NARUTAKA OZAWA

Abstract. This treatise is based on the lecture given by Narutaka Ozawa at
the University of Tokyo during the winter semester 2006-2007. It includes an
elementary theory of orbit equivalence via type II1 von Neumann algebras,
L¨uck’s dimension theory [6] and its application toL^{2} Betti numbers [5], con-
vergence of the semigroup associated to a derivation and a theorem of Popa
on embeddability of subalgebras.

Contents

1. Introduction 1

1.1. Orbit equivalence 1

1.2. Preliminaries on von Neumann algebras 3

1.3. Crossed products 4

1.4. von Neumann algebras of orbit equivalence 4

2. Elementary theory of orbit equivalence 5

2.1. Essentially free action of countable discrete groups 5

2.2. Inclusion of von Neumann algebras 6

2.3. Theorem of Connes-Feldman-Weiss 9

3. L^{2}-Betti numbers 12

3.1. Introduction 12

3.2. Operators affiliated to a finite von Neumann algebra 14 3.3. Projective modules over a finite von Neumann algebra 15

3.4. Application to orbit equivalence 18

4. Derivations on von Neumann algebras 21

4.1. Densely defined derivations 21

4.2. Semigroup associated to a derivation 24

Appendix A. Embeddability of subalgebras 29

References 31

1. Introduction 1.1. Orbit equivalence.

Definition 1.1. LetY be a topological space,B_{Y} theσ-algebra of the Borel sets of
Y. WhenY is a separable complete metric space, (Y, BY) (or, by abuse of language,
Y) is said to be a standard Borel space (standardσ-algebra).

Remark 1.2. WhenX is a standard Borel space, X is either (at most) countable or isomorphic to the closed interval [0,1].

2000Mathematics Subject Classification. 46L10;37A20.

1

Definition 1.3. A standard Borel space with a Borel probability measure is said to be a (standard) probability space. A pointxof a probability space (X, µ) is said to be an atom of (X, µ) when µ(x)>0. A probability space (X, µ) is said to be diffuse when it has no atom.

Example 1.4. (Examples of standard probability spaces) (1) The infinite product (Q

n∈N{0,1},⊗nµn), where µn is a probability mea- sure on{0,1} for eachn∈Nis standard.

(2) WhenGis a separable compact group, the normalized Haar measure onG makesGinto a standard probability space.

When (X, µ) is a probability space, we obtain a (w^{∗}-) separable von Neumann
algebra L^{∞}X and a normal state (also denoted byµ) on it. To each isomorphism
φ: (X, µ) →(Y, ν) of probability spaces, we obtain an isomorphism φ_{∗}:L^{∞}Y →
L^{∞}X, f 7→f◦φsatisfyingµ◦φ^{∗}=ν.

Theorem 1.5. (von Neumann)

(1) When (X, µ) and (Y, ν) are diffuse probability spaces, there is an isomor-
phism (L^{∞}(X, µ), µ)'(L^{∞}(Y, ν), ν).

(2) For each isomorphism σ:L^{∞}Y →L^{∞}X with µσ=ν, there exists a Borel
isomorphismφ:X→Y such thatφ^{∗}µ=ν andφ∗=σ.

Proof. (Outline): (1) We may assume thatY =Q

N{0,1}, µ=⊗_{N}(^{1}_{2},^{1}_{2}). SinceX
is diffuse, we have a decompositionX =X0`

X1by Borel sets withµ(X0) =^{1}_{2}. We
can continue this procedure as X0=X00`

X01, µ(X00) = ^{1}_{4}, so on. The partition
byX_{∗∗···}can be made fine enough because there is a separating family (Bn)_{n∈N} in
BX, which will imply the desired isomorphism betweenL^{∞}X andL^{∞}Y compatible
with the normal states.

(2) Letλdenote the Lebesgue measure on the closed interveal [0,1]. Since there
exists an isomorphism (L^{∞}Y, ν) ' (L^{∞}[0,1], λ), we may assume that Y = [0,1]

and ν =λ here. For each r ∈ Q∩[0,1], put E_{r} = σ(χ_{[0,r)}). Define a mapping
φ:X → [0,1] by φ(x) = inf{r:x∈E_{r}}. The inverse image of [0, t) under φ is
equal to ∪r<tE_{r}. The latter is obviously Borel, which means that φ is a Borel
map. By σ(χ_{[0,r)}) = φ^{∗}(χ_{[0,r)}) for r ∈ Q∩[0,1], we have σ = φ^{∗} and φ_{∗}µ =
Lebesgue measure.

It remains to replace φ by a Borel isomorphism. Let (B_{n})_{n∈}_{N} be a separating
family of X. For each n, there exists Fn ∈ BY such that φ_{∗}χF_{n} = χB_{n}. Thus
N =∪nBn4φ^{−1}Fn is a null set. OnX\N, the conditionx∈Bn is equivalent to
φ(x)∈Fn. If xand y are distinct points ofX \N, there exists an integer nsuch
thatx∈Bn whiley6∈Bn. Thusφ(x)6=φ(y) andφis injective onX\N. We may
assume thatN andY \φ(X\N) are uncountable so that there is an isomorphism

ofN toY \φ(X\N).

Let Γy(X, µ) be a measure preserving action by a discrete countable group.

(We may assume that it acts by Borel isomorphisms.) Let s be an element of Γ.

Whenf is a complex Borel function defined onX, putαs(f) :x7→f(s^{−1}x). This
induces a µ-preserving ∗-automorphism on L^{∞}X. This way we obtain an action
α: ΓyL^{∞}(X, µ) preserving the stateµ.

Definition 1.6. Two actions Γy(X, µ) and Γy(Y, ν) are said to be conjugate when there exists an probability space isomorphismφ: (X, µ)→(Y, ν) witch is a.e.

Γ-equivariant. This is equivalent to the existence of a Γ-equivariant state preserving
isomorphismσ:L^{∞}(Y, ν)→L^{∞}(X, µ).

Definition 1.7. Let Γ y (X, µ) be an action by measure preserving Borel iso- morphisms. The subsetRΓy(X,µ) ={(sx, x) :s∈Γ} ofX×X is called the orbit equivalence relation of the action.

Definition 1.8. Two actions Γ y (X, µ) and Λ y (Y, ν) are said to be orbit equivalent when there exists a measure preserving Borel isomorphism φ:Y →X satisfying Γφ(y) =φ(Λy) for a.e. y∈Y.

Definition 1.9. A partial Borel isomorphism onX is a triple (φ, A, B) consisting ofA, B∈BX and a Borel isomorphismφofA ontoB.

Definition 1.10. A measure preserving standard orbit equivalence is a subsetR ofX×X satisfying the following conditions:

(1) R is a Borel subset with respect to the product space structure.

(2) R is an equivalence relation onX.

(3) For eachx∈X, theR-equivalence class ofxis at most countable.

(4) Any partial Borel isomorphismφwhose graph is contained inR,φpreserves measure.

Theorem 1.11. (Lusin) Let X, Y be standard spaces.

(1) When φ: X →Y is a countable-to-one Borel map, φ(X) is Borel. More-
over there exists a Borel partition X = `X_{n} such that φ|Xn is a Borel
isomorphism ontoφ(X_{n}).

(2) When R is a standard orbit equivalence, R = ∪_{n}G(φ_{n}) where φ_{n} is a
partial Borel isomorphism for eachn.

Lemma 1.12. LetAbe a subset of a standard spaceX,φa mapping ofAintoX.

φ andA are Borel if and only if the graphG(φ) ={(φx, x) :x∈A} ofφ is Borel inX×X.

Proof. ⇐is an immediate consequence of Theorem 1.11.

⇒: Let (Bn)_{n∈N} be a separating family ofX. The conditiony6=φ(x) is equiv-
alent to (y, x)∈ ∪n({Bn)×φ^{−1}(Bn). Thus G(φ) ={(∪({Bn)×φ^{−1}(Bn)).

1.2. Preliminaries on von Neumann algebras. Let H be a Hilbert space, B(H) the involutive Banach algebra of the continuous endomorphisms of H, A a ∗-subalgebra of B(H). (typically A generates a von Neumann algebra M of our interest.) In the followingA is often assumed to admit a cyclic tracial vector ξτ∈H, i.e. kξτk= 1,Aξτ is dense inH, and that the vector stateτ(a) =haξτ, ξτi is tracial.

Remark 1.13. A state τ is tracial means that by definition the two sesquilinear
formsτ(ab^{∗}) andτ(b^{∗}a) in (a, b) are same. To check this property, by polarization
it is enough to show τ(aa^{∗}) =τ(a^{∗}a). Under the assumption aboveξ_{τ} becomes a
separating vector for A^{00}. Indeed,aξ_{τ} = 0 implies τ(bc^{∗}a) = 0 for b, c∈A, which
meansτ(c^{∗}ab) = 0 and in turnhaH, Hi= 0.

Notation. Let ˆadenoteaξτ. (Hence we havehˆa,ˆbi=τ(ab^{∗}).)

Remark 1.14. We have a conjugate linear mapJ:H →H determined by ˆa7→ab^{∗}.
Then we haveJ aJˆb=bac^{∗} which impliesJ AJ ⊂A^{0} andJ A^{00}J ⊂A^{0}. On the other
hand, for anyx∈A^{0} anda∈A

hJ xξτ, aξ_{τ}i=hJ aξτ, xξ_{τ}i=ha^{∗}ξ_{τ}, xξ_{τ}i=hx^{∗}ξ_{τ}, aξ_{τ}i.

ThusJ xξτ =x^{∗}ξτ, thence ξτ is a cyclic tracial vector forA^{0}. The J-operator for
(A^{0}, ξ_{τ}) is exactly equal to the originalJ. Doing the same argument as above, we
obtainJ A^{0}J ⊂A^{00}.

Remark 1.15. The mapA^{00}→A^{0}, a7→J aJ is a conjugate linear∗-algebra isomor-
phism.

1.3. Crossed products. Let Γ y (X, µ) be a measure preserving action of a
discrete group on a standard probability space X. Recall that we have an action
ΓyL^{∞}X induced byαs(f) =f(s^{−1}−) fors∈Γ.

On the other hand, we get a unitary representationπ: ΓyL^{2}(X, µ) given by
the same formula πsf = αsf as the one on L^{∞}X. Note that πsf π_{s}^{∗} = αs(f) for
s∈Γ and f ∈L^{∞}X.

Definition 1.16. Letλ: ΓyB(`_{2}Γ) denote the regular representation. The von
Neumann algebraL^{∞}XoΓ onL^{2}(X)⊗`_{2}Γ is generated by the operatorsπ⊗λ(s)
fors∈Γ and f⊗1 forf ∈L^{∞}X is called the crossed product ofL^{∞}X byα.

LetAdenote{P

finitefs⊗1·π⊗λ(s)} ⊂L^{∞}XoΓ. By abuse of notation, in the
followingf stands forf⊗1 andλ(s) forπ⊗λ(s). Nowξ_{τ} =1⊗δ_{e}∈L^{2}X⊗`_{2}Γ is a
cyclic tracial vector forA. Indeed, it is obviously cyclic, whileτ(f λ(s)) =δ_{e,s}µ(f)
implies the tracial property:

τ(f λ(s)gλ(t)) =δ_{st,e}f α_{s}(g) =δ_{ts,e}α_{t}(f)g=τ(gλ(t)f λ(s)).

Note that the above expressions are nonzero only ifs=t^{−1}.

Let V denote the isometry L^{2}(X) → L^{2}(X)⊗`_{2}Γ, f 7→ f ⊗δ_{e}. Then the
contractionE:L^{∞}X oΓ →B(L^{2}(X)), a 7→ V^{∗}aV has imageL^{∞}X, i.e. E is a
conditional expectation (see Definition 2.6) of L^{∞}X oΓ onto L^{∞}X. Note that
τ=µ◦E.

1.4. von Neumann algebras of orbit equivalence. LetRbe a standard orbit equivalence on X. Hence it is a countable disjoint union `

nG(φn) of the graphs of partial isometries. We may assume that φ0 = IdX. We will define a “Borel probability measure” onR.

Observe that when f:R →C is a Borel function, X →C, x7→P

yf(y, x) = P

nf(φ_{n}x, x) is also Borel. Define a measureν onR by putting
Z

R

ξdν= Z

X

X

yRx

ξ(y, x)dµ(x)

for each Borel function ξ on R. Thus when B is a Borel subset of R, ν(B) =
R|π^{−1}_{r} (x)∩B|dµ(x) for the second projectionπ_{r}:R→X, (y, x)7→x.

We get a pseudogroupJRKwhose underlying set is

{φ: partial Borel isomorphism,G(φ)⊂R}.

The composition φ◦ψ of φ and ψ is defined as the composition of the maps on
ψ^{−1}dom(φ). In particular, the identity maps of the Borel sets are the units ofJRK,
andφ∈JRKimpliesφ^{−1}∈JRK.

For each φ ∈ JRK, define a partial isometry vφ ∈ B(L^{2}(R, ν)) byvφξ(y, x) =
ξ(φ^{−1}y, x). Thus vφvψ = v_{φ◦ψ}. On the other hand, the set

χ_{G}_{(φ)}:φ∈JRK is
total inL^{2}(R, ν) andvφχ_{G}ψ=χ_{G}_{φ◦ψ}. Moreover, we have

hvφχ_{G}ψ, χ_{G}θi=
Z

G(φψ)∩G(θ)dν=µ{x:φψx=θx}=hχ_{G}ψ, v_{φ}^{−1}χ_{G}θi,
which impliesv^{∗}_{φ}=v_{φ}−1.

Definition 1.17. The von Neumann algebravNRonL^{2}(R, ν) generated by{vφ:φ∈JRK}
is called the von Neumann algebra ofR.

ξτ =χ_{G}(Id_{X})is a cyclic tracial vector forvNR: in fact,
τ(v_{φψ}) =µ({x:φ◦ψ(x) =x})

=µ({y:ψφy=y}) (y=φ^{−1}x)

=τ(vψφ).

Note that L^{∞}X is contained “in the diagonal” of vNR, subject to the relation
vφf = (f◦φ^{−1})vφ. We have a conditional expectationE:vNR→L^{∞}X,a7→V^{∗}aV
implemented by the “diagonal inclusion” isometry V: L^{2}X → L^{2}R. We have
E(vφ) =χ_{{x:φx=x}}.

2. Elementary theory of orbit equivalence

2.1. Essentially free action of countable discrete groups. Suppose we are given a measure preserving action Γ y(X, µ) by a discrete group on a standard probability space. As in the last section we get two inclusions of von Neumann algebras:

(1) L^{∞}X ⊂L^{∞}XoΓ inB(L^{2}X⊗`_{2}Γ).

(2) L^{∞}X ⊂vN(RΓy(X,µ)) in B(L^{2}R).

In general these are different, e.g. when the action is trivial.

Definition 2.1. An action Γy(X, µ) is said to be essentially free when the fixed point set ofshas measure 0 for any s∈G\ {e}.

Theorem 2.2. When the action Γ y (X, µ) is essentially free, the above two inclusions of von Neumann algebras are equal.

Remark 2.3. Jvˆφ=vφˆ^{−1} impliesJ ξ(x, y) =ξ(y, x).

Proof of the theorem. Identification of the representation Hilbert spaces is given by
U:L^{2}X⊗`2Γ→L^{2}R,g⊗δt7→g·χ_{G}_{(t)}. When we have an equalityf χ_{G}_{(s)}=gχ_{G}_{(t)}
of nonzero vectors inL^{2}R,smust be equal totby the essential freeness assumption.

Now,

U^{∗}v_{s}U(g⊗δ_{t}) =U^{∗}α_{s}(g)v_{s}χ_{G}_{(t)}=U^{∗}α_{s}(g)χ_{G}_{(st)}=α_{s}(g)⊗δ_{st}.

This showsU^{∗}vsU =π⊗λ(s). On the other hand,U^{∗}f U =f ⊗1 is trivial. Thus,

viaU,L^{2}XoΓ is identified toL^{2}R.

Definition 2.4. Let M be a finite von Neumann algebra, A a von Neumann
subalgebra (in the following A is often assumed to be commutative). The sub-
set NA = {u∈ UM :uAu^{∗}=A} of UM is called the normalizer of A. Likewise
N^{p}A={v∈M : partial isometry,v^{∗}v, vv^{∗} ∈A, vAv^{∗}=Avv^{∗}}is called the partial
normalizer ofA.

Lemma 2.5. For any v ∈ N^{p}A, there exist u∈ NA and e ∈Proj(A) such that
v = ue. For any φ ∈ JRK, there exists a Borel isomorphism φ˜ whose graph is
contained inR andφ|˜domφ=φ.

Proof. We prove the second assertion as the demonstration of the first one is an
algebraic translation of it. Put E = domφand F = ranφ. Whenµ(E4F) = 0,
there is nothing to do. Whenµ(E4F)6= 0,∃k >0 such thatφ^{k}(E\F)∩(F\E)
is non-null. If not, φ^{k}(E\F)⊂F∩{(F \E) =F∩E ⊂E up to a null set and
φ^{k+1}can be defined a.e. on E\F. Thus we would get a sequence (φ^{k}(E\F))_{k∈}_{N}
of subsets with nonzero measure. For any pair m < n, φ^{m}(E\F)∩φ^{n}(E\F) is
equal toφ^{m}(φ^{n−m}(E\F)∩(E\F)) which is null. This contradicts toµ(X) = 1.

Now, given such k, put φ_{1} = φ`(φ^{−k}|_{φ}k(E\F)∩(F\E)). Then we can use the
maximality argument (Zorn’s lemma) to obtain a globally defined Borel isomor-

phism.

2.2. Inclusion of von Neumann algebras.

Definition 2.6. LetM ⊂N be an inclusion of von Neumann algebras. A unital completely positive mapE :N →M is said to be a conditional expectation when it satisfiesE(axb) =aE(x)b fora, b∈M andx∈N.

Fact. WhenN is finite with a faithful tracial stateτ, there exists a unique con-
ditional expectation Ethat preservesτ. Then we obtain an orthogonal projection
eM :L^{2}N→M ξτ 'L^{2}M extendingE.

Remark 2.7. (Martingale) If we are given N_{1} ⊂ N_{2} ⊂ · · · ⊂ M with N = ∨iN_{i}
or M ⊃ N_{1} ⊃ N_{2} ⊃ · · · with N = ∩_{i}N_{i}, together with conditional expectations
E_{n}:M → N_{n} and E:M → N, e_{n} → e in the strong operator topology implies
kE(x)−E_{n}(x)k_{2}→0.

For example, let A ⊂ M be a finite dimensional commutative subalgebra, ei

(1 ≤i ≤ n) the minimal projections of A. Then EA^{0}∩M(x) = Pn

i=1eixei. If we
have a sequenceA1⊂A2⊂ · · · ⊂M of finite dimensional commutative subalgebras
andA=∨Ai, we haveEA^{0}_{n}∩M →EA^{0}∩M. The latter is equal toEA if and only if
Ais a maximal abelian subalgebra.

Definition 2.8. A von Neumann subalgebraA⊂M is said to be a Cartan subal-
gebra ofM when it is a maximal abelian subalgebra inM andN(A)^{00}=M. (Then
we also haveM =N^{p}(A)^{00}.)

Theorem 2.9. L^{∞}X ⊂vNR is a Cartan subalgebra.

Proof. Since the generatorsv_{φ} are inNA, it is enough to show thatL^{∞}X is max-
imal abelian invNR. Recall that R=`G(φ_{n}) with φ_{0} = Id_{X}. Then let abe an

element of the relative commutant ofL^{∞}X. ˆacan be written asP

nfnχ_{G}(φ_{n}). By
assumptionf a=af for anyf ∈L^{∞}X. Thus,

cf a=X

f fnχ_{G}_{(φ}_{n}_{)}, afc=Jf J¯ ˆa=X

f◦φ^{−1}_{n} ·fnχ_{G}_{(φ}_{n}_{)}.

Hence f f_{n} = f ◦ φ_{n}f_{n} for any n and any f, which implies f_{n} = 0 except for

n= 0.

Definition 2.10. R is said to be ergodic when anyR-invariant Borel subset ofX is of measure either 0 or 1. An action Γy(X, µ) is said to be ergodic whenRΓyX

is ergodic.

Corollary 2.11. vNR is a factor if and only ifR is ergodic.

Proof. The Cartan subalgebraL^{∞}X contains the center ofvNR. The central pro-
jections are the characteristic functions of theR-invariant Borel subsets.

Letv ∈ N^{p}L^{∞}, E, F ∈BX the Borel sets (up to null sets) respectively repre-
senting the projections v^{∗}v and vv^{∗} in A. The map L^{∞}E → L^{∞}F, f 7→ vf v^{∗}e
is a∗-isomorphism. Thus there exists a Borel isomorphismφ_{v}:E →F such that
vf v^{∗} =f ◦φ^{−1}_{v} . (v=σv_{φ}_{v} for some σ∈ UL^{∞}F.)

Theorem 2.12. In the notation as above, vξv^{∗} = ξ(φ^{−1}_{v} (y), x) ν-a.e. for any
v∈ N^{p}L^{∞}and anyξ∈L^{∞}R. In particular, φv∈JRKup to a null set. Moreover,
we have L^{∞}∨J L^{∞}J=L^{∞}R.

Proof. Put A = L^{∞}X. First, f J gJ ∈ L^{∞} for f, g ∈ A: indeed, f J gJ is the
multiplication by the functionf(y)g(x) onR.

vf J gJ v^{∗}=vf v^{∗}J gJ=f◦φ^{−1}_{v} J gJ (J M J =M^{0}).

Hence vξv^{∗}(y, x) = v(φ^{−1}_{v} y, x) for ξ ∈ A∨J AJ. It remains to show χ_{G}_{(Id}_{X}_{)} ∈
A∨J AJ. Because, if this is satisfied, we will have χ_{G}_{(φ}_{v}_{)}=vχ_{G}_{(Id)}v^{∗}∈L^{∞}R.

Take an increasing sequenceA1 ⊂A2 ⊂ · · · of finite dimensional algebras with A=∨Ak. The conditional expectation En:vNR →An is equal toP

ke^{(n)}_{k} J e^{(n)}_{k} J
(as an operator on L^{2}R) for the minimal projections (e^{(n)}_{k} )_{k} of A_{n}. Now, (E_{n})_{n}
converges to the conditional expectationE_{A} ontoAwhich is equal to the multipli-
cation byχ_{G}_{(Id}_{X}_{)}in the strong operator topology. Henceχ_{G}_{(Id)}∈A∨J AJ.
Remark 2.13. (2-cocycle [4]) Suppose we are given a map σ_{φ,ψ}: ran(φψ)→Tfor
each pair φ, ψ ∈ JRK, satisfying σ_{φ,ψ}σ_{φψ,θ} = (σ_{ψ,θ}◦φ^{−1})σ_{φ,ψθ}. Then v_{φ}^{σ}v^{σ}_{ψ} =
σ_{φ,ψ}v^{σ}_{φψ} determines an associative product on CJRK with a trace τ. The GNS
representation gives an inclusion L^{∞}X ⊂ vN(R, σ) ⊂ B(L^{2}R) of von Neumann
algebras.

Fact. Any Cartan subalgebra ofvN(R, σ) is isomorphic toL^{∞}X.

Theorem 2.14. LetR(resp. S) be an orbit equivalence onX(resp. Y),F:X →
Y a measure preserving Borel isomorphism. The induced isomorphismF_{∗}: L^{∞}X →
L^{∞}Y can be extended to a normal ∗-homomorphism vNR → vNS if and only if
FR⊂S up to aν-null set.

Proof. For simplicity we identify Y with X by means of F. If JRK ⊂ JSK, the
required homomorphism is induced by the isometry L^{2}R →L^{2}S. Conversely, if
π:vNR→vNS is an extension ofF∗, for anyφ∈JRKwe have

π(v_{φ})π(f)π(v_{φ})^{∗}=π(f◦φ^{−1}) =f ◦φ^{−1},

which impliesπ(vφ) =σφvφ for someσφ∈L^{∞}X.

LetM be a finite von Neumann algebra with traceτ, identified to a subalgebra of
B(L^{2}M). SupposeAis a von Neumann subalgebra ofM. LeteAbe the projection
onto the span ofAξτ and puthM, Ai= (M ∪ {eA})^{00}.

For anyx∈M and ˆa∈L^{2}A,

e_{A}xˆa=e_{A}xac=E\_{A}(xa) =E\_{A}(x)a
which implieseAxeA=EA(x)eA. In particular, we have

hM, Ai=nX

x_{j}e_{A}y_{j}+z:x_{j}, y_{j}, z∈Mo^{wop}
.
Now,

eAJ xJ eAˆa=eAdax^{∗}=E\A(ax^{∗}) =aE\A(x^{∗}) =J EA(x)Jaˆ

implieshM, Ai^{0} =M^{0}∩ {eA}^{0} =J AJ, consequently hM, Ai= (J AJ)^{0}. Note that
whenAis commutativeeAJ aJ =a^{∗}eA fora∈A.

We have the “canonical trace” Tr onhM, Aiwhich is a priori unbounded defined byP

ix_{i}e_{A}y_{i}7→τ(P

ix_{i}y_{i}). Still, Tr is normal semifinite, and its tracial property
is verified as follows:

XxieAyi

2

2,Tr= Tr(X

y_{i}^{∗}eAx^{∗}_{i}xjeAyj) =X

τ(y_{i}^{∗}EA(x^{∗}_{i}xj)yj)

=X

τ(EA(yjy_{i}^{∗})EA(x^{∗}_{i}xj)) =kyieAx^{∗}_{i}k^{2}_{2,Tr}.
SupposeA⊂M is Cartan. Put ˜A={A, J AJ}^{00}⊂ hM, Ai.

Example 2.15. WhenA=L^{∞}X, M =vNR, we have ˜A=L^{∞}R, eA=χ∆ and
Tr|A˜=R

dν onL^{∞}R. Indeed,
Tr(f eA) =τ(f) =

Z

∆

f dµ= Z

f dν (f ∈L^{∞}X)
implies

Tr(uf eAu^{∗}) = Tr(f eA) =
Z

∆

f dµ= Z

uf eAu^{∗}dν (f ∈L^{∞}X, u∈ NA).

Remark 2.16. WhenA⊂M is Cartan andp∈Proj(A),A_{p}⊂pM pis also Cartan
sinceN_{pM p}^{p} (Ap) =pN_{M}^{p}(A)p.

Example 2.17. WhenY ⊂X, the restricted equivalenceR|Y =Y ×Y ∩Rgives
vN(R|Y) =p_{Y}(vNR)p_{Y}.

Exercise 2.18. Show that whenAis a Cartan subalgebra of a factorM,τ p1=τ p2

forp_{1}, p_{2}∈Proj(A) implies the existence ofv∈ N^{p}Asuch thatp_{1}∼p_{2}viav. This
implies that given an ergodic relation R on X, subsets Y_{1} and Y_{2} of X with the
same measure, one would obtain (A_{p}_{Y}

1 ⊂M_{p}_{Y}

1)'(A_{p}_{Y}

2 ⊂M_{p}_{Y}

2) viav.

2.3. Theorem of Connes-Feldman-Weiss.

Definition 2.19. A discrete group Γ is said to be amenable when`∞Γ has a left Γ invariant state.

Example 2.20. Commutative groups, or more generally solvable groups are amenable.

The union of an countable increasing sequence of amenable groups are again amenable.

Definition 2.21. A cartan subalgebraA⊂M is said to be amenable when there
exists a state m: ˜A → C invariant under the adjoint action of NA. An orbit
equivalenceRonX is said to be amenable whenL^{∞}X ⊂vNR is amenable.

Remark 2.22. Let Γ y X be a measure preserving essentially free action. Since Γ is assumed to be discrete, R can be identified to Γ×X as a measurable space and an invariant measure onR is nothing but a product measure on Γ×X of an invariant measure on Γ times an arbitrary measure onX. Thus, Ris amenable if and only if Γ is amenable.

Definition 2.23. A von Neumann algebra M onH is said to be injective when there exists a conditional expectation Φ :B(H)→M.

Fact. The above condition is independent of the choice of a faithfull representa- tionM ,→B(H). Moreover, M is injective if and only if it is AFD [2].

Theorem 2.24. (Connes-Feldman-Weiss [3]) Let M be a factor with separable predual,A a Cartan subalgebra ofM. The following conditions are equivalent:

(1) The pairA⊂M is amenable.

(2) This pair is AFD in the sense that for any finite subset F of NA and a positive real number > 0, there exists a finite dimensional subalgebra B of M such that

• B has a matrix unit consisting of elements of N^{p}A.

• kv−EB(v)k< for anyv∈F.

(3) (A, M)is isomorphic to (D,⊗M¯ 2Cwhere D = ¯⊗D2 for the diagonal sub-
algebra D2 ⊂M2. (Note that N^{p}D is generated by the “matrix units” of
M2^{∞} =⊗M2.)

(4) M is injective.

Lemma 2.25. In the assertion of (2),Bmay be assumed to be isomorphic toM_{2}N

for someN.

Proof of the lemma. Perturbing a bit, we may assume that τ(e^{(d)}_{ij} ) ∈ 2^{−N}N for
large enough N where (e^{(d)}_{ij} )_{d,1≤i,j≤n}_{d} is a matrix unit ofB =⊕_{d}M_{n}_{d}. By taking
a partition if necessary, we may assume thatτ(e^{(d)}_{ii} = 2^{−N} for any dandi. Then,
sinceM is a factor, we havee^{(d)}_{ii} ∼e^{(f)}_{jj} inM for anyd, f, iandj. This means that
B is contained in a subalgebra of M which is isomorphic toM_{2}N.
Proof of (2) ⇒ (3): Note that there is a total (with respect to the 2-norm)
sequence (vk)k∈N ⊂ N^{p}A. We are going to construct an increasing sequence of
subalgebras (B_{k})_{k}inMwith compatible matrix units (e^{(k)}_{i,j})_{i,j}satisfyingB_{k} 'M_{2}_{Nk}
andkEBk(v_{l})−v_{l}k^{2}<_{k}^{1} forl≤k.

Suppose we have constructed B1, . . . , Bk. Applying the assertion of (2) to the
finite set F^{0} = n

e^{(k)}_{i,r}vle^{(k)}_{r,1}o

, we obtain a matrix units (fij)i,j in N^{p}A such that
Pfii=e^{(k)}_{11} and

Espanf_{ij}(x)−x

< 1
n(k)^{2}(k+ 1)

wheren(k) denotes the size ofBk. By the assumption thatAis a maximal abelian
subalgebra inM, the projections ofN^{p}Aare actually contained inA. Thus we ob-
tain an inclusionD⊂A(hence the equality between them) under the identification
M ' ⊗_{N}M2= (∪Bk)^{00}.

Proof of (3)⇒(4): By assumptionM = (∪Bn)^{00}whereBnare finite dimensional
subalgebras ofM,M^{0} = (∪J BnJ)^{00}. Let Φn denote the conditional expectation of
B(H) onto (J BnJ)^{0}: Φn(x) =R

U(J B_{n}J)uxu^{∗}du wheredu denotes the normalized
Haar measure on the compact groupU(J BnJ). For eachx, the sequence (kΦn(x)k)n

is bounded above bykxk. Thus we can take a Banach limit Φ(x) of (Φn(x))n, which
defines a conditional expectation ofB(H) onto∩n(J BnJ)^{0}= (∪J BnJ)^{0} =M.

Proof of (4) ⇒ (1): Put H =L^{2}M and let Φ be a conditional expectation of
B(H) ontoM. Then τΦ is an AdUM-invariant state onB(H) . NA is obviously
contained inUM and so is ˜A inB(H).

Remark 2.26. WhenA⊂M is an amenable Cartan subalgebra andeis a projection inA, the Cartan subalgebraAe⊂Meis also amenable.

We are going to complete the proof of Theorem 2.24 by showing (1)⇒(2).

Lemma 2.27. Let φbe a measure preserving partial Borel isomorphism on a stan-
dard probability space(X, µ). LetE_{0}denote the fixed point setX^{φ}={x∈domφ:φx=x}.

There exist Borel setsB_{1}, B_{2}, B_{3} of X satisfyingX =`

0≤i≤3E_{i} and φE_{i}∩E_{i} is
null fori >0.

Proof. TakeE_{1}to be a Borel set with a maximal measure which satisfiesφE_{1}∩E1=

∅. Put E_{2} = φE_{1}. Then φE_{2} ∩E_{2} = ∅ by the injectivity of φ. Finally, put
E_{3}={(∪_{0≤i≤2}E_{i}). ThenφE_{3}∩E_{3} is null by the maximality ofE_{1}.
Corollary 2.28. For any finite setF of N^{p}A, there exist projections q1, . . . , qm

ofA (m= 4^{|}^{F}^{|}) satisfying Pqk= 1 and thatqkvqk is either0 or inUAq_{k} for any
v∈F.

Lemma 2.29. (Dye) For any finite subsetF ⊂ NAand >0, there existsa∈A˜_{+}
withTr(a) = 1andP

u∈Fkuau^{∗}−ak_{1,Tr}< . (Here,kxk_{1,Tr} = Tr(|x|).)

Proof. Letm: ˜A→Cbe an AdNA-invariant state. SinceL^{1}is w^{∗}-dense in (L^{∞})^{∗},
there exists a netai∈A˜+satisfying Tr(ai) = 1 and Tr(aix)→m(x) for anyx∈A.˜
Then, for anyu∈ NAandx∈A˜

Tr((uaiu^{∗}−ai)x) = Tr(aiu^{∗}xu)−Tr(aix)→m(u^{∗}xu)−m(x) = 0.

Thusua_{i}u^{∗}−a_{i} is weakly convergent to 0. By Hahn-Banach’s theorem, by taking
the convex closure of the sets{ua_{i}u^{∗}−a_{i}:k < i}, we find a sequence (b_{i})_{i}as convex
combinations of thea_{i} satisfyingkub_{i}u^{∗}−b_{i}k_{1,Tr}→0 uniformly foru∈F.

Lemma 2.30. (Namioka) Let F, be as above. There exists a projection pof A˜ satisfyingTr(p)<∞andP

u∈Fkupu^{∗}−pk^{2}_{2,Tr}< kpk^{2}_{2,Tr}.

Proof. Let a ∈ A˜+ be an element given by Lemma 2.29. For each r > 0 put
Pr=χ_{(r,∞)}(a). We have

kuau^{∗}−ak_{1,Tr} =
Z ∞

0

kuP_{r}u^{∗}−P_{r}k_{1,Tr}dr 1 =kak_{1,Tr}=
Z ∞

0

kP_{r}k_{1,Tr}dr.

Hence

Z ∞ 0

X

u∈F

kuPru^{∗}−Prk_{1,Tr}dr <

Z ∞ 0

kPrk_{1,Tr}dr.

Thus there exists rsuch thatp=Pr satisfiesPkupu^{∗}−pk_{1,Tr} < kpk_{1,Tr}. Since
the summands are differences of projections, k−k_{1,Tr} is approximately equal to

k−k^{2}_{2,Tr}.

Lemma 2.31. (Local AFD approximation by Popa) Let F, be as above. There
exists a finite dimensional subalgebra B ⊂ M with matrix units in N^{p}A, satis-
fying

EB(eue)−(u−e^{⊥}ue^{⊥})

2

2 < kek^{2}_{2} for every u ∈ F, where e denotes the
multiplicative unit ofB andEB the conditional expectation eM e→B.

Proof. We may assume 1∈F. Takep∈A˜+ as in Lemma 2.30. Since Trp <∞, we may assume that pcan be written asPn

i=1vieAv_{i}^{∗}forvi∈ N^{p}A. By Corollary
2.28, there exist projections (qk)k inAwithPqk= 1 and eachqkv^{∗}_{i}uvjqk is either
0 or is inU(Aqk) for 1≤i, j ≤ n, u∈F. Taking finer partition if necessary, we
deduce that dist(qkv_{i}^{∗}uvjqk,Cqk)<p

/n.

On the other hand, X

u∈F,k

k(upu^{∗}−p)J qkJk^{2}_{2,Tr}= X

u∈F

kupu^{∗}−pk^{2}_{2,Tr}< kpk^{2}_{2,Tr}=X

k

kpJ qkJk^{2}_{2,Tr}.

Hence for somek, q=q_{k} satisfies Pk(upu^{∗}−p)J qJk^{2}≤kpJ qJk^{2}. BypJ qJ =
PvieAJ qJ v_{i}^{∗} = PviqeAv^{∗}_{i} since A is commutative, replacing vi by viq, we may
assumev_{i}^{∗}vj=δi,jqandpJ qJ=p. (Note thatp=PvieAv_{i}^{∗}is a projection, which
means that the ranges ofvi are mutually orthogonal.)

This way we obtain P

kupu^{∗}−pk^{2} ≤ kpk^{2}, each v_{i}uv_{j}^{∗} ∈ A_{q} is close to a
constantzij byp

/n, and (vi)i is a matrix unit inAq. Pute=P

viv_{i}^{∗}. Thus,
kpk^{2}_{2,Tr}= Tr(X

vieAv_{i}^{∗}) =τ(X

viv^{∗}_{i}) =kek^{2}_{τ}.
Consequently,

kupu^{∗}−pk^{2}_{2,Tr}= 2 Trp−2 Tr(upu^{∗}p) = 2τ(e)−2 Tr(X

uvieAv_{i}^{∗}u^{∗}vjeAvj)

= 2τ(e)−2τ(X

uviv_{i}^{∗}u^{∗}vjv^{∗}_{j}) = 2τ(e)−2τ(ueu^{∗}e)

=kueu^{∗}−ek^{2}_{2,τ}.
Hence P

u∈Fkue−euk^{2}_{2} < kek^{2}_{2}. Noweue =P

viv_{i}^{∗}uvjv_{j}^{∗} ≈P

zijviv^{∗}_{j} ≈kek^{2}
ink−k^{2}_{2,τ}. Hence

keue−E_{B}(eue)k^{2}_{2,τ} < kek^{2}_{2,τ}

E_{B}(eue)−(u−e^{⊥}ue^{⊥})

2

2,τ <2kek^{2}_{2,τ}.

When we have a family (Bi) of mutually orthogonal finite dimensional algebras satisfying the assertion of the lemma,e=P

1B_{i} satisfies
E_{⊕B}_{i}(eue)−(u−e^{⊥}ue^{⊥})

2

2,τ <2kek^{2}_{2,τ}.
Lemma 2.32. In the notation of Lemma 2.31, e= 1.

Proof. Otherwise we can apply Lemma 2.31 toA_{e}⊥ ⊂M_{e}⊥ andF^{0}=e^{⊥}Fe^{⊥}, to
obtain a finite dimensional algebra B0 ⊂M_{e}^{⊥} satisfying the assertion of Lemma

2.31. Use the Pythagorean equality.

Proof of (1)⇒ (2): TakeB1, . . . , Bm satisfyingkP

m1Bik^{2}_{2}>1−. PutB =

⊕iBi⊕C(P

1Bi)^{⊥}. Then we havekEB(u)−uk^{2}_{2}<3foru∈F.
3. L^{2}-Betti numbers

3.1. Introduction. Let F(Ω, X) denote the set of the mappings of a set Ω into another setX. Let Γ be a discrete group,λthe left regular representation of Γ on

`2Γ. We have the “standard complex” of right Γ modules

0 //`2Γ ^{∂} //F(Γ, `_{2}Γ) ^{∂} //F(Γ^{2}, `2Γ) //· · ·
given by

∂(f)(s_{1}, . . . , s_{n+1}) =λ(s_{1})f(s_{2}, . . . , s_{n+1})+

X

1≤j≤n

(−1)^{j}f(s1, . . . , sjsj+1, . . . , sn+1) + (−1)^{n+1}f(s1, . . . , sn).

Conceptually, the above complex can be regarded as Hom_{C}Γ(P∗,_{C}Γ`2Γ) where P∗

denotes the standard free resolution of the trivial left Γ-moduleC. For eachn∈N,
Pn is the vector space with basis Γ^{n+1} as a vector space over C. Since Γ^{n+1} is a
left Γ-set by s.(s0, . . . , sn) = (s.s0, s1, . . . , sn), Pn has the canonically induced left
action of Γ.

LetH_{i}(Γ, `_{2}Γ) denote the i-th (co)homology group of this complex. Note that
this complex consists of RΓ modules given by the action on `_{2}Γ, with boundary
maps beingRΓ-homomorphisms. The space of 1-cocycles

Z1={b∈F(Γ, `2Γ) :b(st) =b(s) +λ(s)b(t)}

is identified with the space of the derivations from Γ to `2Γ with respect to the trivial right action. Whenb∈Z1the map

s7→

λ(s) b(s)

0 1

of Γ into B(`2Γ⊕C) becomes multiplicative. On the other hand the space of 1-coboundaries

B_{1}={b∈F(Γ, `_{2}Γ) :∃f ∈`_{2}Γ, b(s) =λ(s)f −f}

is identified with the space of the inner derivations. Note that for anyb∈Z1, there
is a functionf ∈F(Γ,C) satisfyingb(s) =λ(s)f−f if we do not require the square
summability of f. Indeed, a vector system (b(s))s∈Γ is a derivation if and only if
we have hb(s), δti=hb(st)−b(t), δ_{e}i for anys, t ∈ Γ, and in such a case we may
putf(s) =hb(s), δsito obtainb(s) =λ(s)f −f.

Remark 3.1. The 0-th homology grouopH0 =Z0 is the space of the Γ-invariant vectors in`2Γ. Thus this becomes the 0-module if and only if Γ is infinite.

In the following we assume that Γ admits a finite generating set S. Let DΓ denote the spaceZ1of the derivations, InnDΓ the spaceB1of the inner derivations.

LetOS denote the mappingb7→(b(s))s∈S ofDΓ into⊕_{S}`2Γ. This is an injective
RΓ-module map. Note that the range of OS is closed. Indeed, (f(s))s∈S is in
ranOS if and only if

f(s_{1}) +λ(s_{1})f(s_{2}) +· · ·+λ(s_{1}· · ·s_{n−1})f(s_{n}) = 0
holds for each relations1· · ·sn=eamong elements ofS.

A sequence (fn)_{n∈N} of unit vectors is said to be an approximate kernel of
the restriction O_{S}|InnDΓ when λ(s)fn−fn tends to zero (in norm) for any s ∈
S. O_{S}|InnDΓ has an approximate kernel if and only if Γ is amenable. Thus
OS(InnDΓ) is closed if and only if Γ is finite or non-amenable.

Let P, Q denote the orthogonal projections onto OS(DΓ) and OS(InnDΓ).

These commute with the diagonal action of RΓ on ⊕_{S}`2Γ, i.e. P, Q ∈ M_{S}LΓ.

We can measure them by the trace ˜τ = Tr⊗τ. The first Betti number β^{(2)}_{1} =
dimLΓH1(Γ, `2) is equal to the difference ˜τ(P)−˜τ(Q).

Example 3.2. When Γ is a finite group, β_{0}^{(2)} = _{|Γ|}^{1} while β^{(2)}_{i} = 0 for 0 < i
because any CΓ module is projective. On the other hand when Γ is equal to the
free groupFngenerated by a setS consisting ofnelements, ranO_{S} =⊕_{S}`_{2}Γ and
β_{1}^{(2)}=n−1.

We omit the injection OS and identify DΓ with a subspace of ⊕_{S}`2Γ. Thus

∂^{0}: `_{2}Γ → F(Γ, `_{2}Γ) factors through⊕_{S}`_{2}Γ and ∂^{0}:`_{2}Γ → ⊕_{S}`_{2}Γ is written as
f 7→(λ(s)f −f)_{s∈}_{S}.

Let ^{(2)}_{1} : ⊕_{S} `_{2}Γ → `_{2}Γ denote the adjoint of ∂. Thus ^{(2)}_{1} is expressed as
(ξ_{s})_{s∈}_{S} 7→P

s∈S(λ(s^{−1})−1)ξ_{s}and the orthogonal complement of ker^{(2)}_{1} is equal
to the closure of ran∂= InnDΓ.

Proposition 3.3. When we identifyCΓwith the space of vectors with finite support
in`_{2}Γ, we haveDΓ = (ker^{(2)}_{1} ∩ ⊕_{S}CΓ)^{⊥}.

Proof. The spaceCΓ hasF(Γ,C) as its algebraic dual. A vector systemb∈ ⊕_{S}`_{2}
is in DΓ if and only if there is an f ∈ F(Γ,C) such that b(s) = λ(s)f −f. The
latter implies

∀ξ∈ker^{(2)}_{1} ∩ ⊕_{S}CΓ,hξ, bi=X

s

hξ(s), b(s)i=X

s

h(λ(s^{−1})−1)ξ(s), fi= 0.

Conversely, when (b(s))_{s∈}_{S} is orthogonal to ker^{(2)}_{1} ∩ ⊕_{S}CΓ, the functionalhb,−i
on⊕_{S}CΓ is induced by a functionalf on the kernel of the mapCΓ→C. Thisf
can be extended to a linear map on the wholeCΓ, and we haveb(s) =λ(s)f −f,

i.e. b∈DΓ.

Remark 3.4. The i-th cohomology group H^{i}(Γ, `2Γ) is dimension isomorphic to
Tor^{C}_{i}^{Γ}(C, `2Γ). This is seen by considering the exact functors E → E^{∗} on the
category of LΓ-modules and that of LΓ-bimodules, where E^{∗} denotes the dual
module of the weak closure ofE. We have functors (A, B)→A⊗_{C}ΓBand (A, B)→

Hom_{C}Γ(A, B) ofCΓ-mod×LΓ-bimod intoLΓ-mod. Then the functor equivalence
(A⊗_{C}ΓB)^{∗} 'Hom_{C}Γ(A, B^{∗}) up to dimension implies the dimension equivalence
between the derived functors Torp(A, B)^{∗} ' Ext^{p}(A, B^{∗}). The case A = C and
B=`_{2}Γ describes the desired isomorphism.

For example, we have a flat resolutionP_{·}of the trivial Γ-moduleCwithP_{0}=CΓ
andP1=CΓ⊗_{C}CS, withd1(a⊗b) =ab−a. The first torsion group Tor^{C}_{1}^{Γ}(`2Γ,C)
is by definition the quotient ker(id_{`}_{2}_{Γ}⊗d_{1})/`_{2}Γ⊗kerd_{1}. Now id_{`}_{2}_{Γ}⊗d_{1} = ^{(2)}_{1}
implies ker(id_{`}_{2}_{Γ}⊗d_{1}) = InnDΓ^{⊥} while `_{2}Γ⊗kerd_{1} = ker^{(2)}_{1} ∩ ⊕_{S}CΓ implies

`2Γ⊗kerd1=DΓ^{⊥}.

3.2. Operators affiliated to a finite von Neumann algebra. Let (M, τ) be a
finite von Neumann algebra with a faithful normal tracial state (τ is unique ifM is
a factor),L^{2}M the induced HilbertM-M module. For eachn∈Nput ˜τ=τ⊗Tr
onM⊗M_{n}C'M_{n}M.

Definition 3.5. LetH be a left Hilbert module overM. A densely defined closed
operator T onH is said to be affiliated to M, written asT ∼M, when we have
uT = T u for any u ∈ U(M^{0}). Here the equality entails the agreement of the
domains, i.e. udomT = domT.

Remark 3.6. An operator T is affiliated toM if and only if for the polar decompo-
sitionT =v|T|the partial isometryvand the spectral projections of|T|are inM.
Note that in such casesτ takes the same value on the left supportl(T) =vv^{∗}ofT
and the right supportr(T) =v^{∗}v.

We consider the case H = L^{2}M. Suppose T ∼ M. It is said to be square
integrable when ˆ1∈domT. This condition is equivalent to

τ(|T|^{2}) =kTˆ1k^{2}=
Z

t^{2}dτ(E)<∞
for the spectral measure T = R

tdE of T. For each ξ ∈ L^{2}M let L^{◦}_{ξ} denote the
unbounded operator defined by domL^{◦}_{ξ} = ˆM ⊂L^{2}M andL^{◦}_{ξ}x=ξx.

Proposition 3.7. The operator L^{◦}_{ξ}xis closable and its closure Lξ is affiliated to
M. Moreover we have L^{∗}_{ξ} = LJ ξ. If T is affiliated to M and square integrable,
T =L_{T}ˆ1.

Proof. We show the inclusionL^{◦}_{J ξ}⊂(L^{◦}_{ξ})^{∗}. For any elementsx, y∈M,
hL^{◦}_{ξ}x,ˆ yiˆ =hξx, yi=hJy, J(ξx)iˆ =hˆ1y^{∗}, x^{∗}J ξi=hˆx,(J ξ)yi.

On the other hand, whenu∈ URM,uL^{◦}_{ξ} =L^{◦}_{ξ}uimpliesuL_{ξ}=L_{ξ}u.

Next we show the inclusion (Lξ)^{∗}⊂LJ ξ. Letη∈dom(L^{◦}_{ξ})^{∗}. Consider the polar
decomposition LJ ξ = v|LJ ξ| and the spectral decomposition |LJ ξ| = R∞

0 λdeλ.
Then eλv^{∗}LJ ξ = eλ|LJ ξ| is bounded (i.e. is in M+) for any λ. By definition,
LJ ξ(yˆ1) = (J ξ)y for y ∈ M. Hence eλv^{∗}LJ ξ(yˆ1) = eλv^{∗}((J ξ)y) = (eλv^{∗}J ξ)y.

Puttingy= 1, we obtaine_{λ}v^{∗}L_{J ξ}ˆ1 =e_{λ}v^{∗}J ξ ∈M.ˆ1 for anyλ >0.

Thus, by definition of (Lξ)^{∗}, we have

h(Lξ)^{∗}η,(e_{λ}v^{∗})^{∗}yˆ1i=hη, Lξ(e_{λ}v^{∗})^{∗}yˆ1i=hη, ξ(eλv^{∗})^{∗}yi=hη, J y^{∗}(e_{λ}v^{∗})J ξi

=hη, J y^{∗}eλv^{∗}LJ ξˆ1i (by using above)

=hη,(eλv^{∗}LJ ξ)^{∗}yˆ1i.

Henceeλv^{∗}(Lξ)^{∗}η=eλv^{∗}LJ ξη=|LJ ξ|eληfor anyλ >0. By lettingλ→ ∞,eλη→
η and |LJ ξ|eλη → v^{∗}(Lξ)^{∗}η. Since |LJ ξ| is a closed operator, η ∈ dom(|LJ ξ|) =
dom(LJ ξ). Hence (Lξ)^{∗}⊂LJ ξ and|LJ ξ|=v^{∗}(Lξ)^{∗}.

Finally, let us prove the last part. Let T ∼ M with the polar decomposition
v|T|=T. Note that ˆv^{∗}= ˆ1v^{∗}∈domT, ˆ1∈domT^{∗},T^{∗}ˆ1 =|T|vˆ^{∗}. Putξ=Tˆ1, η=
T^{∗}ˆ1. SinceT ∼M,L^{◦}_{ξ} ⊂T,L^{◦}_{η}⊂T^{∗} and we obtainLξ⊂T ⊂LJ η.
3.3. Projective modules over a finite von Neumann algebra. Let m, n ∈
N. We have an isomorphism Mor(M^{⊕m}, M^{⊕n}) = Mm,n(M) by multiplication of
matrices on column vectors.

Definition 3.8. An left M-module V is said to be finitely generated projective module when it is a projective object in the category of theM-modules and has a finite set generating itself.

Remark 3.9. Any finitely projective M module is isomorphic to someM^{⊕m}.P for
a natural numbermand an idempotent matrix P in MmM.

Lemma 3.10. In the above we may replacePwith an orthogonal projectionP^{∗}=P
without changing the value of ˜τ(P).

Proof. LetP0 be the right support ofP. P(P−P0) = 0 implies P0(P−P0) = 0.

ThusS= Id +(P−P0) is invertible. With respect to the orthogonal decomposition
Id =P_{0}⊕P_{0}^{⊥}, these operators are expressed as

P0=

Id 0 0 0

, P =

Id 0

? 0

, S =

Id 0

? Id

.

The operatorSP0=SP0S^{−1} is self adjoint.

Remark 3.11. WhenM^{⊕m}P andM^{⊕n}Qare isomorphic, ˜τ(P) = ˜τ(Q).

Definition 3.12. For each finitely projectiveM-moduleV isomorphic to M^{⊕m}P
where P is a orthogonal projection in MmM, dimMV −˜τ(P) is called the τ-
dimension

Lemma 3.13. LetV be a submodule ofM^{⊕n}. WhenV is closedM^{⊕n} with respect
to the L^{2}-norm (V is weakly closed), V is finitely generated and projective.

Proof. TheL^{2}completion ¯V^{k·k}^{2}⊂L^{2}M^{⊕n}is written asL^{2}M^{⊕}P for an orthogonal

projectionP. ThenV is equal toM^{⊕n}P.

Lemma 3.14. For each T ∈ Mor(M^{⊕m}, M^{⊕n}), its kernel and range are finitely
generated projective modules.

Proof. Obviously the kernel ofT is weakly closed inM^{⊕m}. On the other hand for
the projection P such that kerT =M^{⊕m}P, T induces an isomorphism M P^{⊥} →

ranT.

Remark 3.15. When a submodule V ⊂M^{⊕m}is finitely generated, V is projective.

In fact,V =M^{⊕m}Afor some A∈Mm,n(M). Thus we have
V 'M^{⊕n}l(A)'M^{⊕m}r(A)'V .¯
Hence dimMV = dimMV¯.

Remark 3.16. If W ⊂ V are finitely generated projective modules, dim_{M}W ≤
dim_{M}V.

Definition 3.17. LetV be anM-module. Put

dim_{M}V = sup{dim_{M}W :W ⊂V, W is projective} ∈[0,∞].

Remark 3.18. Note that the above definition of dimM is compatible with the pre-
vious one for finitely generated projective modules. In general, W ⊂ V implies
dim_{M}W ≤dim_{M}V and (V_{i})_{i∈I} ↑V (V =∪i∈IV_{i}) implies dim_{M}V = lim_{i}dim_{M}V_{i}.
Theorem 3.19. (L¨uck[6]) When

0 //V_{0} ^{ι} //V_{1} ^{π} //V_{2} //0
is exact, we have dimMV1= dimMV0+ dimMV2.

Proof. When W ⊂V2 is finitely generated and projective, π^{−1}W is identified to
W ⊕ιV0. Hence dimV1 ≥ dimV0+ dimV2. Conversely, let W ⊂ V1 be finitely
generated projective. The weak closure ιV0∩W is closed in a finite free module,
hence is projective. From the sequence ιV0∩W → W → W/ιV0∩W, we have
dimW = dimιV_{0}∩W + dimW/ιV_{0}∩W. Note that there is a natural surjection
W/ιV0∩W → W/ιV0∩W. By the first part of the argument this implies the
dimension inequality dimιV_{0}∩W ≤dimιV_{0}∩W. On the other handW/ιV_{0}∩W

is identified to a submodule ofV_{2}.

Corollary 3.20. LetV be a finitely generatedM-module. We have a decomposition
V =Vp⊕VtwhereVp is projective anddimV = dimVp. (HencedimVt= 0.)
Proof. We have a surjection T: M^{⊕m} → V. Note that kerT may not be closed
since we have no matrix presentation ofT. Nonetheless,V 'M^{⊕m}/kerT and the
next lemma imply thatV_{p}=M^{⊕m}/kerT satisfies

dimV =m−dim kerT =m−dim kerT = dimVp.
Lemma 3.21. LetW be a subset of a finite free moduleM^{⊕m}. We havedimW =
dim ¯W.

Proof. PutL={A∈MmM :M^{⊕}.A⊂W}. This is a left ideal ofMmM. We get
a right approximate identity Ai ofL. For the orthogonal projection P such that
W¯ =M^{⊕m}P, the right support r(Ai converges toP in strong operator topology
(for any normal representation, thus, in the ultrastrong topology). Thus for any
>0, P_{,i}−χ_{[,1]}(A^{∗}_{i}A_{i}) is in L and converges to P in the ultrastrong operator

topology.

Proposition 3.22. For anyLΓ-moduleV,dimV = 0 is equivalent to

∀ξ∈V, >0,∃P ∈ProjM :τ P >1− andP ξ= 0.

Proof. ⇒: Letξ∈V. Consider the exact sequence 0→L→M →M.ξ→0 where L is the annihilator of ξ. dimL = dimM implies the existence of projectionsPi

convergent to 1 in the ultrastrong topology.

⇐: IfV ⊃M.Q,P satisfiesτ P >1−τ QandP Q6= 0.

Definition 3.23. A homomorphism φ: V → W of L-modules is said to be a
dimension isomorphism when dim_{M}kerφ= dim_{M}cokφ= 0.

Remark 3.24. The torsion N-modules T = {V : dimNV = 0} form a Serre sub-
category ofN-mod. AnalyzingN-modules up to dimension isomorphisms amounts
to considering the localizationN-mod/T ofN-mod byT. Thus, in general, when
a morphism V∗ → W∗ of complexes is a dimension isomorphism at each degree,
the induced homomorphism between the cohomology groups is also an dimension
isomorphism because it factors through an isomorphism in the localization category
C^{∗}(N-mod/T) of theN-module complex category over the torsion module category.

Lemma 3.25. The standard inclusionM →L^{2}(M)is a dimension isomorphism.

Proof. Letξ∈L^{2}M. We get the corresponding square integrable operator affiliated
withM. PutPn=χ_{[0,n]}(ξξ^{∗})∈ProjM. ThenPnξ∈MandPn→1, thusPn[ξ]−0

in the quotientL^{2}M/M.

When H is a Hilbert M-module, i.e. a normal representation of M on H,
H 'L^{2}M^{⊕n}.P for some cardinalnand an idempotentP inM_{n}M.

Lemma 3.26. In the above notation,dim_{M}H = ˜τ(P).

Proof. We have the following commutative diagram
M^{⊕n} _.P //

^{p.b.}

L^{2}M^{⊕n} _ .P //

cok _

M^{⊕n} //L^{2}M^{⊕n} //cok.

The cokernel in the lower row has dimension 0, thus so does the one in the upper

row.

Definition 3.27. βn^{(2)}(Γ) = dim_{LΓ}Tor^{C}_{n}^{Γ}(LΓ,Ctriv) is called the n-th L^{2}-Betti
number of Γ.