Library of theorems, definitions, problems and examples in the theory of von Neumann algebras

Library of theorems, definitions,

problems and examples

in the theory of von Neumann algebras

Andreas Naes Aaserud

December 3, 2013

Abstract

This document consists of a list of theorems, definitions, problems andexamples mainly in the theory of II1 factors with references to the mostcomplete proofs.

Contents

I Theorems and examples 4

1 General information 41.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 von Neumann algebras . . . . . . . . . . . . . . . . . . . . 41.1.2 Extreme points of the unit ball . . . . . . . . . . . . . . . 51.1.3 Finite von Neumann algebras . . . . . . . . . . . . . . . . 6

1.2 Comparison of projections . . . . . . . . . . . . . . . . . . . . . . 61.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Normal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Normality and related notions . . . . . . . . . . . . . . . . 81.4.2 Decompositions of functionals . . . . . . . . . . . . . . . . 101.4.3 Vector states . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.4 The predual . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Conditional expectations . . . . . . . . . . . . . . . . . . . . . . . 121.7 Some useful estimates . . . . . . . . . . . . . . . . . . . . . . . . 141.8 Actions, representations, and crossed products . . . . . . . . . . 151.9 Unbounded operators and Borel functional calculus . . . . . . . . 16

1.9.1 Polar decomposition and spectral theory . . . . . . . . . . 161.9.2 Operators associated to elements of L2(M) . . . . . . . . 181.9.3 Operators associated to elements of L1(M) . . . . . . . . 18

1.10 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1

2 Special von Neumann algebras 192.1 Uncountably many non-isomorphic factors . . . . . . . . . . . . . 192.2 Amalgamated free products . . . . . . . . . . . . . . . . . . . . . 20

3 Orthogonal subalgebras 20

4 Masas 214.1 Masas in type I von Neumann algebras . . . . . . . . . . . . . . . 214.2 Normalizing groups and groupoids . . . . . . . . . . . . . . . . . 224.3 Hereditary properties . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Projections in masas . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Unitary conjugacy of semiregular masas . . . . . . . . . . . . . . 244.6 Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 Uniqueness of Cartan subalgebras . . . . . . . . . . . . . . . . . . 254.8 Singular masas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.9 Special masas in separable factors . . . . . . . . . . . . . . . . . . 264.10 Examples of masas . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.10.1 Examples in group von Neumann algebras . . . . . . . . . 274.10.2 Masas in non-separable von Neumann algebras . . . . . . 29

5 Property Gamma 295.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Relationship with inner amenable groups . . . . . . . . . . . . . . 305.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Finitely generated von Neumann algebras 31

7 Standard non-commutative results 317.1 Some C*-algebraic results . . . . . . . . . . . . . . . . . . . . . . 317.2 Powers-Strmer type inequalities . . . . . . . . . . . . . . . . . . 327.3 Rokhlin type results . . . . . . . . . . . . . . . . . . . . . . . . . 327.4 Radon-Nikodym type results . . . . . . . . . . . . . . . . . . . . 33

8 Perturbation results 338.1 Distance between subalgebras . . . . . . . . . . . . . . . . . . . . 338.2 Basic perturbation results . . . . . . . . . . . . . . . . . . . . . . 348.3 Perturbation of masas . . . . . . . . . . . . . . . . . . . . . . . . 348.4 General perturbation . . . . . . . . . . . . . . . . . . . . . . . . . 35

9 Bimodules 359.1 Dimension and index . . . . . . . . . . . . . . . . . . . . . . . . . 369.2 Intertwining-by-bimodules . . . . . . . . . . . . . . . . . . . . . . 36

10 Classical invariants 3710.1 Fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2

11 Approximation properties 3711.1 Haagerup property for groups . . . . . . . . . . . . . . . . . . . . 3711.2 Relative property H . . . . . . . . . . . . . . . . . . . . . . . . . 3811.3 Amenable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4 Amenable factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.4.1 Definitions and uniqueness . . . . . . . . . . . . . . . . . 3911.4.2 Maximal injective subalgebras . . . . . . . . . . . . . . . . 41

11.5 Relative amenability . . . . . . . . . . . . . . . . . . . . . . . . . 4111.6 Weakly amenable groups . . . . . . . . . . . . . . . . . . . . . . . 42

12 Rigidity properties 4212.1 Kinds of rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.2 Relative property (T) for groups . . . . . . . . . . . . . . . . . . 4212.3 Relative property (T) for algebras . . . . . . . . . . . . . . . . . 4312.4 Cocycle superrigidity . . . . . . . . . . . . . . . . . . . . . . . . . 4412.5 W-superrigidity of groups . . . . . . . . . . . . . . . . . . . . . . 4412.6 W-superrigidity of actions . . . . . . . . . . . . . . . . . . . . . 45

13 Indecomposability properties 4513.1 Strong solidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.2 Solidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3 Absence of Cartan subalgebras . . . . . . . . . . . . . . . . . . . 4613.4 Primeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

14 C*-algebraic properties 4614.1 Nuclearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.2 Exactness of C*-algebras . . . . . . . . . . . . . . . . . . . . . . . 4614.3 Property A and exactness of groups . . . . . . . . . . . . . . . . 4714.4 Bi-exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.5 Cohomological properties . . . . . . . . . . . . . . . . . . . . . . 47

II Definitions 47

3

Part I

Theorems and examples

1 General information

1.1 Basic facts

1.1.1 von Neumann algebras

Theorem 1.1.1 (Double commutant theorem). Let M be a unital -sub-algebra of B(H). Then M is a von Neumann algebra if and only if M = M .

Proof. [Zhu93], Theorem 18.6, p. 110.

Theorem 1.1.2 (Kaplansky density theorem). Let A be a unital (concrete)C-algebra acting on B(H) and let M be its SO-closure, which is a von Neumannalgebra. Then the norm-closed unit ball of M is the SO-closure of the norm-closed unit ball of A. In symbols,

((A)1)SOT

= (ASOT

)1.

Proof. [Aas13], Theorem 3.3.6, p. 88.

Theorem 1.1.3 (Vigiers theorem). If (s) is an increasing net of self-adjoint operators and there exists a self-adjoint operator t such that s tfor all , then s s (SOT), where s = sup s.Proof. [Top71], 1, p. 7.Theorem 1.1.4 (Representation theory of von Neumann algebras).Let Mi B(Hi), i = 1, 2, be two von Neumann algebras and let pi : M1 M2 be a surjective normal -homomorphism. Then there exists a Hilbert spaceK (separable if H2 is separable), a projection e

M 1B(K), and a unitaryoperator U : e(H1K) H2 such that

x M1 : pi(x) = U(x 1)eU.


Theorem 1.1.5 (Unitary implementation). Any -isomorphism betweenvon Neumann algebras that each has a cyclic and separating vector is of theform Ad(U) for some unitary operator U .

Proof. [KR86], Theorem 7.2.9, p. 469.

Theorem 1.1.6 (Universal representation). Every C-algebra has a univer-sal representation pi : A B(H) with the property that there is a linear isometrypi of A onto pi(A) which is a weak -weak homeomorphism.

4


Theorem 1.1.7 (Dixmier approximation theorem). Let M be a von Neu-mann algebra. If a M then

Z(M) co{uau : u U(M)} 6= .Proof. [Aas13], 1.7.1, p. 20.Theorem 1.1.8. If A B(H) is an abelian von Neumann algebra, then

A co{uTu : u U(A)} 6= for all T B(H).Proof. [KR86], Corollary 8.3.12, p. 527. (Note that [KR86], Proposition 8.3.11,p. 526, gives more general conditions on A under which this conclusion holds.)

Theorem 1.1.9. Let M N B(H) be von Neumann algebras, and let p Mand q M be projections. Then

(i) (pMp) pNp = (M N)p;(ii) Z(pMp) = pZ(M);

(iii) (qM) qB(H)q = qM q;(iv) Z(qM) = qZ(M).

Proof. [SS08], Theorem 5.4.1, p. 92, and Theorem 5.4.3, p. 93.

Theorem 1.1.10. Let a be a normal element with resolution of identity E andlet C. Then

(i) 1{}(a) = PKer (a);

(ii) E({}) > 0 if and only if is an eigenvalue for a.Proof. [Aas13], Proposition 1.3.6, p. 14.

Theorem 1.1.11. If {en}n=1 is a sequence of cyclic projections in a von Neu-mann algebra M converging strongly to e M , then e is a cyclic projection.Proof. [KR86], Proposition 7.3.10, p. 479.

1.1.2 Extreme points of the unit ball

Theorem 1.1.12. The set of extreme points of (A)1, for a C-algebra A, con-

sists precisely of those partial isometries V A such that (1V V )A(1V V ) =0.


Corollary 1.1.13. The set of extreme points of (M)1, for a factor M , is U(M).

5

1.1.3 Finite von Neumann algebras

Theorem 1.1.14 (Center-valued trace). Any finite von Neumann algebrahas a center-valued trace ctr with the following properties:

(i) ctr is a linear map M Z(M);(ii) ctr is positive;

(iii) ctr(1) = 1 and ctr 1;(iv) ctr is tracial;

(v) ctr is a conditional expectation onto Z(M);(vi) ctr is faithful;

(vii) ctr is the unique linear, norm-continuous, tracial projection onto Z(M).Proof. [Aas13], 1.7, p. 19 (specifically Proposition 1.7.13).Theorem 1.1.15. Let M be a finite von Neumann algebra. If Z(M) is -finite,then M is -finite.

Proof. [KR86], Corollary 8.2.9, p. 519.

1.2 Comparison of projections

Theorem 1.2.1. If e and f are finite projections in a von Neumann algebraM , then e f = sup{e, f} M is finite.Proof. [KR86], Theorem 6.3.8, p. 414.

Theorem 1.2.2 (Schroder-Bernstein theorem). Let e and f be projec-tions in a von Neumann algebra M . If e f and f e, then e f .Proof. [Top71], 6, p. 41. (See also [KR86], Proposition 6.2.4, p. 406.)Theorem 1.2.3 (Addition of equivalence). Let {e} and {f} be familiesof pairwise orthogonal projections in a von Neumann algebra M .

(i) If e f for all , thene

f;

(ii) If e f for all , thene

f.

Proof. [KR86], Proposition 6.2.2, p. 406.

Theorem 1.2.4 (Comparison Theorem). Let e and f be projections in avon Neumann algebra M . Then there exists a central projection z M suchthat ze zf and (1 z)f (1 z)e.Proof. [Zhu93], Theorem 25.4, p. 147.

6

Theorem 1.2.5. Let M be a von Neumann algebra. Then M is a factor if andonly if, given projections e, f M , either e f or f e.Proof. See [KR86], Proposition 6.2.6, p. 408 for the only if-direction (or sim-ply use the Comparison Theorem). Conversely, if M is not a factor, let z bea non-trivial central projection in M . If 1 z z, then there exists a partialisometry v M such that 1 z = vv and vv z. Then

vv = v(vv)v = (vv)vv vv = 1 z

so that v = 0, a contradiction.

Theorem 1.2.6. Let e and f be projections in a von Neumann algebra M .Then the following conditions are equivalent:

(i) CeCf 6= 0;(ii) eMf 6= 0;

(iii) e and f have non-zero equivalent subprojections.

Proof. [Top71], 3 (Corollary 4 and Lemma 8), p. 21 and 22.Theorem 1.2.7. Let M B(H) be a von Neumann algebra, and let , Hbe given. Then [M ] [M ] (in M) if and only if [M] [M] (in M ).Proof. [KR86], Theorem 7.2.12, p. 471.

Theorem 1.2.8 (Formulae for central support projection). Let a be anelement of a von Neumann algebra M , and let p M be a projection.

(i) Ca = inf{e P(Z(M)) : ea = a};(ii) Cp = inf{e P(Z(M)) : e p} = sup{upu : u U(M)};

(iii) Ran(Ca) = [MMRan(a)] = [MRan(a)] = [xMRan(xa)].

Note that (iii) easily implies that equivalent projections have equal centralsupport projections. Moreover, (i) is usually given as the definition.

We dont usually have the implication Ce = Cf e f (and it couldnt bemore wrong in finite factors!), but we do have the following fact:

Theorem 1.2.9. Under the following sets of assumptions, Cf Ce impliesf e:

(i) e is properly infinite and f is countably decomposable;

(ii) f is abelian.

Proof. [KR86], Theorem 6.3.4, p. 413, and Proposition 6.4.6, p. 421.

7

Theorem 1.2.10 (Divisibility lemma). Let M be a von Neumann algebrawith no type I summand and let n N be given. If p M is a projection, thenthere exist n equivalent and mutually orthogonal projections p1, . . . , pn such that

p = p1 + p2 + + pn.Proof. [Aas13], Proposition 1.4.3.

Theorem 1.2.11. Let e and f be projections in a finite von Neumann algebraM . The following conditions are equivalent:

(i) e f ;(ii) 1 e 1 f ;

(iii) ueu = f for some u U (M);(iv) ctrM (e) = ctrM (f), where ctrM is the unique center-valued trace on M .

1.3 Ideals

Theorem 1.3.1. If I is a WO-closed left (or right) ideal in a von Neumannalgebra M , then I = Me (or I = eM) for some projection e M . If I is atwo-sided ideal, then e is a central projection.

Proof. [KR86], Theorem 6.8.8, p. 443 (or see [Tak79], II.3 (Proposition 3.12),p. 76).

Theorem 1.3.2. Let I be a two-sided ideal in a von Neumann algebra M . Thenevery projection in I, the norm-closure of I, belongs to I.

Proof. Let e be a projection in the norm-closure of I. Let I 3 xn e in normso that exne e. Thus exne is invertible in the C-algebra eMe for sufficientlylarge n. It follows that e = exne(exne)

1 (eIe)(eMe) I. (This solvesexercise 6.9.46 in [KR86].)

Theorem 1.3.3. The set J of operators with finite range projection in a vonNeumann algebra M is a two-sided ideal in M . If M is a factor, then eachnon-zero, two-sided ideal in M contains J .


Theorem 1.3.4. All finite factors are simple.

Proof. [JS97], Proposition A.3.1, p. 135.

1.4 Normal maps

1.4.1 Normality and related notions

Theorem 1.4.1. If is a state on B(H), then there exists a net (Ti) of positivetrace-class operators with Tr(Ti) = 1 and Tr(xT ) (x) for all x B(H).Proof. [Aas13], Proposition 1.11.1, p. 30.

8

Theorem 1.4.2. Let M be a von Neumann algebra with a faithful normal state. Denote by 2, the 2-norm arising from , i.e., x22, = (xx) forx N . The following holds:

(i) Given a norm-bounded net (x) and an element x in M , x x in theSOT if and only if x x with respect to the 2-norm 2,;

(ii) (M)1 is a complete metric space with respect to the 2-norm 2,.Theorem 1.4.3. Let A be a unital C-algebra with a faithful state . If (A)1is complete with respect to 2,, then A is a von Neumann algebra (acting onL2(A, )) and is normal.

Proof. [SS08], Lemma A.3.3, p. 320.

Theorem 1.4.4. Let be a state on a von Neumann algebra M B(H). Thefollowing conditions are equivalent:

(i) is normal;

(ii) There exists a sequence (n)n=1 in H such that (x) =

n=1xn, n for

all x N ;(iii) There exists an orthogonal sequence (n)

n=1 in H such that

(x) =n=1xn, n for all x N ;

(iv) is completely additive, i.e., (E) =

(E) for any family

{E} of pairwise orthogonal projections in N ;(v) is WO-continuous on bounded sets;

(vi) is SO-continuous on bounded sets.


Corollary 1.4.5. The family of normal states in a von Neumann algebra isnorm-closed.

Corollary 1.4.6. Let : M N be a normal u.c.p. map between von Neu-mann algebras. Then is SO-continuous on norm-bounded sets. In partic-ular, any normal unital -homomorphism between von Neumann algebras isSO-continuous on bounded sets.

Proof. [Aas11], Lemma 2.2.5, p. 4.

Theorem 1.4.7. Let M be a von Neumann algebra, and let : M B(H) bea unital -homomorphism.

(i) If is normal, then is SO-SO and WO-WO continuous on boundedsets, and (M) is a von Neumann algebra.

(ii) If is an isomorphism onto a von Neumann algebra, then |(M)1 is aWO-WO and a SO-SO homeomorphism onto ((M))1.

Proof. [KR86], Proposition 7.1.15 and Corollary 7.1.16, p. 464.

9

1.4.2 Decompositions of functionals

Theorem 1.4.8 (Polar decomposition of linear functionals). If is anormal linear functional on a von Neumann algebra M , then there exists apositive normal linear functional on R and an extreme point u (M)1 (so uis unitary if M is a factor, cf. Theorem 1.1.12) such that

(x) = (ux) and (x) = (ux)

for all x M .Proof. [KR86], Theorem 7.3.2, p. 474.

Theorem 1.4.9 (Positive and negative parts of linear functionals). If is a normal hermitian functional on a von Neumann algebra M , then attainsits norm at an element 2e 1 (M)1, where e is a projection in M . Moreover,

= + ,

where+(x) = (exe) and (x) = ((1 e)x(1 e))

for all x M . Finally, the linear functionals are normal and positive, and

= ++ .


1.4.3 Vector states

Theorem 1.4.10. Let M be a von Neumann algebra with a separating vector.Then all normal states on M are vector states.

Proof. [KR86], Theorem 7.2.3, p. 467. (See [SS08], Lemma B.5.2, p. 364, for aproof in the special case where M is a finite von Neumann algebra, equippedwith a faithful normal trace , and represented on L2(M, ).)

Theorem 1.4.11. Let be a linear functional on a von Neumann algebra M .If is the norm-limit of a sequence of vector states on M , then is itself avector state.


1.4.4 The predual

Theorem 1.4.12. Let M be a finite von Neumann algebra, equipped with adistinguished faithful normal trace . Then the predual M of M is isometricallyisomorphic to L1(M, ).

Proof. [Tak79], V.2, p. 320, or [SS08], section B.5, p. 363.

10

Theorem 1.4.13. Let M be a finite von Neumann algebra, equipped with a dis-tinguished faithful normal trace . Then the following conditions are equivalent:

(i) M has separable predual M;

(ii) L1(M, ) is a separable Banach space;

(iii) L2(M, ) is a separable Hilbert space;

(iv) (M)1 is separable with respect to 2, ;(v) The strong operator topology on (M)1 is separable.


Theorem 1.4.14. Let M be a von Neumann algebra. Then M is separable ifand only if there exists a faithful normal -representation of M on a separableHilbert space.

Proof. [JS97], Proposition A.2.1, p. 134.

Theorem 1.4.15. All von Neumann algebras with separable predual are gener-ated by a countable family of projections.

Proof. [SS08], Lemma 2.3.5, p. 10.

Theorem 1.4.16. Let M be a von Neumann algebra. Then M has separablepredual if and only if

(i) M is countably generated, and

(ii) M is countably decomposable (= -finite).

Proof. [Top71], 9, p. 64.

1.5 Structure theorems

Theorem 1.5.1. Let M be a finite-dimensional von Neumann algebra. Thenthere exist positive integers n1, . . . , nk such that

M = Mn1(C) Mnk(C).

Proof. [Aas13], Proposition 1.5.1, p. 18.

Theorem 1.5.2. If A is a separable diffuse abelian von Neumann algebraequipped with a faithful normal trace , then there is a -isomorphism

(A, ) (L([0, 1]),m),

where m is (integration against) Lebesgue measure.

Proof. [SS08], Theorem 3.5.2, p. 35.

11

Theorem 1.5.3. If A is a separable abelian von Neumann algebra, then thereis a -isomorphism

A L(K,),where K is some compact Hausdorff topological space and is a finite positiveBorel measure on K with supp() = K.

Proof. [Zhu93], Theorem 22.6, p. 132 (and Theorem 22.5, p. 131).

Theorem 1.5.4. If A is an abelian von Neumann algebra, then there is a -isomorphism

A L(X, ),where (X, ) is some locally finite measure space.

Proof. [Bla06], III.1.5.18, p. 237.

Theorem 1.5.5. If M is a type I von Neumann algebra, then so is M .

Proof. [Top71], 5, p. 37.Similar results hold for von Neumann algebras of the other two types ([KR86],

Theorem 9.1.3, p. 588).

Theorem 1.5.6. Let M be a von Neumann algebra. Then M is semifinite ifand only if M is semifinite.

Proof. [KR86], Corollary 9.1.4, p. 589.

1.6 Conditional expectations

Theorem 1.6.1. If M is a von Neumann algebra equipped with a faithful normaltrace , and if B M is a von Neumann subalgebra, then the conditionalexpectation EB of M onto B is the restriction of the orthogonal projection eBof L2(M, ) onto L2(B). It is a linear normal trace-preserving contractive ucpB-bimodular map and is uniquely determined as follows:

(i) It is the unique map : M B which is B-bimodular and -preserving.(ii) It is the unique map : M B which is a -preserving linear contractive

projection (i.e., (b) = b for all b B).The fact that (ii) implies (i) is basically Tomiyamas theorem (see [Aas11],Theorem 2.4.2, p. 11 for the precise statement).

Proof. [SS08], Lemma 3.6.2, p. 39, and [BO08], Theorem 1.5.10, p. 12.

Note that the existence of EB under the assumptions given above was firstproved by Hisaharu Umegaki in 1954.

Lemma 1.6.2. The projection eB has the following properties:

12

(i) CeB = 1 in M {eB} and, for all x M , eBxeB = EB(x)eB.(ii) M {eB} = B.

(iii) M {eB} = {xeBy : x, y M}.(iv) M {eB} = (JBJ).(v) eB(M {eB})eB = BeB.

Proof. [Aas13], Lemma 1.12.1, p. 31.

Theorem 1.6.3. In certain situations we have formulae for the conditionalexpectation:

(i) If H G are countable groups, M = L(G) L(H) = B, then EB is theunique normal extension to a map M B of the map C[G] C[H] thatthrows away elements from G \H.

(ii) If {bn}n=1 is an orthonormal basis for L2(B), then

EB(x) =n=1

(xbn)bn.

In particular, if B is finite-dimensional abelian so that L2(B) has a finiteorthogonal basis {ej}nj=1 consisting of the minimal projections in B, then

EB(x) =nj=1

(ej)1(xej)ej .

Another special case is that of a diffuse abelian subalgebra B of a II1 factorM , in which case B is generated by a Haar unitary u and

EB(x) =

n=(xun)un.

Finally, if B = Mn(C) has a family of matrix units {eij}, then

EB(x) =n

i,j=1

(ejj)1(xeji)eij .

(iii) If p is a projection in B, then

EpBp = EB |pNp.

(iv) If p is a projection in B M , thenEpB(x) = pEB(EB(p)1x),

where the inverse is to be understood appropriately.

13

(v) EBM (x) is the unique element of smallest 2-norm in the WO-closedconvex hull of

{uxu : u U(B)}.

(vi) If B is finite-dimensional abelian with minimal projections e1, . . . , en, then

EBM (x) =nj=1

ejxej .

Proof. [SS08], Remark 3.6.4, p. 42, Lemma 3.6.5, p. 44, and Remark 3.6.7, p.46.

Lemma 1.6.4. If A is the WO-closure of the union of an increasing sequenceof abelian subalgebras A1, A2, . . . of M , then

(i) EAn(x) EA(x)2 0 for all x M , and(ii) EAnM (x) EAM (x)2 0 for all x M .

Proof. [SS08], Lemma 3.6.8, p. 46.

1.7 Some useful estimates

For more estimates than those listed here, cf. Dixmier 1969, III.3, p. 310, and[SS08], Lemmas 5.2.6, 5.2.9, 5.2.10 and 5.3.1, pp. 8489.

Theorem 1.7.1 (Equivalent projections). Let M be a von Neumann algebra,and let e and f be equivalent finite projections in M . Then there exist a partialisometry w M and u U(M) such that

(i) ww = e and ww = f ;

(ii) w and u commute with |e f |;(iii) |e w|, |w f | 2|e f |;(iv) ueu = f ;

(v) |1 u| 2|e f |.Proof. [SS08], Theorem 5.2.5, p. 82.

Theorem 1.7.2 (Polar decomposition). Let M be a von Neumann algebra,equipped with a faithful normal trace . Let x M have polar decompositionx = vs, where s = (xx)1/2, and put p = vv, q = vv. If e M is a projectionwith ex = x, then

(i) p s2 e x2;(ii) e q2 e x2;

14

(iii) e v2 2e x2.Proof. [SS08], Lemma 5.2.7, p. 85.

Theorem 1.7.3. Let B M be von Neumann algebras, where M is equippedwith a faithful normal trace . If e1, e2 M are two projections, then there aretwo orthogonal projections f1, f2 B such that

ej fj2 2(21/2 + 1)ej EB(ej)2 + 21/2e1e22.Proof. [SS08], Lemma 5.3.3, p. 91.

1.8 Actions, representations, and crossed products

Lemma 1.8.1. Let be a trace-preserving action of the countable discrete group on the von Neumann algebra B. Then

(i) is free B o B = Z (B);(ii) if is free and ergodic, then B o is a factor;

(iii) if B is abelian and B o is a factor, then is ergodic.


Given an action : y (X,) of a group , we define the correspondingaction : y L(X,) by [(g)f ](x) = f((g)1x) for g , x X, andf L(X,). Note that if is pmp, then is trace-preserving.Lemma 1.8.2. Let : y (X,) be a pmp action, where (X,) is a probabilityspace, and let : y L(X,) be the corresponding trace-preserving action.

(i) is free is free,(ii) is ergodic is ergodic.


See [Vae07], Proposition D.2, p. 49, for characterizations of weakly mixingactions on finite von Neumann algebras.

See [Pet11], Proposition 1.7.5, p. 23, and [Pop], Lemma 1.3.3, p. 6, forcharacterizations of weakly mixing unitary representations.

See [Pet11], Lemma 1.7.10 and Theorem 1.7.11, p. 25, for characterizationsof compact unitary representations.

Theorem 1.8.3 (Dixmier). Let pi and be unitary representations of . Denoteby pi and the unique extensions to -homomorphisms on C(). Then thefollowing are equivalent:

(i) pi , i.e., pi is weakly contained in ;(ii) ker ker pi;

15

(iii) There exists a -homomorphism : C() Cpi() such that =pi, where the mentioned C-algebras are generated by the images of therespective representations.

Theorem 1.8.4 (Fell absorption principle). Let pi be a unitary representa-tion of . Then pi is unitarily equivalent with 1.Proof. [BO08], Theorem 2.5.5, p. 44.

1.9 Unbounded operators and Borel functional calculus

1.9.1 Polar decomposition and spectral theory

Lemma 1.9.1. Let T be a densely defined operator on a Hilbert space H. ThenT is closed and, moreover, T is closable if and only if T is densely defined, inwhich case T = T .

Proof. [Con07], Proposition 1.6, p. 305, or [SS08], Lemma B.2.2, p. 345.

Theorem 1.9.2 (Polar decomposition of operators). Let T be a denselydefined operator.

(i) If T is a positive operator, then T has a unique positive square rootT .

(ii) If T is a closed operator and |T | = T T , then there exists a uniquepartial isometry v that satisfies T = v|T | (called the polar decompositionof T ) and has initial space Ran|T | and final space RanT .

(iii) If T is closed with polar decomposition T = v|T |, then T = |T |v.(iv) If T is a closed operator with polar decomposition T = v|T | and T is affil-

iated with the von Neumann algebra M , then v M and |T | is affiliatedwith M .

Proof. [SS08], Theorem B.3.7, p. 354.

Theorem 1.9.3 (Borel functional calculus for positive operators). LetT be a densely defined positive operator on a Hilbert space H. Then, to eachf Bloc([0,)), there is assigned a closed densely defined operator f(T ) withthe following properties:

(i) f(T ) = I if f(t) 1;(ii) f(T ) = T if f(t) = t;

(iii) f(T ) B(H) whenever f B([0,)), and the map B([0,)) 3 f 7f(T ) B(H) is a -homomorphism;

(iv) (f + g)(T ) = f(T ) + g(T ) and (fg)(T ) = f(T )g(T ) whenever f, g Bloc([0,));

(v) f(T ) = f(T ) and f(T )f(T ) = |f |2(T ) whenever f Bloc([0,));

16

(vi) if T is affiliated to a von Neumann algebra M B(H), then so is f(T )for each f Bloc([0,)).


Theorem 1.9.4. Let T be a closed symmetric (i.e., T T ) operator. Thenprecisely one of the following conditions is satisfied:

(i) (T ) = C;

(ii) (T ) is the closed upper half plane;

(iii) (T ) is the closed lower half plane;

(iv) (T ) R.Moreover, the following conditions are equivalent:

(i) T is self-adjoint;

(ii) (T ) R;(iii) ker(T i) = ker(T + i) = {0}.

Proof. [Con07], Theorem 2.8 and Corollary 2.9, p. 311.

Theorem 1.9.5 (Spectral theorem for unbounded normal operators).If N is a normal operator on a Hilbert space H, then there exists a uniquespectral measure E defined on the Borel subsets of C such that:

(i) N =z dE(z);

(ii) E() = 0 if (N) = ;(iii) E() 6= 0 whenever is an open set such that (N) 6= ;(iv) if A B(H) satisfies AN NA and AN NA then A dE (

dE)A.

Proof. [Con07], The Spectral Theorem 4.11, p. 323.

See also [Bla06], I.7.4, for the special case of self-adjoint unbounded oper-ators.

Theorem 1.9.6 (Stone 1932). If U is a strongly continuous 1-parameter unitarygroup, then there exists a self-adjoint operator A such that U(t) = exp(itA).

Proof. [Con07], Stones Theorem 5.6, p. 330.

17

1.9.2 Operators associated to elements of L2(M)

In this section M is a given von Neumann algebra, equipped with a faithfulnormal trace . We use the notation

L2(M)+ = (N+)12

.

Given L2(M), we let L be the closure of the operator ` with domain Mand defined by `(x) = x = Jx

J.

Proposition 1.9.7. Let L2(M) be given. Then(i) L = LJ;

(ii) L2(M)+ L 0;(iii) |L| = L for some L2(M)+.


Theorem 1.9.8. The map 7 L is a bijection of L2(M) onto the set ofclosed, square integrable operators on L2(M) that are affiliated with M .


1.9.3 Operators associated to elements of L1(M)

In this section M is a given von Neumann algebra, equipped with a faithfulnormal trace . Given N, we let M be the closure of the operator mwith domain

{y M : |(yz)| kyz2 for some ky > 0 and all z M}

and defined bym(y), z = (yz).

Moreover, (x) = (x)

Proposition 1.9.9. Let M be given. Then(i) M = M ;

(ii) 0 M 0;(iii) |M| = M for some M+ .

Proof. [SS08], Proposition B.5.4, p. 365, and Proposition B.5.6, p. 367.

Theorem 1.9.10. The map 7 M is an injection of M into the set ofclosed, densely defined operators on L2(M) that are affiliated with M .

Proof. [SS08], Lemma B.5.5 and Proposition B.5.6, p. 367.

18

The isometric isomorphism between L1(M) and M identifies L1(M)with the normal linear functional (), where denotes the unique continuousextension of to L1(M). We put

L1(M)+ = { L1(M) : () M+ }.Note also that the product map M M M extends uniquely to a continuousmap L2(M) L2(M) L1(M). By identifying L2(M) with the image inL1(M) of (, 1), we may regard L2(M) as a linear subspace of L1(M).

Theorem 1.9.11. Let L1(M) be given.(i) L1(M)+ if and only if there exists a sequence {xn}n=1 in M+ such

that limn xn 1 = 0;(ii) If L2(M) then L = M;

(iii) The product map L2(M) L2(M) L1(M) is surjective;(iv) L1(M)+ if and only if there exists L2(M) such that is the image

of (, ) under the product map;

(v) There exists L1(N)+ and a partial isometry v M such that = v.Proof. [SS08], Theorem B.5.9, p. 371.

1.10 Derivations

Theorem 1.10.1. Let A be a C-algebra, and let : A A be a derivation.Then is bounded and, if A B(H), extends to an ultraweakly continuousderivation : A A, where A is the WOT-closure of A. Moreover, annihilatesthe center of A and leaves invariant any norm-closed two-sided ideal in A.

Proof. [SS95], Theorem 2.2.1, p. 61, Theorem 2.2.2, p. 62, and Lemmas 2.2.3and 2.2.4, p. 64.

Theorem 1.10.2 (Kadison-Sakai 1966). Let M be a von Neumann algebra, andlet : M M be a derivation. Then is an inner derivation. More precisely,there exists m M such that m = /2 and (x) = xmmx for all x M .Proof. [SS95], Theorem 2.5.1, p. 70, and Corollary 2.5.5, p. 75.

2 Special von Neumann algebras

2.1 Uncountably many non-isomorphic factors

Note first that property allows one to distinguish the hyperfinite II1 factor Rfrom the free group factors, cf. subsection 5.3.

Theorem 2.1.1 (McDuff 1969). If H is a separable Hilbert space, then B(H)contains uncountably many non-isomorphic II1 factors.

Proof. [McD69].

19

2.2 Amalgamated free products

Theorem 2.2.1 (Compression Formula Dykema-Radulescu 2000). If(Mi)iI is an infinite family of II1 factors, then

(iIMi)t = iIM ti .

Proof. [DR00].

3 Orthogonal subalgebras

Let M be a von Neumann algebra, equipped with a faithful normal trace .Given a non-empty subset S M , put

S = {x M : (xs) = 0 for all s S}.

If B M is a von Neumann subalgebra, then (B) = B, cf. [SS08], p. 104.Lemma 3.0.2 (Popa 1983). Let M be as above, and let B1, B2 M be vonNeumann subalgebras. The following conditions are equivalent (to each otherand to the corresponding statements with B1 and B2 interchanged):

(i) If x B1, y B2 and (x) = (y) = 0, then (xy) = 0;(ii) If x B1 and y B2, then (xy) = (x)(y);

(iii) If x B1 and y B2, then xy2 = x2y2;(iv) If z M , then EB1EB2(z) = (z)1;(v) EB1(B2) = C1.

Proof. [SS08], Lemma 6.3.1, p. 104.

This notion of orthogonality was used in [Pop83] to prove the following facts(as well as results concerning M and L(FS) for uncountable S, cf. subsection4.10.2 as well as Theorems 13.3.4 and 13.3.5):

Theorem 3.0.3 (Popa 1983). Let B M be von Neumann algebras, where Mis equipped with a faithful normal trace , and let u U(M) be given. If thereexists a diffuse von Neumann subalgebra B0 B such that uB0u B, thenu NM (B).Proof. [Pop83], Corollary 2.6, p. 257. See also [SS08], Lemma 6.3.2, p. 105.

Theorem 3.0.4 (Popa 1983). If n 2, then Mn(C) contains two orthogonalmasas. If n is prime, then Mn(C) contains at least n + 1 pairwise orthogonalmasas.

Proof. [Pop83], Theorem 3.2, p. 258.

20

Conjecture 3.0.5 (Popa 1983). If n is prime and A0 and A1 are orthogonalmasas in Mn(C), then there exist u0 U(A0) and u1 U(A1) such that u1u0 =e2pii/nu0u1.

The previous conjecture is labeled as Conjecture 3.4 on p. 258 of [Pop83].

Theorem 3.0.6 (Popa 1983). Let G0 G1 G be infinite discrete groups.Suppose that

gG0g1 G0 = {1}

for any g G \ G1. If B L(G0) is a diffuse von Neumann subalgebra, thenNL(G)(B) L(G1). If, in addition, G0 is a normal subgroup of G1, thenNL(G)(L(G0)) = L(G1).Proof. [Pop83], Proposition 4.1, p. 260.

Theorem 3.0.7. Let G be a discrete group, and let G1 and G2 be infinitesubgroups of G. Suppose that

gG2g1 G1 = {1}

for all g G. If B1 L(G1) and B2 L(G2) are diffuse von Neumannsubalgebras, then there exists no u U(L(G)) such that uB2u = B1.Proof. [Pop83], Corollary 4.3, p. 260.

4 Masas

4.1 Masas in type I von Neumann algebras

Lemma 4.1.1. Let H be a separable Hilbert space. Every masa A B(H) hasa cyclic (and, hence, separating) vector.

Proof. [Aas13], Lemma 2.1.14, p. 40.

Theorem 4.1.2. Let H be a separable Hilbert space. Every diffuse masa A B(H) is spatially isomorphic to the masa L([0, 1]) acting on L2([0, 1]).

Proof. [SS08], Lemma 2.3.6, p. 10.

Theorem 4.1.3. Let H be a separable Hilbert space. Every masa A B(H)is spatially isomorphic to a direct sum of L([0, 1]) (acting on L2([0, 1])) and adiagonal masa (acting on some B(H) with dimH


Corollary 4.1.5. Each masa in a finite type I von Neumann algebra is unitarilyequivalent to a direct sum of diagonal masas.

Corollary 4.1.6. Each masa in a finite type I von Neumann algebra containsan abelian projection whose central support is 1.

Proof. [SS08], Corollary 2.4.6, p. 16.

4.2 Normalizing groups and groupoids

Lemma 4.2.1 (Jones-Popa 1982). Let M be a von Neumann algebra, equippedwith a faithful normal trace , and let A be an abelian von Neumann subalgebraof M .

(i) If v1, v2 GNM (A), then v1v2, v1 GNM (A);(ii) If v1, v2 GNM (A) and v1v2 = v2v1 = 0, then v1 + v2 GNM (A);

(iii) Let v M be a partial isometry. Then v GNM (A) if and only if thereexists a projection e A and u NM (A) such that v = ue = (ueu)u;

(iv) GNM (A) spanNM (A) so that GNM (A) = NM (A).Proof. [SS08], Lemma 6.2.3, p. 99.

Lemma 4.2.2 (Jones-Popa 1982). Let M be a von Neumann algebra, equippedwith a faithful normal trace , and let A be a masa in M .

(i) Let A B M be a von Neumann algebra. If v GNM (A), then thereexists a projection e A such that e vv and EB(v) = ve. Moreover,e A is a maximal projection with respect to the conditions ve B ande vv. Hence

EB(GNM (A)) = GNB(A).

(ii) Let be an automorphism of A, and let v M be a partial isometrysuch that va = (a)v for all a A. Put e = vv and f = vv. Thenv GNM (A), (e) = f , and there exists u U(M) such that v = ue.

Proof. [SS08], Lemma 6.2.4, p. 101.

4.3 Hereditary properties

Theorem 4.3.1 (Jones-Popa 1982, Popa 1985). Let A B M be von Neu-mann algebras, where A is a masa in M and M is equipped with a faithfulnormal trace.

(i) If A is a Cartan subalgebra of M , then A is a Cartan subalgebra of B;

(ii) If A is a singular masa in M , then A is singular in B;

22

(iii) If A is a simple masa in M , then A is a simple masa in B;

(iv) Any Cartan subalgebra is simple;

(v) Even if A is a semiregular masa in M , A need not be semiregular in B.

Proof. (i): [SS08], Corollary 6.2.5, p. 102.(ii): Obvious.(iii): [Aas13], Lemma 2.1.5, p. 35.(iv): [Pop85], Corollary 3.2, p. 178.(v): [JP82], Example 2.6.

4.4 Projections in masas

Lemma 4.4.1. If A is a masa in M and p is a projection in A, then pA is amasa in pMp. Moreover, if A is a Cartan subalgebra of M , then pA is a Cartansubalgebra of pMp.


Lemma 4.4.2 (Popa). Let A be a semiregular masa in a II1 factor M . Ifp, q A are projections of equal trace, then there exists a partial isometry v GNM (A) such that p = vv and q = vv. (Indeed, there exists u NM (A) suchthat p = uqu.)

Proof. [SS08], Lemma 6.2.6, p. 103. ([Aas11], Lemma 6.2.2, p. 75.)

Theorem 4.4.3 (Popa). If M is a finite von Neumann algebra containing amasa A, then every projection in M is equivalent to a projection from A.


Theorem 4.4.4. Let B be a masa in a II1 factor M . If e is a projection in M ,then there is a projection f B such that e f in M , and

e f2 23/2e EB(e)2Proof. [SS08], Lemma 5.3.2, p. 90.

Open problem 4.4.5. Find the best possible constant in the previous theorem.(Source: [SS08], p. 90.)

Theorem 4.4.6 (Kadison 1983). Let B be a masa in a type II1 von Neumannalgebra M , and let p B be a non-zero projection. If h Z(M) and 0 h ctr(p), then there is a projection q B such that q p and ctr(q) = h.Proof. [SS08], Theorem 5.6.2, p. 96.

23

4.5 Unitary conjugacy of semiregular masas

Theorem 4.5.1 (Popa). Let A and B be semiregular masas in a II1 factor M .If there exists a non-zero partial isometry v M such that vv A, vv B,and vAv = Bvv, then there exists u U(M) such that uAu = B.Proof. [SS08], Theorem 6.2.7, p. 103.

More generally, if B M A (i.e., a corner of B embeds into a corner ofA, cf. Intertwining-by-bimodules, below), then we get the same conclusion (cf.[Aas11], Proposition 6.2.6, p. 77).

4.6 Cartan subalgebras

Open problem 4.6.1. Give an abstract property of II1-factors that is equiv-alent to the existence of a Cartan subalgebra.

Open problem 4.6.2. Construct a II1 factor with exactly n Cartan subalgebras(up to conjugation by a unitary).

Theorem 4.6.3. Let M be a separable II1 factor and let A be a Cartan subal-gebra of M . Then there exists an irreducible hyperfinite subfactor R of M suchthat A is a Cartan subalgebra of R.


Theorem 4.6.4 (Popa 1985). Let A be a Cartan subalgebra in a II1 factorM , and let N be a factor with A N M . Then there exists a family{uj} NM (A) such that (a) Nui Nuj whenever i 6= j, and (b) M =

Nuj.


Corollary 4.6.5 (Popa 1985). Let M be a separable II1 factor and let N Mbe a subfactor containing a Cartan subalgebra of M . Then [M : N ] N {}.Proof. [Pop85], Corollary 2.4, p. 176.

Corollary 4.6.6 (Popa 1985). Let A be a Cartan subalgebra in a separableII1 factor M . Then there exists a sequence {uk}k=1 NM (A) such that (a)Auj Aui whenever i 6= j, and (b) M =

Auk.

Proof. [Pop85], Corollary 2.5, p. 176.

Corollary 4.6.7 (Popa 1985). Let A be a Cartan subalgebra in a separable II1factor M . Then L2(M) has a countable orthonormal basis {uk}k=1 NM (A).Proof. [Pop85], Corollary 2.6, p. 177.

Open problem 4.6.8. Do all II1-factors have an orthonormal basis consistingof unitary elements?

24

Theorem 4.6.9 (Popa 1985). Let M be a separable II1 factor containing amasa A. If A is a Cartan subalgebra, or more generally a simple masa, then Mis singly generated.

Proof. [Pop85], Theorem 3.4, p. 180.

4.7 Uniqueness of Cartan subalgebras

See also subsection 4.5 and note that Cartan subalgebras may not be unique aswas first observed in [CJ82].

Theorem 4.7.1 (Connes-Feldman-Weiss 1981). If A and B are Cartan sub-algebras of the hyperfinite II1 factor R, then there exists an automorphism Aut(R) such that (A) = B.Proof. [Pop85], Theorem 4.1, p. 181.

Open problem 4.7.2. Does L(X)o have a unique Cartan subalgebra when is non-amenable and the action is a Bernoulli action?

Open problem 4.7.3. Do there exist mixing actions of on X, with non-amenable, such that L(X)o does not have a unique Cartan subalgebra?

4.8 Singular masas

Theorem 4.8.1 (Characterization of singular masas Popa 2001, Sin-clair-Smith 2002). Let A be a masa in a separable II1 factor M . The followingconditions are equivalent:

(i) A is singular;

(ii) A is strongly singular;

(iii) A has the weak asymptotic homomorphism property;

(iv) For each partial isometry v M with orthogonal initial and final projec-tions in A,

sup{(I EA)(vxv)2 : x (A)1} = v22.


Let us also mention the following related results:

Theorem 4.8.2. Let A be a masa in a separable II1 factor M , and let > 0be given.

(i) If u U(A) satisfies EA EuAu,2 , then there exists u NM (A)such that u u2 5.

25

(ii) The inequality

5EA EuAu,2 u ENM (A)(u)2holds for all u U(M).


Corollary 4.8.3. Let A be a singular masa in a separable II1-factor M . ThenA has the weak asymptotic homomorphism property in M and is stronglysingular.


Theorem 4.8.4. Let A be a masa in a separable II1 factor M . Then A issingular in M if and only if A is singular in M.


4.9 Special masas in separable factors

Theorem 4.9.1 (Popa 1981, Popa 1983). Let M be a separable II1 factor, andlet B be a subfactor with B M = C1.

(i) There is a singular masa A of M contained in B.

(ii) There is a masa A of M and a hyperfinite subfactor R of B such that Ais a Cartan subalgebra of R.

(iii) If B is a hyperfinite subfactor of M , then there is a masa A in M suchthat A is a Cartan subalgebra of B.

Note that (i)(iii) are immediate consequences of the following theorems.Compare the previous theorem to Theorem 4.3.1.

Theorem 4.9.2 (Popa 1981). Let M0 be a von Neumann subalgebra of a sep-arable II1 factor M with M

0 M M0. If B is a finite-dimensional abelian

-subalgebra of M0, then there is a masa A of M with B A M0.Proof. [SS08], Theorem 12.2.4, p. 229.

Theorem 4.9.3 (Popa 1981). Let M be a separable II1 factor.

(i) If M0 is a subfactor of M with M0M = C1 and B is a finite-dimensional

abelian -subalgebra of M0 with all its minimal projections of equal trace,then there exists a masa A in M containing B and a hyperfinite subfactorR of M0 such that A is a Cartan subalgebra of R and R

M = C1.(ii) If R is a hyperfinite subfactor of M with R M = C1, then there exists

a masa A in M such that A is a Cartan subalgebra of R.


26

Theorem 4.9.4 (Popa 1983). Let M be a II1 factor and let M0 be a subfactorof M with M 0 M = C1. If B is a finite-dimensional abelian -subalgebra ofM0, then there is a singular masa A in M such that B A M0.Proof. [SS08], Theorem 12.4.3, p. 237.

Theorem 4.9.5 (Christensen-Pop-Sinclair-Smith 2003). Let M be a separableII1 factor with property . There exists a masa A in M and an irreduciblehyperfinite subfactor R of M such that A is a Cartan subalgebra of R and,if x1, . . . , xm M , n N and > 0, then there exist pairwise orthogonalprojections p1, . . . , pn A, each of trace 1/n, such that

pixj xjpi2 <

for all 1 i n and 1 j m.Proof. [SS08], Theorem 13.5.4, p. 255.

4.10 Examples of masas

4.10.1 Examples in group von Neumann algebras

Lemma 4.10.1. Let H G be an inclusion of ICC countable discrete groups.If #{hgh1 : h H} = for all g G \H, then L(H) is a masa in L(G).Moreover,

(i) L(H) is a Cartan subalgebra of L(G) if H is a normal subgroup of G.

(ii) L(H) is a singular masa in L(G) if, for each finite set B G \H,

BHB (G \H) 6= .

Proof. [SS08], Lemma 3.3.1, p. 22, and Lemma 3.3.2, p. 23. See [SS08], Lemma11.2.4, p. 203, for a shorter proof of (ii) using Theorem 4.8.1, and remarks onp. 106 of [SS08] for a proof using the notion of orthogonality (cf. section 3).

Example 4.10.2. Consider the groups

G =

{(a x0 1

): a Q, x Q

},

H1 =

{(1 x0 1

): x Q

},

H2 =

{(a 00 1

): a Q

}.

Then G is amenable, which implies that L(G) = R, and by checking the con-ditions in the previous lemma one can verify that L(H1) is a Cartan masa inL(G) while L(H2) is a singular masa in L(G).

27

Proof. [SS08], Example 3.3.3, p. 24.

Example 4.10.3. Consider the groups G = Fn = g1, . . . , gn and H = g1(with n 2). By checking the conditions in the previous lemma one can verifythat L(H) is a singular masa in L(G). Moreover, it can be shown (cf. [SS08],Theorem 14.1.1, p. 257) that L(H) is a maximal injective subalgebra of L(G).


Theorem 4.10.4. Let F2 be the free group with generators a and b. The Lapla-cian algebra in L(F2) generated by a+a1 +b+b1 is a singular masa in L(F2).

Proof. [SS08], Theorems 11.6.3 and 11.6.4, p. 221.

Lemma 4.10.5. Let H G K be inclusions of countable discrete groups.Assume that G and K are ICC groups, that H is an abelian normal subgroup ofG, and that #{hkh1 : h H} = for all k K \H. If one of the followingtwo conditions is satisfied, then L(H) is a semi-regular masa in L(K) whosenormalizer generates L(G):

(i) If k K \ G and a finite set B K are given, then there exists h Hsuch that k1hk K \H and, for all b B, hbh1 = b or hbh1 / B.

(ii) If k K \G and a finite set F K \H are given, then there exists h Hsuch that k1hk K \H and hFh1 F = .

Proof. [SS08], Lemma 3.3.4, p. 25.

Example 4.10.6. Consider the groups

K = PGL2(Q),

G =

{(a x0 1

): a Q, x Q

},

H =

{(1 x0 1

): x Q

}.

As G and H intersect the center of GL2(Q) trivially, we have inclusions H G K. By checking the conditions in the previous lemma one can verify thatL(H) is a semi-regular masa in L(K) whose normalizer generates L(G).


Lemma 4.10.7. Let H and K be infinite abelian discrete groups and let : K Aut(H) be a group homomorphism such that k(h) 6= h whenever h 6= e andk 6= e. Then G = H o K is an ICC group, L(H) is a Cartan masa in L(G),and L(K) is a singular masa in L(G).

Proof. [SS08], Lemma 3.4.1, p. 28.

See also [Aas11], 3.5.2, p. 32, for a discussion of crossed products arisingfrom semi-direct products of groups.

28

Example 4.10.8. Consider the groups K = Z and H = {h : K {1} :#{x K : h(x) 6= 1} < }. For k K define k Aut(H) by k(h)x =h(x k), x K. One can check that the conditions in the previous lemma aresatisfied.


4.10.2 Masas in non-separable von Neumann algebras

Theorem 4.10.9 (Popa 1983). Let M be a separable II1 factor.

(i) Every masa in M is non-separable;

(ii) If A and B are separable diffuse abelian von Neumann algebras of M,then there exists u U(M) such that uAu = B;

(iii) If A is a separable diffuse abelian von Neumann subalgebra of M, thenthere exists a hyperfinite subfactor R of M such that A is a Cartansubalgebra of R.

Proof. (i): [SS08], Theorem 15.2.3, p. 270.(ii): [SS08], Theorem 15.2.5, p. 271.

Theorem 4.10.10 (Popa 1983). If S is an uncountable set, then all abelianvon Neumann subalgebras of L(FS) are separable.


See also Theorem 13.3.4.

5 Property Gamma

5.1 Characterizations

Theorem 5.1.1. Let M be a II1 factor. The following conditions are equivalent:

(i) M has property ;

(ii) Given x1, . . . , xk M , there exists a sequence {un}n=1 in U(M) such thatlimn (un) = 0 and limn unxi xiun2 = 0 for 1 i k.


Theorem 5.1.2 (Connes). Let M be a separable II1 factor. The followingconditions are equivalent:


(ii) M M 6= C1;(iii) M M is diffuse;

29

(iv) There exists a sequence {n} of unit vectors in L2(M) C1 such that[n, x]2 0 for each x M ;

(v) The only compact operator in M M B(L2(M)) is 0.Proof. (i) (ii): [SS08], Lemma A.7.3, p. 340.

See [Hou11] for a complete proof of the previous theorem.

Note that the sequence in (iv) consists of almost invariant vectors for theadjoint representation of U(M) on L2(M) C1. That condition is thereforeknown as the spectral gap condition for property .

Theorem 5.1.3 (Dixmier 1969). Let M be a separable II1 factor. The followingconditions are equivalent:


(ii) Given x1, . . . , xm M , k N and > 0, there exist mutually orthogonalprojections p1, . . . , pk M each with trace 1/k such that

pixj xjpi2 <

for 1 j m and 1 i k.Proof. [SS08], Theorem A.7.5, p. 341.

Theorem 5.1.4. Let M be a II1 factor, and let r N. Then M has property if and only if Mr(M) has property .


5.2 Relationship with inner amenable groups

Edward Effros introduced the notion of inner amenability in ca. 1975 in anattempt to describe property on the group level. Indeed, Effros proved thatG is inner amenable if L(G) has property , but the converse turned out to befalse as Stefaan Vaes recently (cf. [Vae12]) came up with a counterexample.

5.3 Examples

Example 5.3.1. The hyperfinite II1 factor R and any McDuff factor MRhas property .

Proof. [SS08], Example 13.4.8, p. 252. (Approximate by increasing sequence oftensor products.)

Example 5.3.2. The free group factors L(Fk), k = 2, 3, . . ., do not have prop-erty . In particular, L(Fk) 6= R.Proof. [SS08], Theorem A.7.2, p. 339.

30

6 Finitely generated von Neumann algebras

Note that a von Neumann algebra is generated by a single element (singlygenerated) if and only if it is generated by a set of two self-adjoint elements.

Open problem 6.0.3 (Finite generation problem). Show that L(F) isnot finitely generated as a von Neumann algebra (or give another example ofsuch a II1 factor).

Theorem 6.0.4. Let M be a finitely generated von Neumann algebra. If M =M2(M), then M is singly generated.

Proof. [Top71], 9, p. 77.Theorem 6.0.5 (Wogen 1969). All separable properly infinite von Neumannalgebras are singly generated.

Proof. [Top71], 10, p. 78.If a separable II1 factor contains a Cartan subalgebra, then it is singly gen-

erated, cf. Theorem 4.6.9.

Theorem 6.0.6 (Gaboriau-Popa 2005). These factors are singly generated:

(i) tensor products of separable II1 factors;

(ii) separable II1 factors with property .

Proof. [SS08], Theorem 16.8.1, p. 312, and Theorem 16.8.5, p. 313. Warning:Both proofs use Shens generation invariant.

Open problem 6.0.7. Are the free group factors L(Fn), 3 n < , singlygenerated?

It is known that the minimal number of self-adjoint generators of L(Fn) isat most n.

7 Standard non-commutative results

7.1 Some C*-algebraic results

Theorem 7.1.1 (Stinespring). Let A be a C-algebra and let : A B(H) be ac.p. map. Then there exists a Hilbert space K, a -representation pi : A B(K),and a bounded linear map V : H K such that

(a) = V pi(a)V

for all a A.Proof. [SS95], Theorem 1.2.1, p. 12.

31

Theorem 7.1.2 (Glimm). Let A B(H) be a separable C-algebra such thatAK(H) = {0}. If is a state on A, then there exist orthonormal vectors {n}such that an, n (a) for all a A.Proof. [BO08], Lemma 1.4.11, p. 8.

Below, we say that a state can be excised (resp., by projections) if thereexists a net (ei) with 0 ei 1 (resp., e2i = ei = ei ), (ei) = 1, andlimi eiaei (a)e2i = 0 for all a A.Theorem 7.1.3 (Excision). Let A be a C-algebra.

(i) Any pure state on A can be excised;

(ii) If A is unital, simple, and infinite-dimensional of real rank zero, meaningthat any self-adjoint element can be approximated in norm by self-adjointelements with finite spectrum, then any state on A can be excised by pro-jections.

Proof. [BO08], Theorem 1.4.10, p. 8, and Proposition 11.4.2, p. 328.

Other important theorems include: Existence of approximate identities (withadditional properties), Tomiyamas Theorem [BO08], existence of multiplicativedomains [BO08], Arvesons Extension Theorem, and Voiculescus Theorem.

7.2 Powers-Strmer type inequalities

Theorem 7.2.1. Let M be a finite von Neumann algebra, equipped with afaithful normal trace . If a, b M+ then

a1/2 b1/222 2a b1.

Proof. [SS08], Lemma B.5.1, p. 364.

7.3 Rokhlin type results

Lemma 7.3.1 (Popa 1986). Let A be a masa in a tracial von Neumann algebra(M, ). If f A is a non-zero projection, y1, . . . , yt M , and > 0, then thereexists a projection e A, e f , such that

eyie ie2 < e2,

where i = (eyie)/(e).


32

7.4 Radon-Nikodym type results

Theorem 7.4.1 (Sakai-Radon-Nikodym theorem). If and 0 are normalpositive linear functionals on a von Neumann algebra M and 0 , then thereexists h (M)1 M+ such that

0(x) = (hxh)


Theorem 7.4.2 (Linear Radon-Nikodym theorem). If is a normal pos-itive linear functional on a von Neumann algebra M , and 0 is a positive linearfunctional such that 0 , then there exists k (M)1 M+ such that

0(x) =1

2(kx+ xk)


8 Perturbation results

8.1 Distance between subalgebras

The most useful norm for comparing subalgebras arises from the following op-erator norm: If : M M is a map on a II1 factor M , then put

,2 = sup{(x)2 : x 1},i.e., the operator norm of the map considered as a map from (M, ) into(M, 2).

Next we put d2(A,B) = EAEB,2 for any two von Neumann subalgebrasA and B of M .

Theorem 8.1.1 (Christensen). The set of all von Neumann subalgebras of M ,equipped with the norm just defined, is a complete metric space. Moreover, thefollowing sets are closed subspaces of this metric space:

(i) The set of abelian subalgebras;

(ii) The set of masas;

(iii) The set of singular masas;

(iv) The set of subfactors;

(v) The set of irreducible subfactors.

Proof. [SS08], Lemma 6.4.1, p. 107, and Corollary 6.4.5, p. 111.

33

8.2 Basic perturbation results

Theorem 8.2.1 (Popa-Sinclair-Smith 2004). Let A B be abelian von Neu-mann algebras, where B is equipped with a faithful normal trace . If B Afor some > 0 (i.e., supb(B)1 bEA(b)2 ), then there exists a projectionp B such that Ap = Bp and (p) > 1 42.Proof. [SS08], Theorem 4.4.4, p. 74. Uses basic construction, averaging viaamenability of U(B), and spectral theorem.Theorem 8.2.2 (Christensen). Let A and B be von Neumann subalgebra ofa II1 factor M . If EA EB < 1/8, then there exists u U(M) such thatuAu = B and 1 u 8EA EB.Proof. [SS08], Theorem 9.3.1, p. 157.

Theorem 8.2.3. Let M be a II1-factor, and A M a von Neumann subalgebra.If : AM is a unital -homomorphism with I , then there is a v U(M) such that (a) = vav for all a A, and v1 2(1(12)1/2)1/2.Proof. [SS08], Theorem 5.2.11, p. 88.

8.3 Perturbation of masas

Theorem 8.3.1 (Popa). Let A and B be masas in a II1 factor M . The followingconditions are equivalent:

(i) There exists w U(M) such that (I EB)EwAw,2 < 1;(ii) There exists a non-zero h A M, eB+ such that Tr(h),Tr(h2) 0 such that

ni,j=1

EB(yi uyj)22

for all u U(A).Proof. [SS08], Theorem 9.5.1, p. 174.

Compare this theorem to the Intertwining-by-bimodules theorem (Theorem9.2.1). It follows from these two theorems that if A and B are masas such thatd2(A,B) < 1, then A M B and B M A.Corollary 8.3.2 (Popa). Let A and B be semiregular masas in a II1 factor M .The following conditions are equivalent:

34

(i) There exists w U(M) such that (I EB)EwAw,2 < 1;(ii) There exists a non-zero h A M, eB+ such that Tr(h),Tr(h2) 0 be given, and letB0 and B be von Neumann subalgebras of a II1 factor M .

If EB EB0,2 , then there exist projections q0 B0, q B, q0 B0M , q BM , p0 = q0q0 and p = qq, and a partial isometry v M suchthat vp0B0p

0v = pBp, vv = p, vv = p0, 1 v2 69, 1 p2 35,

and 1 p02 35.If we assume that B0 M = Z(B0) and B M = Z(B), we may arrange

that p0 B0 and p B.Proof. [SS08], Theorem 10.4.1, p. 193.

9 Bimodules

Much more information can be found in [JS97], [Pop86b], and [Aas11]. Somecomments on the use of bimodules as a non-commutative representation the-ory (based on lectures by Thomas Sinclair) may be found in [Aas13], subsection2.12, p. 83, and subsection 3.2, p. 86.

35

9.1 Dimension and index

See [Aas11], subsection 4.2, p. 36, for the definition of the dimension of a module.

Open problem 9.1.1. Is M 7 dimM (H) locally constant?Proposition 9.1.2. Let N M P be II1 factors, and let H be separable leftM -module such that dimM H

10 Classical invariants

10.1 Fundamental groups

Theorem 10.1.1 (Connes 1980). Let be an ICC countable discrete group. If has Kazhdans property (T), then F(L()) and Out(L()) are countable.Proof. [Con80].

10.2 Examples

Theorem 10.2.1. The following examples are known:

(i) [Murray-von Neumann 1943] F(R) = R+;(ii) [Radulescu 1991] F(L(F)) = R+;

(iii) [Popa 2001] F(L(SL2(Z)o Z2)) = {1}.Proof. (i): [vNM43].

(ii): [Rad91].(iii): [Pop04]. See also [Aas11], 6.4, pp. 8183.The following two results are from [IPP08] and are corollaries of analogs of

isomorphism and subgroup theorems from Bass-Serre theory for amalgamatedfree product groups that are valid for amalgamated free product II1 factors.

Theorem 10.2.2 (Ioana-Peterson-Popa 2007). If S R+ is an infinite sub-group and P is a w-rigid II1 factor with F(P ) = {1} (as in (iii) above), then

F (sSP s) = S.Proof. [IPP08], Corollary 0.3.

Theorem 10.2.3 (Ioana-Peterson-Popa 2007). If K is a compact abelian group,then there exists a separable II1 factor M with F(M) = {1} and Out(M) = K.Proof. [IPP08], Corollary 0.7.

11 Approximation properties

11.1 Haagerup property for groups

Theorem 11.1.1 (Haagerup). A countable discrete group that acts properly ona tree has the Haagerup property.

Proof. [BO08], Theorem 12.2.5, p. 356.

The previous theorem applies to amalgamated free products of finite groups(e.g. SL2(Z)), via their action on the Bass-Serre tree (cf. [Aas11], 5.5, pp.6066), and finitely generated free groups, via their action on the Cayley graph.

37

Theorem 11.1.2. Let be a countable discrete group. The following conditionsare equivalent:

(i) There exists a net (i)iI of positive definite functions on such thati(1) = 1 and i vanishes at infinity for each i I and limiI i(x) = 1for all x ;

(ii) There exists a proper 1-cocycle on .

Proof. (i) (ii): [BO08], Theorem 12.2.4, p. 354.More characterizations are proved in [CCJ+01].

11.2 Relative property H

Some characterizations can be found in [Pop06a].

Proposition 11.2.1 (Popa 2001). Let be a countable discrete group with theHaagerup property acting on a von Neumann algebra B, equipped with a faithfulnormal trace , via the -preserving action : Aut(B). Then (B Bo)has the relative property H.


11.3 Amenable groups


(i) (Existence of invariant mean.) There exists a state on `() such that(x.f) = (f) for all f `(), where (x.f)(y) = f(x1y);

(ii) (Existence of almost invariant mean.) There exists a net (fi)iI in Prob() ={f `1() : f(x) 0 for all x , f1 = 1} such that x.fi fi1 0for all x ;

(iii) (Flner condition.) There exists a net (Fi)iI (called a Flner net) offinite sets Fi such that

limiI|FixFi||Fi| = 0

for all x ; (Equivalently, there exists a Flner sequence.)(iv) The left regular representation of has almost invariant vectors, i.e.,

1 ;(iv) All unitary representations pi of are weakly contained in the left regular

representation of , i.e., pi ;

38

(v) (Absence of spectral gap.) For any finite set F , we have that 1|F |sF

s

= 1;

(vi) There is a net (i)iI of finitely supported positive definite functions on such that limiI i(x) = 1 for all x ;

(vi) (The Brown-Guentner criterion.) There is a net (i)iI of positive definitefunctions in `p1() such that limiI i(x) = 1 for all x ;

(vii) The canonical -epimorphism C() C() is injective;(viii) The map C[] C, gg 7g, extends to a character on C();

(viii) C() has a finite-dimensional representation;

(ix) C() is nuclear;

(ix) C() is nuclear;

(x) L() is semidiscrete;

(xi) The product map : C() C() B(`2()) is continuous withrespect to the minimal tensor product norm.

Proof. (iv) (iv): [BdlHV08], Corollary F.3.3.(vi) (vi): [Aas13], Remark 2.6.3, p. 74.The rest of the proof can be found in [BO08] (cf. Theorem 2.6.8, p. 50, and

Exercise 3.6.3, p. 91) and [Pet11].

11.4 Amenable factors

11.4.1 Definitions and uniqueness

Theorem 11.4.1 (Connes 1976). Let M be a factor with separable predual.The following conditions are equivalent:

(i) M is injective;

(ii) M is approximately finite-dimensional;

(iii) M is semidiscrete;

(iv) M has property P (cf. Definition 14.5.10 and compare Theorem 1.1.8).

Proof. [Con76], Theorem 6, p. 74.

Theorem 11.4.2. Let (M, ) be a separable II1-factor. The following conditionsare equivalent:

(i) M is injective, i.e., there is a conditional expectation E : B(L2(M))M ;

39

(ii) M has a hypertrace (an invariant mean), i.e., there is a state B(L2(M)) such that |M = and ([x, T ]) = 0 for all x M andT B(L2(M));

(iii) M is hyperfinite, i.e., for all > 0 and x1, . . . , xn M there exists a finite-dimensional subalgebra N such that xj EN (xj)2 < for j = 1, . . . , n;

(iv) M is approximately finite-dimensional, i.e., M is the WO-closure of anincreasing union of countably many finite-dimensional -subalgebras;

(v) M satisfies the Flner condition, i.e., given > 0 and x1, . . . , xm Mthere exists a projection p M and a matrix algebra N M supported onp (i.e., p N is the identity element) such that EN (pxjp) (xj (1 p)xj(1 p))2 < p2 for j = 1, . . . ,m;

(vi) M is semidiscrete, i.e., idM is the point-ultraweak limit of maps of theform , where : M Mn and : Mn M are c.c.p. maps;

(vii) M is algebraically injective, i.e., any c.p. map V M , where V is aclosed self-adjoint unital subspace of a C-algebra A, extends to a c.p.map AM .

(viii) There exists a net (n) of Hilbert-Schmidt operators on L2(M) of unit

norm such that xn, n (x) for all x M and [n, u]2 0 for allu U(M).

Proof. The theorem is mostly due to Connes (cf. his seminal paper [Con76]),who proved parts thereof in more generality. Note however that (i) (vii)is a consequence of the Arveson extension theorem, and that the equivalence of(v) with the other conditions is due to Popa, cf. [Pop86a]. Some brief remarkson the proof can be found in [Aas13], Theorem 2.7.3, p. 78.

Theorem 11.4.3 (Haagerup). Let M be a II1-factor. Then a von Neumannsubalgebra N M is amenable if and only if, for any projection p Z(N) andany finite set F U(pN), we have 1|F |

uFu u

= 1.

Proof. [Haa85], Lemma 2.2.

Theorem 11.4.4. Up to -isomorphism, there is a unique injective II1-factor.Proof. [Con76], Theorem 1, p. 73.

Theorem 11.4.5. Up to -isomorphism, there is a unique injective II-factor.Proof. [Con76], Corollary 4, p. 74.

Theorem 11.4.6. Let be an ICC countable discrete group. Then L() is thehyperfinite II1 factor if and only if is amenable.

Proof. [Con76], Corollary 7.2, p. 112.

40

11.4.2 Maximal injective subalgebras

Note first that L(F2) contains a natural maximal injective von Neumann sub-algebra, cf. Example 4.10.3.

Theorem 11.4.7. Let Fn y R by outer automorphisms (for some n 2), andlet H be the abelian subgroup generated by a single generator. Then R oH isan irreducible maximal injective subfactor in Ro Fn.


Theorem 11.4.8 (Shen 2006). Let aj denote some generator of Fnj for j =1, 2, . . . (finitely or infinitely many). Then {a1, a11 }{a2, a12 } is max-imal injective in L(Fn1)L(Fn2) .Proof. [She06].

Theorem 11.4.9 (Fang 2006). Let B1 and B2 be maximal injective von Neu-mann subalgebras of M1 and M1, respectively, where M1 has separable predualand B1 has atomic center. Then B1B2 is a maximal injective von Neumannsubalgebra of M1M2.Proof. [Fan07].

Open problem 11.4.10 (Popa 1983). Is it true that if B1 and B2 are maximalinjective von Neumann algebras of M1 and M2, respectively, then B1B2 ismaximal injective in M1M2?

11.5 Relative amenability

Theorem 11.5.1 (Popa-Ozawa 2010). Let N and Q be von Neumann subalge-bras of a von Neumann algebra M , equipped with a faithful normal trace. Thefollowing conditions are equivalent:

(i) N lM Q (i.e., N is amenable relative to Q inside M);

(ii) There exists an N -central state M, eQ such that |M is normal and|Z(N M) is faithful;

(iii) There exists a conditional expectation : M, eQ N onto N such that|M = EN ;

(iv) There exists a net (n) in L2M, eQ such that xn, n (x) for all

x M and [u, n]2 0 for all u N .Proof. [OP10].

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11.6 Weakly amenable groups

Theorem 11.6.1 (Ozawa 2008). All hyperbolic groups are weakly amenable.

Proof. [Oza08], Theorem 2, p. 272.

Weakly amenable (ICC) groups are always exact?

Theorem 11.6.2. If a group acts properly on a tree, then it is weaklyamenable with cb() = 1.

Proof. [BO08], Corollary 12.3.5, p. 362.

12 Rigidity properties

12.1 Kinds of rigidity

Property (T); Spectral gap, e.g.,

1. Non-amenability, cf. Theorem 11.3.1(v), or

2. Non-, cf. Theorem 5.1.2(iv);

OE/W (super)rigidity.

12.2 Relative property (T) for groups

Theorem 12.2.1. Let be an inclusion of countable discrete groups. Thefollowing conditions are equivalent:

(i) If a unitary representation pi : B(H) has almost invariant vectors,then it has a -invariant vector;

(ii) There exists a finite 6= Q and > 0 such that if a unitary rep-resentation pi : B(H) has a (Q, )-invariant vector, then it has a-invariant vector; (Such a pair (Q, ) is called a Kazhdan pair for theinclusion .)

(iii) There exists a finite 6= Q and > 0 such that if a unitary repre-sentation pi : B(H) has a (Q, )-invariant vector, then it has a finite-dimensional -invariant subspace;

(iv) The restriction to of every complex-valued function on which is con-ditionally negative definite is bounded.

(v) The restriction to of every real-valued function on which is condition-ally negative definite is bounded.

42

(vi) (Definition with continuity constants.) For every > 0, there is aKazhdan pair [cf. (ii)] (Q, ) for with the following property: forevery unitary representation pi : B(H) with a (Q, )-invariant unitvector , its orthogonal projection onto H = { H : pi(){} ={}} satisfies

.(vii) If (i)iI is a net of positive definite functions on such that limiI i(x) =

1 for all x [and i(1) = 1 for all i I], then i| 1 uniformly;(viii) Every 1-cocycle on is bounded on .

If is a normal subgroup, the previous conditions are also equivalent to

(ix) There is a Kazhdan pair [cf. (ii)] (Q, ) with the following property: forany > 0 and for every unitary representation pi : B(H) with a(Q, )-invariant unit vector , its orthogonal projection onto H ={ H : pi(){) = {}} satisfies

.

If = , the previous conditions are also equivalent to

(x) is finitely generated, and if S is a finite generating set for then thereexists > 0 such that (S, ) is a Kazhdan pair [cf. (ii)].

Proof. [Jol05a] and [BO08].

Conjecture 12.2.2 (Connes Rigidity Conjecture). If L() = L() and has property (T), then = .

If the above conjecture were to be proved for a large class of property(T) groups, e.g., PSLn(Z) for n = 3, 4, 5, . . ., it would be considered a majorbreakthrough.

12.3 Relative property (T) for algebras

Many alternate characterizations (including independence of the choice of )have been shown, cf. the papers [PP05] and [Pop06a].

Proposition 12.3.1. Let N M be an inclusion of finite von Neumann al-gebras and fix a faithful normal trace on M . The following conditions areequivalent:

(1) (N M) has the relative property (T);(2) For any > 0 there exist a finite set F M and > 0 satisfying the

following property: If H is an M -M -bimodule containing a unit vector such that a, H = (a) = a, H for all a M and xxH < forall x F , then there exists 0 H such that 0H and a0 = 0afor all a N .

43

Moreover, if N = L() and M = L() for some inclusion of countablediscrete groups, then the above conditions are equivalent to the following:

(3) has the relative property (T).Proof. [Aas11], Theorem 5.5.2, p. 51.

Proposition 12.3.2. Let A M be an inclusion of finite von Neumann al-gebras, where M is equipped with the faithful normal trace . The followingconditions are equivalent:

(i) (A M) has the relative property (T);(ii) For any > 0 there exist a finite set F M and > 0 satisfying the

following property: If : M M is a subtracial, subunital c.p. map suchthat (x) x2, < for all x F , then (a) a2, a for alla A.


Proposition 12.3.3. Let M be a II1-factor with unique trace , let N be a vonNeumann subalgebra of M , and let p N be a non-zero projection. If (N M)has the relative property (T), then so does (pNp pMp).Proof. [Aas11], Proposition 5.2.6, p. 55.

Conjecture 12.3.4 (Analogue for primeness). If M has property (T), thenM cannot be written as the tensor product of n II1-factors for n sufficientlylarge.

12.4 Cocycle superrigidity

Theorem 12.4.1 (Popa 2005). Let G be a countable discrete group with aninfinite normal subgroup H such that H G has the relative property (T). Let Gact strongly malleably on (X,) (e.g., Gy X is a generalized Bernoulli action)and suppose that the restriction of this action to H is weakly mixing. Then any1-cocycle G X K, where K is a closed subgroup of the unitary group of afinite von Neumann algebra with a trace, is cohomologous to a homomorphismG K.Proof. [Vae07], Theorem 4.1, p. 15.

12.5 W-superrigidity of groups

Theorem 12.5.1 (Ioana-Popa-Vaes 2011). Let 0 be a non-amenable group,and let S be an infinite amenable group. Consider the action of the wreathproduct = S0 o S on the set I = /S by left multiplication. Let n be asquare-free integer and consider the generalized wreath product

G = (Z/n)I o .

44

If is any countable group and there exists a -isomorphism L() L(G)t forsome t > 0, then t = 1 and = G.

If n {2, 3}, then the -isomorphism pi is necessarily group-like: Thereexists a group isomorphism : G, a character : T, and a unitaryelement w U(L(G)) such that

pi(vs) = (s)wu(s)w

for all s , where (vs)s and (ut)tG denote the canonical generators.Proof. [IPV13], Theorem 1.1, p. 231.

12.6 W-superrigidity of actions

Theorem 12.6.1 (Ioana 2010). Let be a countable ICC group that admits aninfinite normal subgroup 0 such that (0 ) has the relative property (T).Let (X0, 0) be a non-trivial probability space and let y (X,) = (X0, 0)be the Bernoulli action. Put M = L(X) o and let p M be a projection.Let y (Y, ) be a free, ergodic, pmp action of a countable group. Put N =L(Y )o .

If N = pMp, then p = 1, = , and the actions y (X,) and y (Y, )are conjugate.

Proof. [Ioa11], Theorem A, p. 1177.

13 Indecomposability properties

13.1 Strong solidity

Note that strongly solid von Neumann algebras are always solid and never con-tain Cartan subalgebras.

Theorem 13.1.1 (Chifan-Sinclair 2013). Let be a weakly amenable ICCcountable discrete group. If admits a proper quasi-1-cocycle into some weakly-`2 unitary representation, then L() is strongly solid.

Proof. [CS13], Theorem B, p. 3.

The above theorem yields the following strengthening of Theorem 13.2.1:

Corollary 13.1.2 (Chifan-Sinclair 2013). For any ICC hyperbolic group , theII1 factor L() is s-solid.

13.2 Solidity

Note that solid II1 factors are always prime.

Theorem 13.2.1 (Ozawa 2004). The group von Neumann algebra of any ICChyperbolic group is solid.

Proof. [Oza04], Theorem 1, p. 111.

45

13.3 Absence of Cartan subalgebras

Open problem 13.3.1. Make the following statement precise and prove it:There seems to be a correlation between L() being s-solid (or merely havingno Cartan subalgebras) and L(X)o having a unique Cartan subalgebra forany free ergodic pmp action.

Open problem 13.3.2. Does L(PSLn(Z)) have no Cartan subalgebras?

Conjecture 13.3.3. If has the property that (2)n () > 0 for some n, then

is C-rigid and has no Cartan subalgebras.Theorem 13.3.4 (Popa 1983). If S is an uncountable set, then L(FS) containsno Cartan subalgebras.


Theorem 13.3.5 (Popa 1983). If M is a separable II1 factor, then M contains

no Cartan subalgebras.


13.4 Primeness

Open problem 13.4.1 (Skandalis). Does there exist M non- such that M1M2 = M = N1 N2 N3 with all factors solid (or merely prime)?Theorem 13.4.2 (Fang-Ge-Li 2006). If M is a separable II1 factor, then M

is prime.


14 C*-algebraic properties

See [CS13].

14.1 Nuclearity

14.2 Exactness of C*-algebras

Theorem 14.2.1. Every exact C-algebra is locally reflexive.

Proof. [BO08], Theorem 9.3.1, p. 293.

46

14.3 Property A and exactness of groups


(i) has Guoliang Yus Property A;

(ii) acts amenably on its Stone-Cech boundary;

(iii) the Roe algebra Cu() is nuclear;

(iv) C() is exact;

(v) is an exact group.

Proof. See [CS13], Appendix A, p. 29, for references.

Theorem 14.3.2. All hyperbolic groups are exact.

Proof. See [CS13], p. 2, for a reference.

Moreover, all amenable groups and all linear groups are exact (cf. [DN10],p. 7 [just before 3.2], for references).

14.4 Bi-exactness

14.5 Cohomological properties

Existence of proper 1-cocycles, quasi-1-cocycles, arrays.

Theorem 14.5.1. Every hyperbolic group admits a proper quasi-1-cocycle intothe left regular representation.

Proof. See [CS13], p. 2, for a reference.

Part II

DefinitionsDefinition 14.5.2. A non-zero projection e in a von Neumann algebra M iscalled properly infinite if, given a central projection z M , ze is either infinite(= not finite) or 0.

Definition 14.5.3. A von Neumann algebra is called diffuse if it contains nominimal projections.

Definition 14.5.4. A -representation pi : A B(H) is called universal if,given any -representation : A B(K), there exists a surjective -homomorphism : pi(A) (A) such that = pi.

47

Definition 14.5.5. A masa A in a von Neumann algebra M , equipped with afaithful normal trace, is called simple if A JAJ is a masa in B(L2(M)).Definition 14.5.6 (Sinclair-Smith). A masa A in a II1 factor M is calledstrongly singular if

EA EuAu,2 u EA(u)2for all u U(M)Definition 14.5.7. A masa A in a II1 factor M is said to have the weakasymptotic homomorphism property if, for all > 0 and x1, . . . , xn M , thereexists u A such that

EA(xiuxj ) EA(xi)uEA(xi )2 < for all 1 i, j n.Definition 14.5.8 (von Neumann). A von Neumann algebra (M, ) is said tohave property if, for every > 0 and every finite set {x1, . . . , xn} M ,there exists a unitary element u M such that (u) = 0 and [u, xj ]2 < forj = 1, . . . , n.

Definition 14.5.9 (Effros 1975). A countable discrete group is called inneramenable if there exists a state `() such that `1() ker and ad(g) = for all g , where we set ad(g)f(x) = f(gxg1) for f `() andg, x .Definition 14.5.10. A von Neumann algebra M B(H) is said to have prop-erty P of J. Schwartz if, for each T B(H), the norm-closed convex hull of

{uTu |u U(M)}intersects M .

Definition 14.5.11. A unitary representation of a countable discrete groupG on a Hilbert space H is called

(i) ergodic if it has no non-zero invariant vector;

(ii) mixing if |G| = and limgg, = 0 for all , H;(iii) weakly mixing if, for each finite set F H and > 0, there exists g G

such that |g, | < for all , F .See [Pet11] for more information on the above-mentioned properties.

Definition 14.5.12. An action of a group G on a measure space (X,) isa group homomorphism : G Aut(X). It is said to be probability measurepreserving (abbreviated pmp) if (X,) is a probability space and (g) is measurepreserving for each g G, free if ({t X |(g)t = t}) = 0 for all g G \ {e},and ergodic if (X0) {0, 1} whenever X0 X is measurable with (g)X0 =X0 (modulo a null set) for all g G.

48

Definition 14.5.13. An action of a discrete countable group G on a measurespace (X,) is called mixing if limg |(Ag(B)) (A)(B)| = 0 for allmeasurable sets A and B.

Definition 14.5.14. An action of a discrete countable group G on a measurespace (X,) is called weakly mixing if, whenever f L(X,) satisfies g.f Cf for all g G, it holds that f is constant -almost everywhere.Definition 14.5.15. An action of a group G on a von Neumann algebra B isa group homomorphism : G Aut(B). It is said to be trace-preserving if Bis equipped with a distinguished faithful normal trace and (g) preserves thistrace for all g G, free if a = 0 or g = e whenever a B and g G satisfya(g)b = ba for all b B, and ergodic if p {0, 1} whenever p P(B) satisfies(g)p = p for all g G.

See [Pop] for more information on the above-mentioned properties. Thefollowing two definitions are taken from [Vae07].

Definition 14.5.16. An action of a countable discrete group G on a vonNeumann algebra (B, ) is called mixing if limg |(ag(b)) (a)(b)| = 0for all a, b B.Definition 14.5.17. An action of a countable discrete group G on a vonNeumann algebra (B, ) is called weakly mixing if, for every a1, . . . , an Band > 0, there exists g G such that

|(aig(aj)) (ai)(aj)| <

for all i, j = 1, . . . , n.

Definition 14.5.18 (Popa-Ozawa 2010). Let Q,N M be finite von Neumannalgebras, where M has a faithful normal trace . We say that N is amenablerelative to Q inside M (written N lM Q) if there exists an N -central state M, eQ such that |M = .Definition 14.5.19 (Connes-Jones, Popa). Let N M be an inclusion offinite von Neumann algebras and fix a faithful normal trace on M . Then(N M) is said to have the relative property (T) if, for each > 0, there exista finite set F M and > 0 satisfying the following property: If : M Mis a -preserving u.c.p. map such that (x) x2, < for all x F , then(a) a2, a for all a N .Definition 14.5.20 (Popa 2001). Let N M be an inclusion of finite vonNeumann algebras and assume that M is equipped with a distinguished faithfulnormal trace . Then (N M) is said to have the relative property H ifthere exists a net () of normal c.p. N -bimodular maps on M satisfying thefollowing conditions:

(i) for all ;

49

(ii) For all , the extension T of to a bounded operator on L2(M) sat-

isfies T J (M, eA) = the norm-closed two-sided ideal generated bythe finite projections in M, eA;

(iii) lim (x) x2, = 0 for all x M .Definition 14.5.21 (Ozawa 2004). A von Neumann algebra M is called solidif the relative commutant of any diffuse von Neumann subalgebra is amenable.

Definition 14.5.22 (Popa 2006). A von Neumann algebra M is called stronglysolid (or s-solid) if the normalizer of any diffuse amenable von Neumann sub-algebra of M generates an amenable von Neumann algebra.

References

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