exists acyclic vector ξ ∈ H for M such that τ(x) = ⟨xξ, ξ⟩ defines a tracial state on M . Then τ is faithful.

Proof. Let x ∈ M be such that τ(x∗x) = 0, and let a ∈ M . Then since τ is tracial, we have (usePropositin 1.13)

∥xaξ∥2 = τ(a∗x∗xa) = τ(xaa∗x∗)

≤ ∥a∗∥2τ(xx∗)2 = ∥a∥2τ(x∗x)

= 0.

Therefore xaξ = 0. Since ξ is cyclic for M , {aξ; a ∈ M} is dense in H, whence x = 0 holds.

Proof of (a). Applying Lemma 1.22 to M = R, we get that τ = ⟨ · ξτ , ξτ ⟩ is a faithful normal tracialstate on R.

We go to the proof of (b’). This follows from the next easy lemma:

Lemma 1.23. Let n ∈ N. Then 1nTr(·) is the only tracial state on Mn(C).

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Proof. Let τ be a tracial state on Mn(C). Let {eij ; 1 ≤ i, j ≤ n} be the standard system of matrixunits in Mn(C). That is, the (k, l)-component of eij is 1 precisely when (k, l) = (i, j) and 0 otherwise(1 ≤ k, l ≤ n). It is easy to see that they satisfy

eijekl = δjkeil,n∑

i=1

eii = 1, 1 ≤ i, j, k, l ≤ n.

Then we show that for each 1 ≤ i, j ≤ n, we have

τ(eij) =1

nδij .

Indeed, for each 1 ≤ i, j ≤ n, it holds that (use the trace property of τ)

τ(eii) = τ(eijeji) = τ(ejieij) = τ(ejj).

Therefore since e11 + · · ·+ enn = 1,

1 =n∑

k=1

τ(ekk) = nτ(eii), 1 ≤ i ≤ n.

On the other hand, if i = j, then

τ(eij) = τ(eijejj) = τ(ejjeij) = τ(δjiejj) = 0.

This shows that since each x ∈ Mn(C) is uniquely written as x =∑n

i,j=1 xijeij (xij ∈ C),

τ(x) =n∑

i,j=1

xijτ(eij) =1

n

n∑i=1

xii =1

nTr(x).

This shows that τ = 1nTr.

Proof of (b’): R has unique normal faithful tracial state. Let τ ′ be a normal tracial state on R. Weshow that τ = τ ′. Since τ, τ ′ are normal, it suffices to show that τ and τ ′ agree on the stronglydense *-subalgebra R0 =

∪n∈N M2(C)⊗n. But by Lemma 1.23, there is only one tracial state on

M2(C)⊗n ∼= M2n(C). Therefore τ |M2(C)⊗n = τ ′|M2(C)⊗n for every n ∈ N, which finishes the proof.

Proof of (b): R is a factor. By spectral theory (cf. Corollary 1.5)1, it can be shown that the centerZ(R) is generated by positive elements. So we have only to show that Z(R)+ = R+1. Let h ∈ Z(R)+be a nonzero positive element. By rescaling we may assume that τ(h) = 1, since τ is faithful by (a).Then since h is in the center, the new normal state τ ′ = τ(h·) = ⟨ · ξτ , hξτ ⟩ is tracial (and normal):

τ ′(xy) = τ(hxy) = τ(xhy)trace= τ(hyx) = τ ′(yx) (x, y ∈ R).

Therefore τ ′ = τ by (b’). This means that for every x ∈ R, one has

τ(hx) = τ(x) ⇔ ⟨x, h⟩ = ⟨x, 1⟩.

Since {x;x ∈ R} is dense in the GNS Hilbert space L2(R, τ), h = 1, which shows that h = 1. ThereforeZ(R)+ = R+1, and we are done.

Thus we finally see that R is a hyperfinite II1 factor.

Theorem 1.24 (Murray-von Neumann). Every hyperfinite II1 factor on a separable and infinite-dimensional Hilbert space is *-isomorphic to R.

1Z(R) is spanned by its self-adjoint elements. Decompose a self-adjoint element x = x∗ ∈ Z(R) as a difference of twopositive elements x = x+ − x−, x+x− = 0 using spectral decomposition. Then with a more argument using Corollary1.5 shows that x± ∈ Z(R)+.

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Hence R is the hyperfinite II1 factor.

Remark 1.25. (1) It is a nontrivial fact, that on a II1 factor M , every tracial state is automaticallynormal (and faithful). This does not hold if we drop either a factoriality ofM or tracial condition.

(2) It is a much more important and nontrivial fact, that a von Neumann subalgebra of the hyper-finite von Neumann algebra R is hyperfinite (see [Co76]). Therefore II1 subfactors of R are allisomorphic to R itself. This does not follow from Murray-von Neumann’s uniqueness theorem.

2 Talk 2 (Oct. 8 2014, 10:30-11:15)

We cover the following two topics.

(A) Standard representation

(B) Conditional expectation

(A) Standard representation

Let M be a von Neumann algebra with a faithful normal tracial state τ . Let (L2(M, τ) =L2(M), ξτ , πτ ) be the GNS representation of τ . Namely, (since τ is faithful, Nτ = {x ∈ M ; τ(x∗x) =0} = {0}), L2(M) is the completion of M with respect to

⟨a, b⟩τ = τ(b∗a), a, b ∈ M.

The canonical image of a ∈ M in L2(M) is denoted as a ∈ L2(M), and by definition M = {a; a ∈ M}is a dense subspace of L2(M). For each a ∈ M , π(a) is the uniquely bounded operator in L2(M)determined by the relation

πτ (a)b = ab, b ∈ M.

M ∋ a 7→ πτ (a) ∈ B(L2(M)) is an injective *-homomorphism. To see the injectivity, let πτ (a) = 0.Then 0 = ∥πτ (a)ξτ∥2 = τ(a∗a), whence a = 0 by the faithfulness of τ .

It takes little more work to show that (proof omitted)

Proposition 2.1. πτ is an injective normal *-homomorphism, and πτ (M) is a von Neumann algebraon L2(M) (i.e., πτ (M) is SOT-closed).

Therefore we identify M with πτ (M) and regard M ⊂ B(L2(M)). Because of this identification, itis very important to understand whether we regard a ∈ M as an operator a ∈ B(L2(M) or as a vectora ∈ L2(M). From now on, in order to avoid confusion, we use ∥ · ∥∞ to denote the operator norm (asa bounded operator on L2(M)) and ∥ · ∥2 to denote the Hilbert space norm on L2(M). Therefore fora ∈ M :

∥a∥∞ = sup∥ξ∥2≤1

∥aξ∥2, ∥a∥2 = τ(a∗a)12 .

In general, it is a nontrivial task to compute the commutant M ′ of a given von Neumann algebra.However, when M is represented in the special Hilbert space L2(M), the commutant is explicitly

described. To describe the commutant, let J be an operator M → M defined as

Ja := a∗, a ∈ M.

Note that ∥a∗∥22 = τ(aa∗) = τ(a∗a) = ∥a∥22, whence J is isometric on the dense subspace M of L2(M).Therefore J is extended to a conjugate-linear isometry J : L2(M) → L2(M) by continuity satisfyingJ ◦ J = idL2(M) (in particular, J is invertible). First we see that

Lemma 2.2. JMJ ⊂ M ′.

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Proof. Let a, b, c ∈ M . Then

JaJbc = Jac∗b∗ = Jac∗b∗ = bca∗,

bJaJc = bJac∗ = bca∗ = bca∗.

This shows that JaJb and bJaJ agree on the dense subspace M of L2(M). Since they are bothbounded, this shows that JaJb = bJaJ . Since b ∈ M is arbitrary, we have JaJ ∈ M ′. ThereforeJMJ ⊂ M ′.

Actually, the opposite inclusion holds:

Theorem 2.3. JMJ = M ′.

Proof. The proof goes in 3 steps.Step 1. Let x′ ∈ M ′. Then Jx′ξτ = (x′)∗ξτ .Given a ∈ M , we have (use x′a = ax′)

⟨Jx′ξτ , a⟩ = ⟨Ja, x′ξτ ⟩ = ⟨a∗, x′ξτ ⟩= ⟨(x′)∗a∗ξτ , ξτ ⟩ = ⟨a∗(x′)∗ξτ , ξτ ⟩= ⟨(x′)∗ξτ , aξτ ⟩ = ⟨(x′)∗ξτ , a⟩.

Since M is dense in L2(M), we have Jx′ξτ = (x′)∗ξτ .

Step 2. M ′ξτ = {x′ξτ ;x′ ∈ M ′} is dense in L2(M).

Let E = M ′ξτ . Then E is a closed subspace of L2(M). Let p = PE be the orthogonal projectionof L2(M) onto E. It suffices to show that p = 1 in order to prove the density. We first see thatE is invariant under M ′, whence E⊥ is also M ′-invariant. This shows that p commutes with everya′ ∈ M ′. Therefore by double commutant theorem, we have p ∈ (M ′)′ = M . Since 1 ∈ M ′, we knowthat ξτ ∈ E, and ξτ = pξτ . Therefore by p ∈ M , it holds that

0 = ∥(1− p)ξτ∥22 = τ((1− p)∗(1− p)).

This shows that 1− p = 0 by the faithfulness of τ , as desired.

Step 3. M ′ ⊂ JMJ .Let x′, y′, z′ ∈ M ′. Then by Step 1, we have

Jx′Jy′(z′ξτ ) = Jx′(z′)∗(y′)∗ξτ = y′z′(x′)∗ξτ

= y′Jx′(z′)∗ξτ = y′Jx′J(z′ξτ ).

By Step 2, M ′ξτ is dense. Therefore by boundedness of involved operators, Jx′Jy′ = y′Jx′J holds.Since y′ ∈ M ′ is arbitrary, we see (again by double commutant theorem) that

Jx′J ∈ (M ′)′ = M ⇔ x′ ∈ JMJ.

Therefore M ′ ⊂ JMJ . Together with Lemma 2.2, we have shown that JMJ = M ′.

As a consequence: we know by construction that M acts on L2(M) from the left by operatormultiplication. On the other hand, we can also define the right M -action on L2(M) by

ξ · x := Jx∗Jξ, x ∈ M, ξ ∈ L2(M)

By Theorem 2.3, these actions commute:

a · (ξ · b) = (a · ξ) · b, a, b ∈ M, ξ ∈ L2(M),

and they are the commutant of each other. It is (probably) important for Prof. Nest’s talks tomorrowto regard L2(M) as an M -M bimodule. Finally, let us see that any normal state on M ⊂ B(L2(M))is a vector state:

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Theorem 2.4. Let φ be a normal state on M ⊂ B(L2(M)). Then there is a vector ξφ ∈ L2(M)+ =

M+∥·∥2

such that φ = ⟨ · ξφ, ξφ⟩.

Unfortunately we need the following difficult result:

Theorem 2.5 (Powers-Størmer inequality). Let ξ, η ∈ L2(M)+. Then if we let ωζ = ⟨ · ζ, ζ⟩ (ζ ∈L2(M)+), one has

∥ξ − η∥22 ≤ ∥ωξ − ωη∥ ≤ ∥ξ − η∥2∥ξ + η∥2.

Proof of Theorem 2.4 assuming Theorem 2.5. Since φ is σ-weakly continuous, there exists a sequence(ξn)

∞n=1 ∈ ℓ2(L2(M)) with

∑∞n=1 ∥ξn∥2 = φ(1) = 1 such that

φ(x) =∞∑

n=1

⟨xξn, ξn⟩, x ∈ M.

Step 1. For every ε > 0, there exists b ∈ M+ such that ∥φ− ωb∥ < ε.

Fix ε > 0. Let φk(x) =∑k

n=1⟨xξn, ξn⟩ (x ∈ M,k ∈ N). Then since positive linear functional attainsits norm at 1,

∥φ− φk∥ =∞∑

n=k+1

∥ξn∥2k→∞→ 0.

Then fix k0 ∈ N such that ∥φ − φk0∥ < ε/2. Since M ⊂ L2(M), there exists an ∈ M (1 ≤ n ≤ k0)such that ∥an − ξn∥2∥an + ξn∥2 < ε/(2k0). Then by Theorem 2.5, we have∥∥∥∥∥φk0 −

k0∑n=1

ωan

∥∥∥∥∥ ≤k0∑n=1

∥ωξn − ωan∥

≤k0∑n=1

∥an − ξn∥2∥an + ξn∥2

< ε/2.

Also, for x ∈ M ,

k0∑n=1

ωan(x) =

k0∑n=1

τ(a∗nxan) = τ

(k0∑n=1

ana∗nx

)= τ(bxb) = ωb(x),

where b :=(∑k0

n=1 ana∗n

) 12 ∈ M+. Therefore ∥φ− ωb∥ < ε.

Step 2. By Step 1, there exists (bn)∞n=1 ⊂ M+ such that ∥φ− ω

bn∥ → 0 (n → ∞). By Theorem 2.5,

we have∥bn − bm∥22 ≤ ∥ω

bn− ω

bm∥ n,m→∞→ 0.

Therefore (bn)∞n=1 is a Cauchy sequence in L2(M)+, whence has a limit

ξφ = limn→∞

bn ∈ L2(M)+.

Again by Theorem 2.5,

∥ωξφ − φ∥ = limn→∞

∥ωξφ − ωbn∥

≤ limn→∞

∥ξφ − bn∥2∥ξφ + bn∥2

= 0,

whence φ = ωφ.

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(B) Conditional Expectations

Let N ⊂ M be an inclusion of von Neumann algberas, and let τ be a normal faithful tracial state onM . As in (A), we let M act on the standard Hilbert space L2(M) by left multiplication (hence N alsoacts on L2(M)). Let H = Nξτ . Then N acts on H and ξτ ∈ H is cyclic for the N -action. Moreover,τ |N (x) = ⟨xξτ , ξτ ⟩ for x ∈ N . Therefore by the uniqueness of the GNS construction, Theorem 1.19,we may identify H = L2(N, τ |N ) = L2(N) and therefore we have an inclusion L2(N) ⊂ L2(M).

Definition 2.6. We define eN to be the orthogonal projection of L2(M) onto L2(N).

Theorem 2.7 (Trace-preserving conditional expectation). There exists a positive linear mapEN : M → N satisfying

(1) eN (x) = EN (x), x ∈ M .

(2) EN (axb) = aEN (x)b, a, b ∈ N, x ∈ M .

(3) τ(EN (x)) = τ(x), x ∈ M .

(4) eNxeN = EN (x)eN , x ∈ M .

EN is faithful and normal.

Remark 2.8. A positive linear map E : M → N satisfying (2)(3) is unique = EN .

Definition 2.9. EN in Theorem 2.7 is called the trace-preserving conditional expectation of M ontoN with respect to τ .

EN will (probably) play an important role in Prof. Nest’s talks. In order to construct EN , we usethe following useful lemma. Recall that M acts on L2(M) from the right by ξ · x = Jx∗Jξ. We canalso think of this as an “left action of ξ on x” (though ξ is a vector, not an operator).

Lemma 2.10. Let M be a von Neumann algebra with a faithful normal tracial state. Then forξ ∈ L2(M), the following two conditions are equivalent.

(1) ξ is left-bounded. That is, there exists a constant c > 0 such that

∥ξ · x∥2 ≤ c∥x∥2, x ∈ M.

(2) ξ ∈ M . That is, there exists a ∈ M such that ξ = a.

Proof. (2)⇒(1) Let ξ = a ∈ M (a ∈ M). Then for x ∈ M , we have

∥ξ · x∥2 = ∥Jx∗Jaξτ∥2 = ∥aJx∗Jξτ∥2 = ∥ax∥2≤ ∥a∥∞ · ∥x∥2,

whence (1) holds with c = ∥a∥ (operator norm of a).

(1)⇒(2) Assume that (1) holds. Then the linear map Lξ : M → L2(M) given by

Lξ(x) := ξ · x, x ∈ M

extends to a bounded linear map Lξ ∈ B(L2(M)) (with operator norm ∥Lξ∥∞ ≤ c). We show thata := Lξ ∈ M . To see this, let x, y ∈ M . Then

LξJxJy = Lξ yx∗ = Jxy∗Jξ = JxJ(Jy∗Jξ) = JxJLξ y.

Since Lξ, JxJ are both bounded, density of M in L2(M) implies that Lξ and JxJ commute. ByTheorem 2.3 and double commutant theorem, we see that

a = Lξ ∈ (JMJ)′ = (M ′)′ = M.

Thereforeξ = Lξ(1) = a ∈ M.

This finishes the proof.

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Proof of Theorem 2.7. (1): In view of Lemma 2.10, we show that eN (x) is a left-bounded vector inL2(N). Let y, z ∈ N . Then

⟨Jy∗JeN (x), z⟩L2(N) = ⟨eN (x), JyJz⟩L2(N) = ⟨x, eN ( zy∗︸︷︷︸∈N⊂L2(N)

)⟩

= ⟨x, zy∗⟩L2(M) = ⟨x, JyJz⟩L2(M)

= ⟨Jy∗Jx, z⟩L2(M) = ⟨xy, z⟩L2(M)

Therefore|⟨Jy∗JeN (x), z⟩L2(N)| ≤ ∥xy∥2∥z∥2 ≤ ∥x∥∞∥y∥2∥z∥2.

This shows that∥Jy∗JeN (x)∥2 ≤ ∥x∥∞∥y∥2.

Hence eN (x) is left-bounded. By Lemma 2.10, there exists EN (x) ∈ N such that eN (x) = EN (x). Itis clear that EN : M → N is linear. We prove the positivity of EN later.(2) We first note that J preserves L2(N), and JeN = eNJ . Therefore for each x ∈ M , we have

JeN (x) = eNJx ⇔ EN (x)∗ = EN (x∗).

ThereforeEN (x)∗ = EN (x∗). (3)

We next show see that EN (ax) = aEN (x) for a ∈ N and x ∈ M . To see this, let b ∈ N . Then

⟨EN (ax), b⟩L2(N) = ⟨eN (ax), b⟩L2(N) = ⟨ax, eN (b)︸ ︷︷ ︸=b

⟩L2(M)

= ⟨x, a∗b⟩L2(M) = ⟨x, eN (a∗b)⟩L2(M)

= ⟨eN (x), a∗b⟩L2(N) = ⟨aeN x, b⟩L2(N)

= ⟨aEN (x), b⟩L2(N).

Therefore by the density of N in L2(N), EN (ax) = aEN (x) hold. Replacing a by b∗ (b ∈ N) and xby x∗ and use (3) to obtain EN (xb) = EN (x)b. This shows (2).(3): This is straightforward, since for every x ∈ M ,

τ(EN (x)) = ⟨EN (x), 1⟩ = ⟨eN (x), 1⟩= ⟨x, eN (1)⟩ = ⟨x, 1⟩= τ(x).

(4): Let x, y ∈ M . Then by (2), we have (use b = EN (y) ∈ N)

eNxeN y = eNxEN (y) = eN xEN (y) = EN (xEN (y))

= EN (x)EN (y) = EN (x)eN (y).

Therefore by the density of M in L2(M), eNxeN = EN (x)eN holds.Finally, we show that EN preserves positivity. Let x ∈ M+. Then for each y ∈ N ,

⟨EN (x)y, y⟩ = ⟨y∗EN (x)yξτ , ξτ ⟩ = ⟨EN (y∗xy)ξτ , ξτ ⟩ = τ(EN (y∗xy))(3)= τ(y∗xy) ≥ 0.

Since y ∈ N is arbitrary, the density of N in L2(N) shows that ⟨EN (x)ξ, ξ⟩ ≥ 0 holds for ξ ∈ L2(N),whence EN (x) ∈ N+. Let x ∈ M be such that EN (x∗x) = 0. Then by (3),

τ(x∗x) = τ(EN (x∗x)) = 0.

This shows that x = 0, by the faithfulness of τ . Normality of EN requires more effort, but it followsfrom the fact that EN preserves the normal faithful trace τ ◦EN = τ (we do not discuss it here).

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Definition 2.11. The von Neuman algebra on L2(M) generated by M and {eN} is denoted as⟨M, eN ⟩, called the Jones’ basic construction, and eN is called Jones projection.

It can be checked directly, that N = M ∩ {eN}′ and ⟨M, eN ⟩ = JN ′J (but we have no time todiscuss it here!).

Appendix: dimM H (Oct. 9 2014)

Let us add one more tool, the von Neumann dimension dimMH, that will be needed but could notbe covered in the last two talks. Since we need the notion of semifinite trace (which we have notdefined), this appendix is not entirely self-contained. Let M be von Neumann algebra with a normalfaithful tracial state τ . We consider the representation theory of M . We denote the opposite algebraof M as Mop. That is, Mop = {xop;x ∈ M} is a copy of M as a set, equipped with the same linearspace and *-operation, but the product is reversed:

xopyop = (yx)op, xop, yop ∈ Mop.

Definition 2.12. A left M -module is a Hilbert spaceH with the leftM -action. That is, H is equippedwith a normal *-homomorphism π : M → B(H) and the left M -action is given by

x · ξ := π(x)ξ, x ∈ M, ξ ∈ H.

Similarly, a right M -module is a Hilbert space K with the right M -action, i.e, a normal *-homomorphism ρ : Mop → B(K) and we write

ξ · x := ρ(xop)ξ, x ∈ M, ξ ∈ K.

We always assume that the underlying *-homomorphisms for left/right actions are faithful. Givenanother von Neumann algebra N , an M -N bimodule is a Hilbert space H which is both a left M -module and a right N -module structure in such a way that two actions commute:

x · (ξ · y) = (x · ξ) · y, x ∈ M,y ∈ N, ξ ∈ H.

Example 2.13. L2(M) is a standard example of an M -M bimodule.

Given left M -modules H1,H2, we say a bouded linear map T ∈ B(H1,H2) is called M -linear, if Tcommutes with M -actions:

T (a · ξ) = a · (Tξ), a ∈ M, ξ ∈ H1.

Denote by MB(H1,H2) the space of all bounded M -linear maps from H1 to H2. We say that H1 andH2 are isomorphic as a left M -module, if there is a unitary u ∈ MB(H1,H2). We also use the notation

MB(H1) := MB(H1,H1).

Lemma 2.14. Two left M -modules H1 and H2 are isomorphic, if and only if there is an invertibleelement T ∈ MB(H1,H2).

Proof. Let T = u|T | be the polar decomposition of T . That is, |T | = (T ∗T )12 ∈ MB(H1) is a positive

self-adjoint operator and u ∈ MB(H1,H2) is a partial isometry which maps Ker(T )⊥ ⊂ H1 ontoRan(T ) ⊂ H2. Therefore u is a unitary if and only if T is invertible. The conclusion follows fromthis.

Definition 2.15. Let H be a left M -module. A M -submodule is a closed subspace K that is stableunder the left M -action.

Theorem 2.16. Let M be a II1 factor with the faithful normal tracial state τ , and let H be a separableleft M -module. Then H is isomorphic to a M -submodule of L2(M)⊗ ℓ2(N). Here, the left M -actionon L2(M)⊕∞(∼= L2(M)⊗ ℓ2(N)) is given by

a · (ξn)∞n=1 = (aξn)∞n=1, a ∈ M, (ξn)

∞n=1 ∈ L2(M)⊕.

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Proof. First assume that there is a cyclic vector ξ ∈ H for the left M -action (i.e., Mξ is dense in H).Then φ = ωξ = ⟨ · ξ, ξ⟩ is a normal state on M . Then by Theorem 2.4, there exists ξφ ∈ L2(M)+such that φ = ωξφ . Then the map V : Mξ → Mξφ given by V (xξ) := xξφ exntends to an isometryV : H → L2(M). It is clear that V commutes with left M -actions on H and 2(M). ThereforeV ∈ MB(H,L2(M)). Therefore by Lemma 2.14, H is isomorphic to the M -submodule V (H) ofL2(M).

Now let H be a general left M -module. Since we assume that H is separable, there exists (useZorn’s lemma) a coutable family (ξn)n∈I of nonzero vectors in H such that Hi = Mξi (i ∈ I) satisfies

Hi ⊥ Hj (i = j), H =⊕i∈I

Hi.

Clearly, each Hi is a left M -submodule of M with a cyclic vector ξi. Thus there exists by the firststep a left M -linear isometry Vi : Hi → L2(M). We then see that V : H =

⊕i∈I Hi → L2(M)⊕I ⊂

L2(M)⊕∞,V (ηi)i∈I = (Viηi)i∈I , (ηi)i∈I ∈ H

defines an isomorphism of H onto V (H) ⊂ L2(M)⊕∞ as a left M -module.

Let H be a left M -module. Then by Theorem 2.16, there is an M -linear isometry V : H →L2(M)⊕∞. We have V V ∗ ∈ MB(L2(M)⊕∞). If we identify L2(M)⊕∞ = L2(M) ⊗ ℓ2(N), thenB(L2(M)⊕∞) = (M ′⊗B(ℓ2)). So the projection p′ = V V ∗ is considered as an infinite matrix p′ = [p′ij ]with entries in p′ij ∈ M ′ = JMJ . If we set pij = Jp′ijJ ∈ M (i, j ∈ N), we see that as a left M -module,

H ∼= p′(L2(M)⊗ ℓ2(N)) = L2(M)⊗ ℓ2(N)p.

Definition 2.17. Define the von Neumann dimension dimMH of H as

dimMH :=∞∑

n=1

τ(pnn) = τ ′ ⊗ Tr(V V ∗).

Here, τ ′ is the unique tracial state on the II1 factor JMJ = M ′ given by τ ′(x′) := τ(Jx′J) (x′ ∈ M ′)and τ ′ ⊗ Tr([x′

ij ]) =∑∞

n=1 τ′(xnn) give us a (what is called, whose definition we omit) semifinite,

normal and faithful tracial weight on the II∞ factor (definition omitted, or think of the next as thedefinition) M ′⊗B(ℓ2(N)).

Note that dimM H does not depend on the choice of an M -linear isometry V . If W : H →L2(M)⊗ ℓ2(N) is another such M -linear isometry, then since τ ′ is a trace,

τ ′ ⊗ Tr(V V ∗) = τ ′ ⊗ Tr(VW ∗WV ∗) = τ ′ ⊗ Tr(WV ∗ · VW ∗)

= τ ′ ⊗ Tr(WW ∗).

Definition 2.18. Given an inclusion of II1 factors N ⊂ M , the Jones index, denoted [M : N ], is thenumber [M : N ] = dimN (L2(M)).

Please consult [JS97] for details.

References

[Ar76] W. Arveson, An Invitation to C*-Algebras (Graduate Texts in Mathematics), Springer (1976).

[Co76] A. Connes, Classification of injective factors, Ann. Math. vol. 104 no. 1 (1976), 73–115.

[JS97] V. F. R. Jones and V. S. Sunder, Introduction to subfactors, London Mathematical SocietyLecture Note Series, vol. 234, Cambridge University Press (1997).

[Pe14] D. Pennys, Introduction to subfactors (May 2014), notes that I found on the internet andfound helpful to get an overview of subfactors:http://www.math.ucla.edu/ dpenneys/PenneysNCGOA2014.pdf

[Su86] V. S. Sunder, An Invitation to von Neumann Algebras (Universitext), Springer (1986).

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