Tauer masas in the hyper�nite II1 fator

Stuart White

Dotor of PhilosophyUniversity of Edinburgh2005

To Blake,What you've been through as a onsequene of my fall, forms a debt I annever repay.

DelarationI delare that this thesis was omposed by myself and that the work ontainedtherein is my own, exept where expliitly stated otherwise in the text.(Stuart White)

AbstratIn 1964, Tauer gave examples of ountably many masas inside the hyper�nite II1von Neumann fator R. These masas were shown to be pairwise non-onjugate inR using a length invariant for the normalisers of semi-regular masas. A lass ofmasas, the Tauer masas, is introdued onsisting of all those masas obtained usingher basi method of onstrution. The main body of this thesis is then onernedwith examining the properties of these Tauer masas. In partiular, the onepts ofsingularity, strong singularity and the weak asymptoti homomorphism propertyoinide for Tauer masas, and all Tauer masas have Pukánszky invariant {1}.Modern methods for alulating von Neumann algebras generated by nor-malisers are used to examine Tauer's original examples, leading to shorter proofsof all of her results. Her initial example of a singular masa is studied in furtherdetail. A generalisation of her semi-regular masas leads to the onstrution of anunountable family of semi-regular masas of in�nite length inside R. Examinationof the Jones index of inlusions of the iterated normaliser algebras demonstratesthat no pair of these masas an be onjugate by an automorphism of R.Centralising sequenes for R lying inside masas are examined, with examplesgiven to show that singular masas an be found ontaining non-trivial entralisingsequenes. An invariant, Γ(A), for a masa inside a II1 fator is introdued as thesize of a maximal ut-down for whih the resulting masa ontains non-trivialentralising sequenes. This invariant is then used to exhibit a d∞,2-ontinuouspath of unountably many strongly singular masas in R with the same Pukánszkyinvariant, no pair of whih is onjugate by an automorphism of R.Various issues arising from these onepts are disussed, suh as possible masasin Rω and the relationship between A-valued entralising sequenes and auto-morphisms of R �xing A pointwise. Possible onnetions between this relativeautomorphism group and the Pukánskzy invariant will also be touhed upon.

AknowledgementsFirst and foremost, my thanks go to Allan Sinlair, who repeatedly o�ered memuh more than one ould expet from a PhD supervisor, ontinuing his role longafter his well deserved retirement. Thanks also to all the sta� and students atthe Shool of Mathematis who have made these four years suh a pleasurableexperiene. In partiular, I would like to pik out Tony Carbery, Jim Wright andToby Bailey for their advie, mathematial and otherwise. During my studies,I have enjoyed the opportunity to visit College Station, Texas on two o

asions.My thanks to Roger Smith, for making this possible and for many helpful on-versations both in the States and in Edinburgh.Enjoying retirement

Allan Sinlair (middle), and friends on Stu a' Chroin, 2005.Photo: UnknownI would like to thank my parents for their onstant support over the last fouryears; they've always been there for me when I've needed it. They should also beongratulated for being brave enough to proof read this thesis! All my �at mates

in Edinburgh, Gregor, Sott, Mike, Andy, Tessa, Blake, Tom and Hamish, haveput up with various foibles. Their tolerane has been muh appreiated. Esapingto the mountains has helped to keep me sane, heers to Pete, Jon, Jonathan andMark and all the yummiks I've been away with, for the good memories.In their natural habitat

Mum and Dad, Bernese Oberland, 2005. Photo: John GayThanks to Tom Banister - without your apable ations the end result onMarh the 6th, 2005 would have muh worse. The members of the LohaberMountain Resue Team, RAF heliopter resue squadron and Belford Hospitalare literally lifesavers - what people would do without you I don't know. Finallythanks to Blake and ountless other people for helping me to start to ome toterms with things sine then. You've all been great.Edinburgh, June 27, 2005.

It was the best of times, it was the worst of times

Blake on Centre Post Diret (V 5), Creag Meagaidh, before the a

ident. 2005.Photo: Saw

1

Table of ContentsChapter 1 Preliminaries 51.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Some topis in the theory of von Neumann algebras . . . . . . . . 91.2.1 Tensor produts . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 The hyper�nite II1 fator . . . . . . . . . . . . . . . . . . . 111.2.3 Orthogonality of subalgebras . . . . . . . . . . . . . . . . . 131.2.4 Property Γ and entralising sequenes . . . . . . . . . . . . 141.2.5 The ultraprodut Nω and entral sequene algebras Nω ∩N ′ 151.2.6 Automorphisms of II1 fators . . . . . . . . . . . . . . . . 181.3 Conditional expetations, the basi onstrution and the Jones index 191.3.1 The basi onstrution and Jones index . . . . . . . . . . . 201.3.2 Inlusions of index 2 . . . . . . . . . . . . . . . . . . . . . 221.3.3 Inlusions of �nite dimensional C∗-algebras . . . . . . . . . 241.3.4 The Wenzl index formula . . . . . . . . . . . . . . . . . . . 261.4 Masas in II1 fators . . . . . . . . . . . . . . . . . . . . . . . . . . 301.4.1 Basi properties of masas . . . . . . . . . . . . . . . . . . . 311.4.2 Normalisers of masas . . . . . . . . . . . . . . . . . . . . . 321.4.3 A metri on masas and strong singularity . . . . . . . . . . 351.4.4 Establishing strong singularity: asymptoti homomorphismproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.4.5 Centralising sequenes lying in masas . . . . . . . . . . . . 451.4.6 Invariants for singular masas . . . . . . . . . . . . . . . . . 46Chapter 2 Tauer's original examples of masas 482.1 De�ning Tauer masas . . . . . . . . . . . . . . . . . . . . . . . . . 492.2 Tauer's singular masa . . . . . . . . . . . . . . . . . . . . . . . . . 512.2.1 Establishing singularity . . . . . . . . . . . . . . . . . . . . 522.2.2 A reformulation and the θ-masas . . . . . . . . . . . . . . 542.3 Tauer's semi-regular masas . . . . . . . . . . . . . . . . . . . . . . 572.3.1 Tauer's length invariant . . . . . . . . . . . . . . . . . . . 572

2.3.2 Tauer's examples of �nite length masas: Setup . . . . . . . 592.3.3 Parts 1 and 2 of the proof of Theorem 2.3.6 . . . . . . . . 612.3.4 Parts 3 − 5 of the proof of Theorem 2.3.6 . . . . . . . . . . 652.3.5 Other semi-regular masas with length 2 . . . . . . . . . . . 702.4 Masas of In�nite Length . . . . . . . . . . . . . . . . . . . . . . . 732.4.1 De�ning the subfators N n (A) . . . . . . . . . . . . . . . 742.4.2 Computing the normaliser of S . . . . . . . . . . . . . . . 802.4.3 De�ning the required in�nite length masa . . . . . . . . . 862.4.4 Thoughts on in�nite length masas . . . . . . . . . . . . . . 90Chapter 3 The Pukánszky invariant and entralising sequenes 933.1 Pukánszky's invariant: de�nition and bakground . . . . . . . . . 943.1.1 Known values of the Pukánszky invariant . . . . . . . . . . 953.1.2 The Pukánszky invariant and normalisers . . . . . . . . . . 963.1.3 Continuity properties of the Pukánskzy invariant . . . . . 973.2 The Pukánszky invariant of a Tauer masa . . . . . . . . . . . . . 1003.2.1 Proof of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . 1003.2.2 Approximating masas by matries . . . . . . . . . . . . . . 1043.3 Unountably many singular masas with the same Pukánszky in-variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.3.1 The plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.3.2 The exeution . . . . . . . . . . . . . . . . . . . . . . . . . 1143.3.3 Transitive masas . . . . . . . . . . . . . . . . . . . . . . . 1233.4 Masas in Rω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.4.1 Dω ∩R′ is a masa in Rω ∩R′ . . . . . . . . . . . . . . . . 1273.4.2 The singular Tauer masa of setion 3.3 with Γ(A) = 1 . . . 1293.5 Some thoughts about automorphisms �xing masas . . . . . . . . . 1363.5.1 A relative automorphism group . . . . . . . . . . . . . . . 1373.5.2 Density of the relative inner automorphism group . . . . . 1383.5.3 Charaterising approximately inner relative automorphisms 141Chapter 4 Singularity and the θ-masas 1444.1 Singularity for Tauer masas . . . . . . . . . . . . . . . . . . . . . 1444.1.1 A onvergene riterion for the singularity of Tauer masas 1444.1.2 Proof of Theorem 4.1.3: 2 ⇒ 3 . . . . . . . . . . . . . . . . 1464.1.3 Proof of Theorem 4.1.3: 3 ⇒ 1 . . . . . . . . . . . . . . . . 1494.2 Singularity of θ-masas of Constrution 2.2.4 . . . . . . . . . . . . 1524.3 Is Tauer's singular masa transitive? . . . . . . . . . . . . . . . . . 1564.3.1 Weak A-onjugay of minimal projetions in At . . . . . . 1573

4.3.2 Finite tensor powers of A . . . . . . . . . . . . . . . . . . . 1594.3.3 In�nite tensor powers of A . . . . . . . . . . . . . . . . . . 162Appendix A Failure of the weak asymptoti homomorphism prop-erty 171Appendix B Radial masas in fators oming from free produts ofertain �nite groups 175Appendix C A note on automorphisms of group von Neumann al-gebras 182Bibliography 186

4

Chapter 1PreliminariesWhat to do?

Dulux Corner (HS 4b), Neist point, Skye, May 2002.Photo: Jon Powell5

1.1 IntrodutionThe study of weakly losed rings of operators ating on a Hilbert spae datesbak to Frank Murray and John von Neumann's series of papers [35, 36, 74, 37℄.These rings are now alled von Neumann algebras and have been extensivelystudied sine. The basi building bloks of the subjet are the fators, those vonNeumann algebras with trivial entre, whih Murray and von Neumann lassi�edinto types In, II1 , II∞ and type III. Throughout we shall study only the II1ase, o

asionally using �nite type In fators1 in our onstrutions. The fous ofthis thesis will be on masas, that is maximal abelian self-adjoint subalgebras, inII1 fators, usually the hyper�nite II1 fator. Extensive bakground material onmasas is given in setion 1.4, and various results in the theory of II1 fators areontained in setions 1.2 and 1.3 - we shall not repeat ourselves here.The main motivation for this thesis is a fairly obsure paper, [70℄, due toSister Rita Tauer in 1965, in whih she gives examples of ountably many di�erentmasas in the hyper�nite II1 fator. Another set of examples of suh masas, due toPukánszky ([51℄), preeded Tauer's work, although the two approahes are verydi�erent. Pukánszky's examples ome from groups, and are now muh more wellknown than Tauer's examples whih arise from elaborate matrix onstrutions.Chapter 2 is wholly onerned with this work of Tauer's. We introdue, in setion2.1, a lass of masas, whih we all Tauer masas, to be those that an be produedusing Tauer's basi method. In setions 2.2 and 2.3 we give a modern a

ountof Tauer's examples using tensor produts rather than large matrix algebras. Upto date methods for alulating von Neumann algebras generated by normalisersallow us to give muh shorter proofs of Tauer's results. Tauer's main idea fordistinguishing between masas was to examine the iterated hain of normalisingalgebras, and she was able to give examples of masas where this hain reahes Rin l �nite steps. An interesting historial point is that the invariant Tauer usedto show that this l did not ompletely determine (up to onjugay) the originalmasa turns out to be an alternative haraterisation2 of an index 2 inlusion ofsubfators. In setion 2.4 we are able to use this index idea, and an extension ofTauer's examples, to present an expliit, elementary onstrution of unountablymany, pairwise non-onjugate semi-regular masas in the hyper�nite II1 fator. Inontrast, the examples produed by Jones and Popa in [25℄ are non-onstrutive.In hapter 3, we examine properties relating to the Pukánszky invariant; theidea Pukánszky used to distinguish between his original examples in [51℄. Fol-1An extravagant way of desribing the n × n matries over C!2At least under an additional �nite index assumption, whih I strongly suspet is unneessary- see subsetion 1.3.2. 6

lowing Popa's work, [45℄, this invariant has beome a useful objet in the studyof masas, [52, 3, 38, 61, 17℄, not least beause for a long time it was one of veryfew methods for showing that a masa is singular. We present the bakgroundand urrent state of play in subsetions 3.1.1 and 3.1.2, while in subsetion 3.1.3we examine how the Pukánszky invariant behaves under d∞,2-limits of masas. Insetion 3.2, we show that the lass of Tauer masas all have Pukánskzy invariant{1}, Theorem 3.2.1. After the Pukánskzy invariant, the presene or absene ofnon-trivial entralising sequenes inside a masa has perhaps been the next mostuseful onept for showing the non-onjugay of masas when we are unable to dis-tinguish them using normalisers. A brief history of entralising sequenes insidemasas an be found in subsetion 1.4.5. In setion 3.3, we use entralising se-quenes and Tauer masas to give an unountable family of pairwise non-onjugatesingular masas in the hyper�nite II1 fator with the same Pukánszky invariant.This extends a result of Størmer and Neshveyev ([38℄) in whih two non-onjugatesingular masas with the same Pukánszky invariant appeared. Furthermore, theresulting family turns out to give us a ontinuous (with respet to the d∞,2-metrion masas) path from the unit interval into the singular masas with Pukánszkyinvariant {1} in R, no two points on whih are onjugate. In subsetion 3.3.3, weintrodue the onept of transitivity for masas, the idea being that two identiallysized ut-downs of a transitive masa should look the same in the underlying II1fator. That is they should be onjugate.In setion 3.4, we look at entralising sequenes further. The main objetiveis to determine whih masas A in R give rise to a masa Aω ∩ R′ in Rω. We areunable to resolve this problem, but give some examples suggesting a onnetionbetween this property and the normalisers of A. Chapter 3 ends with a disussion,setion 3.5, of a relative automorphism group of masas in II1 fators. We statea translation of Connes' haraterisation of approximately inner automorphisms[7, Theorem 3.1℄ in this ontext, Theorem 3.5.9, and ask some questions aboutthe relationship between these automorphisms and the Pukánskzy invariant.Chapter 4, begins by asking the question `When is a Tauer masa singular?'.We resolve this ompletely in Theorem 4.1.3, giving a riterion for singularity ofa Tauer masa in terms of the approximates used to onstrut it. Furthermore,we are able to show that singular Tauer masas have the weak-asymptoti homo-morphism property and therefore are strongly singular. The question of whetherall singular masas are strongly singular appears in [59, 50℄, and the Tauer masasare the �rst large lass of masas for whih this result is known. In setion 4.2, weapply this riterion for singularity to a family of θ-masas introdued in setion2.2, and in setion 4.3 we investigate these masas further. Tauer's original exam-7

ple of a singular masa falls into this sublass, and we are able to see it has someperhaps surprising properties suh as being onjugate to its own in�nite tensorprodut, Theorem 4.3.8, under an identi�ation of R with R⊗∞. We would liketo be able to show that this is an example of a transitive singular masa in R, aswe only have examples of these in the free group fators at present. We presentevidene in setion 4.3 suggesting that this ould be the ase.Finally we turn to the appendies. Appendix A gives an example of an in-lusion of II1 fators M ⊂ R where M fails to have the relative weak asymptotihomomorphism property away fromN (M)′′ = N , a onept de�ned in subsetion1.4.21. We are unable though to deide whether thisM is strongly normalised byN . In Appendix B we do something very di�erent and give a alulation, follow-ing [60℄, showing the strong singularity of the radial masas onsidered by Boaand R dulesu, in [3℄. The thesis ends with Appendix C, where we present a sim-ple observation on automorphisms of group II1 fators dating bak to Kallman,[32℄.We have now reahed the end of the beginning of this thesis, and onlude thisintrodution by making some remarks about the notation used within. CapitalRoman letters will in general refer to von Neumann algebras, by preferene thesewill be N andM . We reserve R, S and o

asionally T for hyper�nite fators, andA and B will usually denote abelian von Neumann algebras. However, J will beonsistently used for the modular onjugation operator, and G and H will alwaysbe disrete groups. We shall o

asionally refer to free groups, these will alwayshave k ≥ 2 generators and will be denoted Fk. Small Roman letters, normallya, b, u, v, x, y, z, will be used for operators in von Neumann algebras, and Greekletters, often ξ and η, for elements of the Hilbert spae they operate on. TheseHilbert spaes, when not appearing as L2 of some N , will be denoted H. SriptA (and o

asionally B) are not to be onfused with Hilbert spaes - they will beused in Chapter 3 to denote the augmented algebra (A∪JAJ)′′ orresponding toa masa A.A substantial number of footnotes an be found in the text. They mainly fallinto one of three ategories: giving itations and ross referenes that would oth-erwise disturb the �ow; points of exessive mathematial pedantry; and remarksthat ould be found humorous. It is left to the reader to deide whih is whih.Addendum 1.1.1. After the ompletion of this thesis, Questions 1.4.18 and1.4.20 on whether all singular masas are strongly singular and have the weakasymptoti homomorphism property respetively, have now been answered posi-tively whih, as noted in the text, gives a partial positive result in the diretionof Question 1.4.29. This result also ompletely supersedes Theorem 4.1.1. The8

proof an be found in [62℄, whih is joint work between Allan Sinlair, RogerSmith, Alan Wiggins and myself.1.2 Some topis in the theory of von NeumannalgebrasWe have endeavored to make this thesis relatively self ontained, in that the vastmajority of the de�nitions needed and statements of existing results used an befound in this hapter. One has to start somewhere though - we shall assumefamiliarity with the basi theory of von Neumann algebras as an be found ineither Kadison and Ringrose's Magnum Opus [30, 31℄ or the �rst volume, [67℄,of Takesaki's extensive a

ount of operator algebras, [67, 68, 69℄. In this setionwe develop a few assorted topis from the theory, the seletion not being quiteas random as it appears. In the next setion we disuss Vaughan Jones' indexfor subfators and �nally, in setion 1.4, we develop the theory of masas in II1fators. We end these remarks with a note on the standard form.Throughout the thesis, we shall only be interested in �nite von Neumann al-gebras, usually fators, but if not they will always ome equipped with a �xedfaithful normal trae, tr. For x in suh a von Neumann algebra N , we write‖x‖2 = tr(x∗x)1/2 and then omplete N with respet to this norm to obtain theHilbert spae L2(N). We shall follow the tradition in the subjet of regardingN as a subset of L2(N) whenever it is onvenient. The standard form is therepresentation N ⊂ B (L2(N)) obtained by letting eah x ∈ N at by left mult-pliation3 on L2(N). The modular onjugation operator is the onjugate linearmap J : L2(N) → L2(N) obtained by extending the onjugation x 7→ x∗ fromN . Then JxJ is the operator of right multipliation on L2(N) by x∗, and theommutant N ′ of N in B (L2(N)) is preisely JNJ .We shall use group von Neumann algebras o

asionally, but not enough tojustify giving a lengthy disussion. Su�e it to say that when G is a ountabledisrete I.C.C. group4, we obtain a II1 fator L (G) to be the von Neumann algebragenerated by the left regular representation of G on ℓ2(G). The trae is givenby tr(x) = 〈xδe, δe〉, where δe is the point mass in ℓ2(G) orresponding to theidentity. The left regular representation is already the standard form of L (G)in that ℓ2(G) = L2(L (G)). Finally, when g is an element of G we shall abusenotation and write g for the element of L (G) orresponding to the image of g3extend y 7→ xy from N to L2(N) by ‖.‖2-ontinuity.4I.C.C. stands for in�nite onjugay lass, the de�nition of whih is that { ghg−1 ∣∣ g ∈ G }is in�nite, whenever h ∈ G is not the identity. 9

under the left regular representation.5Warning 1.2.1. The use of the term separable in this thesis di�ers from the stan-dards in the literature. The appropriate onept of separability for von Neumannalgebras is that of being separably ating, that is being faithfully represented asbounded operators on some separable Hilbert spae, or equivalently having a sep-arable predual. When we talk of a separable von Neumann algebra heneforth,it is these properties we mean and most de�nitively not that it is separable as aC∗-algebra - it almost surely won't be.1.2.1 Tensor produtsThe theory of tensor produts for C∗-algebras is fairly involved as there is not aunique C∗-norm on the algebrai tensor produt of two C∗-algebras. In the vonNeumann ontext, this problem does not arise.6 Given two von Neumann algebrasN1, N2 faithfully represented on the Hilbert spaes H1 and H2 respetively, formthe Hilbert spae tensor produt H1 ⊗ H2. An elementary tensor x1 ⊗ x2 ofoperators from N1 and N2 respetively, ats on this Hilbert spae in the obviousway; with (x1 ⊗x2)(ξ1⊗ ξ2) = (x1ξ1)⊗ (x2ξ2). The von Neumann tensor produt,N1⊗N2 of N1 and N2 is then generated by all these elementary tensors,

N1⊗N2 = { x1 ⊗ x2 | xi ∈ Ni }′′ ⊂ B (H1 ⊗H2) .This de�nition appears to depend on the hoie of the faithful representations,but atually this is not the ase. If we represent N1 and N2 faithfully on someother spaes then the resulting von Neumann tensor produt is ∗-isomorphi tothat obtained from our �rst hoie of representation. In any event, the vonNeumann algebras N onsidered within will all be �nite with a given faithfulnormal trae, tr, and so we have a anonial representation - that on the standardform L2(N, tr). When the von Neumann algebras are �nite dimensional, we shallwrite ⊗ rather than ⊗, as no losure is involved.We shall regularly work with inlusions of von Neumann algebras - obtainingfurther inlusions naturally from the tensor produt. A key result here is theTomita ommutation theorem, for whih we refer to [31, Theorem 11.2.16℄.Theorem 1.2.2 (Tomita ommutation theorem). Let M1 ⊂ N1 and M2 ⊂ N2 beinlusions of von Neumann algebras. Working in N1⊗N2, we have(M1⊗M2)′ ∩ (N1⊗N2) = (M ′1 ∩N1)⊗(M ′2 ∩N2).5We reserve the right to further write g for the point mass δg in ℓ2(G) ∼= L2(L (G)) orre-sponding to G in a

ordane with our poliy of regarding �nite von Neumann algebras N asbeing subsets of L2(N). Hopefully the loation of g will be obvious from ontext!6Of ourse von Neumann algebras are not in general nulear as C∗-algebras, but we are notonsidering them as C∗-algebras. 10

We shall also need in�nite tensor produts of von Neumann algebras. Herewe have to be slightly more areful, as the isomorphism lass of the resulting vonNeumann algebra is no longer independent of the hoie of representations. Weshall atually only work with �nite fators, and follow [69, Setion XIV.1℄. Given�nite fators (Mn)∞n=1, denote by trn the unique faithful normal trae on Mn.Write⊗∞n=1Mn for the algebrai tensor produt of these Mn, that is �nite linearombinations of elementary tensors⊗∞n=1 xn, where xn ∈Mn and all but �nitelymany xn are 1. We have the produt state tr on⊗∞n=1Mn de�ned on elementarytensors by tr( ∞⊗n=1

xn) =∞∏

n=1

tr(xn).Now let π be the representation of⊗∞n=1Mn by left multipliation on the Hilbertspae L2(⊗∞n=1Mn, tr) in the usual way. The in�nite von Neumann tensor prod-ut of the Mn is then the weak-losure of the image of π. This is neessarily a�nite fator, as it has a trae, namely the extension of tr, whih is the unique nor-malised trae on⊗∞n=1Mn. We shall regularly denote this objet as (⊗∞n=1Mn)′′in the sequel without referene to π. When all the Mn are idential to M say, weshall o

asionally be lazy and write M⊗∞ for this in�nite von Neumann tensorprodut. The Tomita ommutation theorem remains true in this in�nite setting.1.2.2 The hyper�nite II1 fatorMurray and von Neumann's de�ning property of hyper�niteness was that of ap-proximate �nite dimensionality (AFD). Following [69, Setion XIV.2℄, a II1 fatorN was said to be AFD when for any x1, . . . , xn ∈ N and strong neighbourhood Vof 0 in N , a �nite dimensional ∗-subalgebraM of N an be found with xi ∈ M+Vfor eah i.Examples are immediately apparent by taking in�nite tensor produts of ma-trix algebras. Let Mn be a algebra of matries for eah n, then the in�nite tensorprodut (⊗∞n=1Mn)′′ produed with respet to the unique normalised trae oneah Mn is a II1 fator, whih is obviously AFD. In [37℄, Murray and von Neu-mann showed that up to isomorphism this is the unique way of obtaining an AFDII1 fator and, in omplete ontrast with the C∗ ase, that the resulting objet isindependent of the size of the matries involved.Theorem 1.2.3 (Murray and von Neumann). Let N be a separable II1 fator.The following two onditions are equivalent:1. N is isomorphi to (⊗∞n=1 Mat2(C))′′;2. N is AFD. 11

Heneforth, we use the term hyper�nite for AFD and denote the hyper�niteII1 fator by R, and o

asionally S and T . The proof of this result involves aareful approximation argument, whih we are unable to irumvent entirely as insetion 3.2.2 we shall dedue an approximation result for masas in the hyper�niteII1 fator using these methods.7 Here we state preisely the tools we shall needlater, from Takesaki's a

ount of Theorem 1.2.3, found in setion XIV.2 of [69℄.Lemma 1.2.4 ([69, Lemma XIV.2.1℄). If e and f are equivalent projetions in a�nite von Neumann algebra N then there exists a unitary u ∈ N with

|u− 1| ≤√

2 |e− f | and ueu∗ = f.8Lemma 1.2.5 ([69, Lemma XIV.2.2℄). Let N be a separable II1 fator. If h ∈ N ,0 ≤ h ≤ 1, satis�es the inequality

∥∥h− h2∥∥

2= δ < 1/4,then the spetral projetion e of h orresponding to the interval [1−√δ, 1] satis�esthe estimate

‖e− h‖2 ≤ 2√δ,

∥∥h1/2 − e∥∥

2≤ 2δ1/4.In the next lemma we fae a pedagogial problem - namely that some ofthe onepts appearing will be de�ned later. See setion 1.3 for the onditionalexpetation map, E, and De�nition 1.4.10 for ‖.‖∞,2. We have also hosen toreplae the use of ǫ⊂ in the original, with in�nity-two norm estimates.Lemma 1.2.6 ([69, Lemma XIV.2.10℄). If V is a �nite dimensional subspae ofan AFD II1 fator R0 and N1 is a subfator of type I2n, then for any ǫ > 0 thereexists a type I2p subfator N2, for some large p, suh that N1 ⊂ N2, and

‖(I − EN2) EV ‖∞,2 ≤ ǫ.Before moving on there is one more observation we should perhaps make.Remark 1.2.7. When we work with R as the in�nite von Neumann tensor prod-ut (⊗∞n=1 Mat2(C))′′ with respet to the unique normalised trae, we shall oftenonsider the �nite dimensional approximates Nn = ⊗nm=1 Mat2(C) ∼= Mat2n(C),whose union is by de�nition weakly dense in R. A priori, it might not be possi-ble to use the Kaplanszky density theorem to weakly approximate unitaries in R7Indeed, this is the only reason this disussion is present at all.8The moduli in this equation are de�ned by |x| = (x∗x)1/2. Our only use of these moduliin the sequel, will be the simple observation that ‖|x|‖2 = ‖x‖2. We have stated the Lemma inthe original form of [69℄ for easy referene. 12

by those in ⋃∞n=1Nn - for this last objet is not norm losed.9 To approximateunitaries using the Kaplanszky density theorem, one expresses them in the formeıh for some self-adjoint h, approximates the h by h1 and takes eıh1 - a proessthat requires norm-losure. In this diret limit ontext we are able to proeed asour h1 will lie in ⋃∞n=1Nn and so in some Nn, whene eıh1 lies in Nn as this is aC∗-algebra.1.2.3 Orthogonality of subalgebrasIn [43℄, Sorin Popa introdued the onept of orthogonality for pairs of subalgebrasin �nite von Neumann algebras.De�nition 1.2.8. Let N be a �nite von Neumann algebra with trae tr. Twovon Neumann subalgebras M1 and M2 of N are said to be orthogonal if M1 ⊖C1is orthogonal to M2 ⊖ C1 in L2(N, tr). We shall write M1 ⊥M2 when this is thease.As noted in Lemma 2.1 of [43℄, there are many alternative formulations of thisde�nition. We brie�y highlight those we shall use later.Proposition 1.2.9 ([43, 2.1℄). Let M1 and M2 be von Neumann subalgebras ofa �nite von Neumann algebra N with trae tr. The following onditions are thenequivalent.1. M1 and M2 are orthogonal von Neumann subalgebras of N ;2. tr(x1x2) = tr(x1)tr(x2) whenever xi ∈Mi;3. ‖x1x2‖2 = ‖x1‖2 ‖x2‖2 whenever xi ∈Mi;4. EM1 (EM2 (x)) = tr(x)1, for all x ∈ N .10The main aim of [43℄ was to give a riterion for alulating the normalisersof ertain masas, whih we brie�y disuss in subsetion 1.4.2. This onnetionbetween orthogonality and singularity will appear later, both in our examinationof Tauer's original examples in setions 2.2 and 2.3, and in onstruting singularmasas in setion 3.3. In this seond ase, we shall need an abundane of mutuallyorthogonal masas in �nite dimensional matrix algebras. Fortunately providene,in the form of Sorin Popa, has provided exatly what we need.9Reall that the Kaplanszky density theorem allows the weak approximation of elementsin the unit ball of a von Neumann algebra by elements in the unit ball of a weakly dense∗-subalgebra, and of self-adjoint elements of this ball by self-adjoint elements in the dense ball.10Here EMi is the unique trae-preserving onditional expetation from N onto Mi, whih weeventually de�ne in setion 1.3. 13

Proposition 1.2.10 ([43, Theorem 3.2℄). For any prime p there exist a familyof p + 1 pairwise orthogonal masas in the algebra of p× p matries.1.2.4 Property Γ and entralising sequenesIn 1943, Frank Murray and John von Neumann, in the fourth part ([37℄) of theirseries of work on rings of operators, demonstrated the existene of non-isomorphiII1 fators. They introdued property Γ to show that none of fators L (Fk) arehyper�nite, where Fk is the free group on k ≥ 2 generators.De�nition 1.2.11 (Murray and von Neumann). A II1 fator N has propertyΓ when, for all ǫ > 0 and x1, . . . , xn ∈ N , there exists a unitary u ∈ N withtr(u) = 0 and

‖uxi − xiu‖2 < ǫ,for eah i = 1, . . . , n.It omes as no surprise to learn, [37, Lemma 6.2.2℄, that it is impossible to�nd unitaries in L (F2) approximately ommuting with the generators a and b ofF2 so that L (F2), and in general the the free group fators L (Fk), do not haveproperty Γ. On the other hand, the hyper�nite II1 fator does have property Γ([37, Lemma 6.1.2℄) as we an see immediately by writing R as an in�nite vonNeumann tensor produt of matrix algebras (⊗∞n=1 Mat2(C))′′. As this idea willappear frequently we shall spell it out. Elements 1⊗n ⊗ u, for some trae-freeunitary u ∈ Mat2(C), ommute with⊗nr=1 Mat2(C) and, as the union of all thesesets is ‖.‖2-dense in R, the laim follows.The onept of entral sequenes, introdued in [13℄, follows naturally fromthe idea of property Γ. A bounded sequene (xn)∞n=1 in a II1 fator N is said tobe a entral sequene if

limn→∞

‖xny − yxn‖2 = 0,for all y ∈ N . Two entral sequenes (x(1)n )∞n=1 and (x(2)n )∞n=1 are equivalent when∥∥∥x(1)n − x(2)n ∥∥∥2→ 0 as n → ∞, and a entral sequene is alled trivial, if it isequivalent to a entral sequene for N lying in the salars C1. The equivalenelasses of entralising sequenes form a C∗-algebra, ensuring that a II1 fator Nhas property Γ if and only if there exist non-trivial entralising sequenes for N .A separable II1 fator N is alled strongly-stable if it is isomorphi to N⊗R,where R is the hyper�nite II1 fator. Dusa MDu� used entralising sequenesto give a riterion, Theorem 3 of [34℄,11 for strong-stability of a II1 fator N . M-Du�'s result is that N is strongly-stable preisely when the entralising sequenesof N give rise to a non-ommutative C∗-algebra.11See also Theorem 4.8 of Takesaki's book [69℄.14

Given a subalgebra B of N , we say that B ontains non-trivial entralisingsequenes for N if there is a non-trivial entralising sequene (xn)∞n=1 for N witheah xn ∈ B. In [1℄ and [2℄, Bish investigated entralising sequenes lying in sub-fators. The �rst paper, [1℄, generalises Theorem 2.1 of Connes' injetive fatorspaper, [7℄, and extends MDu�'s result to give a riterion for the strong-stabilityof an inlusion of II1 fators M ⊂ N ,12 in terms of entralising sequenes forN lying in M . The seond paper, [2℄, gives examples of �nite index inlusionsM ⊂ N of hyper�nite II1 fators, where M ontains suh non-trivial entralis-ing sequenes for N , and examples of suh inlusions where all the entralisingsequenes for N lying in M are trivial.To demonstrate the absene of non-trivial entralising sequenes in a subfa-tor, Bish used an idea whih originally appears in Popa's orthogonality work,[43, Remark 5.4.2℄.13Proposition 1.2.12 (Popa). Let M be a subalgebra of a separable II1 fator N .If there exists a unitary u ∈ N with uMu∗ ⊥ M then any entralising sequenefor N lying in M is trivial.Whether or not non-trivial entralising sequenes an be found inside ertainmasas in II1 fators has often been a usful tool. Unfortunately, this topi isperhaps less well known than it should be and so we shall ollet together thework in this area in subsetion 1.4.5.1.2.5 The ultraprodut Nω and entral sequene algebras

Nω ∩N ′In setion 3.4 we shall have ourse to examine Rω = Rω ∩ R′. The theory of ω-entralising sequenes involved dates bak to [13℄ and is developed in [34℄; manyalternative a

ounts also exist - see for example [69, Setion XIV.4℄. Here weontent ourselves with a brief outline of the situation, for a �xed II1 fator N .Let ω be a non-prinipal ultra�lter on N, that is an element of βN \ N.14 Wehave a traial state on the C∗-algebra ℓ∞(N) of all uniformly bounded sequenesin N given by trω ((xn)) = limn→ω

tr(xn).12Unsurprisingly, this is de�ned as there being an ∗-isomorphism between N and N⊗R whihtakes M onto M⊗R.13See also [2, Lemma 2.1℄.14As is usual, βN denotes the Stone-Cêh ompati�ation of N - an objet of whih we haveno intention of developing the theory. Instead the reader is referred to Lemma XIV.4.2 in [69℄and the preeding disussion, for the properties we need.15

The GNS onstrution gives a representation πω of ℓ∞(N) on some Hilbert spaeHω and a yli vetor ξω, suh that trω((xn)) = 〈πω((xn))ξω, ξω〉. Let Nω =πω(ℓ

∞(N))′′ a �nite von Neumann algebra in B (Hω), whih has trae x 7→〈xξω, ξω〉 whih we ontinue to denote by trω. In this way Nω is a non sepa-rable type II1 von Neumann algebra, alled the ultraprodut of N .15We have the natural inlusion of N into Nω, given by taking x ∈ N to thesequene (xn) with eah xn = x and then applying πω. De�ne Nω = Nω ∩N ′, a�nite von Neumann algebra. In [12℄, Dixmier showed this entral sequene algebrais either trivial or di�use.16 This an also be found in [6℄, where the onnetionbetween the triviality ofNω and automorphisms ofN is developed. We should alsonote that referring to Nω as a entral sequene algebra is reasonable - elementsof Nω ∩N ′ are images under πω of sequenes (xn)∞n=1 whih are ω-entralising inthat

limn→ω

‖xny − yxn‖2 = 0,for all y ∈ N . This is not immediately obvious from the de�nition, sine we havetaken a weak losure in the formation of Nω, but it is well known - see for example[69, Theorem XIV.4.6℄. When two ω-entralising sequenes give the same elementin Nω we shall all them ω-equivalent.Given M ⊂ N we regard Mω as a subset of Nω allowing us to de�ne theentral sequene algebra Mω ∩ N ′. For the following result, whih is surely wellknown, it su�es to follow Takesaki's a

ount of the original in [69, TheoremXIV.4.7℄ and hek that elements an be hosen in M where neessary.Proposition 1.2.13. Let M be a di�use von Neumann subalgebra of a II1 fatorN , then Mω ∩N ′ ⊂ Nω is either trivial or di�use. Elements of Mω ∩N ′ are the(images under πω) of ω-entralising sequenes lying in M . The latter ase o

urspreisely when M ontains non-trivial entralising sequenes for N . Furthermore,unitaries in Mω ∩ N ′ are the πω images of ω-entralising sequenes of unitariesin M .Reently, Fang, Ge and Li have onsidered another objet. In setion 3 of [18℄they examine Nω ∩M ′ when M ⊂ N is an irreduible inlusion of II1 fators,showing that it too is either trivial or di�use. This work also extends a resultof Connes ([7℄), whih was also known to MDu� ([34℄), that the von Neumanntensor produt of two II1 fators has property Γ if and only if at least one of the15This is unfortunate, it would have been nie to pre�x the entire thesis with a dislaimerasserting the separability of everything within.16A di�use von Neumann algebra is one with no minimal projetions.16

two fators does.17 The result of [18℄ shows more.Theorem 1.2.14 ([18, Theorem 4.7℄). Let ω be a free ultra�lter on N. Supposethat M is a non-Γ fator of type II1 and N is another type II1 fator. Then(M⊗N)ω is anonially isomorphi to Nω.Unfortunately we need more still. Like the rest of this setion everything iswell behaved when we onsider inlusions of di�use von Neumann subalgebras.No essential hanges are required to work in this situation, but18 this time wegive the details, as [18℄ is not yet readily available.Theorem 1.2.15. Let M1 ⊂ N1 and M2 ⊂ N2 be inlusions of di�use vonNeumann subalgebras in II1 fators. Suppose that M1 does not ontain non-trivial entralising sequenes for N1, then (M1⊗M2)ω ∩ (N1⊗N2)′ is anoniallyisomorphi to Mω2 ∩N ′2.Proof. SineM1 does not ontain entralising sequenes forN1, we an �ndK > 0and unitaries u1, . . . , ul ∈ N1 suh that

‖x− tr(x)1‖22 ≤ K l∑i=1

‖xui − uix‖22 , (1.2.1)for all x ∈M1.19 Take an ω-entralising sequene (zn)∞n=1 in M1⊗M2 for N1⊗N2.By density we may assume that eah zn is a sum of elementary tensors: i.e.zn =

mn∑

j=1

x(n)j ⊗ y(n)j ,with x(n)j ∈ M1 and y(n)j ∈ M2. Furthermore, for eah n we an demand thatthe y(n)j are orthogonal in L2(N2) and have ‖yj‖2 = 1. We write EN2 for theonditional expetation ontoN2 (regarded as C1⊗N2 - a von Neumann subalgebraof N1⊗N2), and note that

EN2 (zn) =mn∑

j=1

tr(x(n)j )1 ⊗ y(n)j .17Compare with taking an in�nite von Neumann tensor produt of II1 fators - this alwaysprodues a Γ fator.18In ontrast with Proposition 1.2.13.19This is the di�use von Neumann subalgebra version of Lemma 4.6 of [18℄ and, just as there,it follows diretly from the de�nition.17

We now estimate ‖zn − EN2 (zn)‖2, using (1.2.1):‖zn − EN2 (zn)‖22 =

∥∥∥∥∥

mn∑

j=1

(x(n)j − tr(x(n)j )1) ⊗ y(n)j ∥∥∥∥∥2

2

=

mn∑

j=1

∥∥∥x(n)j − tr(x(n)j )1∥∥∥22

(1.2.2)≤ K

mn∑

j=1

l∑

i=1

∥∥∥x(n)j ui − uix(n)j∥∥∥

2

2(1.2.3)

= K

l∑

i=1

‖zn(ui ⊗ 1) − (ui ⊗ 1)zn‖22 . (1.2.4)Here (1.2.2) and (1.2.4) follow from our additional hypotheses on the form of they

(n)j , and (1.2.3) follows from (1.2.1). Sine (zn)∞n=1 is ω-entralising for N1⊗N2,we see that

limn→ω

‖zn − EN2 (zn)‖2 = 0,whih is the ω-equivalene of (zn)∞n=1 and (EN2 (zn))∞n=1, whih lies in (1 ⊗M2)ω.1.2.6 Automorphisms of II1 fatorsThe group of automorphisms of a separable II1 fator N is a well studied objet.Equipped with the so alled u-topology20 of pointwise norm onvergene on thepredual N∗, as de�ned in [22℄, Aut (N) is a Polish spae - that is a ompletemetri spae - see [6℄. We shall prefer to work with pointwise ‖.‖2-onvergene,whih gives the same topology on Aut (N). It is neessary though to ensure wegenuinely work with this only on the automorphism group. Suppose we haveautomorphisms θn of N and some θ suh thatlimn→∞

‖θn(x) − θ(x)‖2 = 0. (1.2.5)Then θ is neessarily an injetive ∗-homomorphism, but not neessarily an au-tomorphism.21 If we know that θ is an automorphism though, then (1.2.5) isequivalent to the onvergene of θn to θ in the u-topology.The normal subgroup of inner automorphisms, written here as Inn (N),22 on-sist of all automorphisms of the form Ad u : x 7→ uxu∗ for some unitary u ∈ N .The quotient Aut (N) /Inn (N) is the outer automorphism group of N , writtenOut (N).20Formally, a net (θα)α in Aut (N) onverges to θ ∈ Aut (N) if and only if ‖φ ◦ θα − φ ◦ θ‖ → 0for all φ ∈ N∗.21Indeed examples an be given in the abelian von Neumann algebra L∞[0, 1] of this failure.22Also denoted Int N in the literature, due to the Frenh tradition in the subjet.18

The approximately inner automorphisms, namely the u-topology losure Inn (N)of the inner automorphisms, was examined by Connes in [6℄. He showed that theinner automorphisms of N are losed in Aut (N) if and only if N fails to haveproperty Γ. In his lassi�ation of injetive fators ([7, Theorem 3.1℄) he went onto haraterise the approximately inner automorphisms preisely, Theorem 1.2.16below. He noted, see [69, Theorem XIV.2.16℄, that all automorphisms of the hy-per�nite II1 fator are approximately inner, and haraterised hyper�niteness [7,Theorem 5.1℄, amongst other ways, by the property that the swap automorphismon N⊗N taking x ⊗ y to y ⊗ x is approximately inner. In setion 3.5, we shallexamine these ideas in the relative ontext of masas inside II1 fators.Theorem 1.2.16 (Connes). Let N be a fator of type II1 with separable predualating on L2(N). Then the following onditions are equivalent for θ ∈ Aut (N):1. θ ∈ Inn (N);2. There exists an automorphism of the C∗-algebra generated by N and N ′ inL2(N) whih is θ on N and the identity on N ′;3. For any unitary operators u1, . . . , un ∈ N and any ǫ > 0 there is a ξ ∈L2(N), with ‖ξ‖2 = 1 and ‖θ(uk)JukJξ − ξ‖2 < ǫ for all k = 1, . . . , n;4. There exists a bounded sequene (xn)∞n=1 in N , not onverging strongly to 0,suh that xny − θ(y)xn onverges to 0 strongly, for any y ∈ N .1.3 Conditional expetations, the basi onstru-tion and the Jones indexThroughout this thesis we will be examining a variety of inlusions 1 ∈M ⊂ N ofvon Neumann algebras with the same unit. A key tool to study this situation is aonditional expetation operator, namely a norm 1, M-bimodule projetion from

N ontoM . For general N and M , the existene of these onditional expetationsis not guaranteed. Furthermore, the theory is also fairly involved - the enthusiastireader will �nd a full a

ount in [65℄. Fortunately when N is a �nite von Neumannalgebra, as it is throughout, things are muh easier. As usual in this situation we�x a normalised faithful normal trae tr on N , and now look only for onditionalexpetations preserving this trae.In this ase a trae preserving onditional expetation ontoM not only exists,it is also unique and an be expliitly onstruted. We write eM for the orthogonalprojetion from L2(N, tr) onto L2(M, tr). If we regard N and M as subspaes ofL2(N, tr) and L2(M, tr) respetively, then it an be easily heked that eM (N)19

is atually ontained in M . We write EM for the restrition of eM to N , the(trae-preserving) onditional expetation from N onto M , whih is the uniquebounded linear map from N into M satisfying• EM (1) = 1.• M-bimodularity, i.e. EM (m1xm2) = m1EM (x)m2 for all m1, m2 ∈ M andx ∈ N .

• tr ◦ EM = trall of whih an be easily veri�ed from the onstrution. The uniqueness of theonditional expetation has many useful appliations, for example it allows us toimmediately dedue thatEuMu∗ (x) = uEM (u

∗xu)u∗, (1.3.1)for any unitary u ∈ N and all x ∈ N . We shall also require a result of Christensen,for omputing onditional expetations, whih an be found in [4℄.Proposition 1.3.1 (Christensen). Let M be a von Neumann subalgebra of theII1 fator N . For eah x ∈ N , let oM(x) denote the onvex hull of the set{ uxu∗ | u a unitary in M }. Let E(x) denote the element of minimal ‖.‖2 in the‖.‖2-losure of oM(x). This E(x) lies in N , and has E(x) = EM ′∩N (x).1.3.1 The basi onstrution and Jones indexThe basi onstrution, whih dates bak to [64℄ and [4℄, and was developedextensively by Vaughan Jones in [28℄, assoiates to an inlusion 1 ∈ M ⊂ N of�nite von Neumann algebras with �xed trae tr on N , the extension 〈N, eM〉,de�ned to be the von Neumann algebra ating on L2(N, tr) generated by N andeM . We shall not be heavily involved with the basi onstrution in the mainbody of this thesis, so ontent ourselves here by just stating the basi propertieswe shall need. A full disussion of these results an be found in a number ofsoures, inluding [28, Setion 3.1℄ and [26, Chapter 3℄.

• eMxeM = EM (x) eM = eMEM (x), for all x ∈ N .• 〈N, eM〉 = JM ′J where, as usual, J denotes the modular onjugation oper-ator on L2(N, tr).• 〈N, eM〉 is a fator if and only if M is a fator.• The entral support of eM in 〈N, eM〉 is 1.20

The Jones index, [N : M ], for our inlusion M ⊂ N an be de�ned usinga variety of di�erent methods: Murray and von Neumann's oupling onstant([36℄) was used in Jones original paper [28℄; other de�nitions involve examiningthe dimension of L2(N, tr) as an M-bimodule, whih is the approah taken inChapter 2 of [26℄. We shall only be interested in the ase where M and N are�nite fators, so we are able to follow [75℄ and de�ne the index, [N : M ], of M inN by

[N : M ] =

{ Tr(eM )−1 〈N, eM〉 is a �nite fator∞ 〈N, eM〉 is an in�nite fator , (1.3.2)where in the �rst ase, Tr denotes the unique normalised faithful trae on the �nitefator 〈N, eM〉. That this de�nition agrees with the more general non fator aseis Proposition 3.1.7 of [28℄. When the index is �nite, we have the Markov propertyTr(eMx) = [N : M ]−1tr(x), (1.3.3)for every x ∈ N . It is also important to note that, see for example [26, Corollary2.3.6(b)℄, when we have a tower of inlusions of II1 fators, M ⊂ P ⊂ N , theindex is multipliative in the sense that

[N : M ] = [N : P ][P : M ].23 (1.3.4)The other elementary result we shall need in the sequel, is how the index interatswith tensor produts. Let M1 ⊂ N1 and M2 ⊂ N2 be inlusions of II1 fators.Working in the Hilbert spae tensor produt L2(N1)⊗L2(N2) ∼= L2(N1⊗N2), theoperator eM1⊗M2 of projetion onto L2(M1⊗M2) fatorises as eM1 ⊗ eM2 . In thisway, 〈N1⊗N2, eM1⊗M2〉 fatorises as 〈N1, eM1〉⊗ 〈N2, eM2〉 and thenTr〈N1⊗N2,eM1⊗M2〉(eM1⊗M2) = Tr〈N1,eM1〉(eM1)Tr〈N2,eM2〉(eM2),or alternatively[N1⊗N2 : M1⊗M2] = [N1 : M1][N2 : M2]. (1.3.5)The main result of [28℄, is the striking observation that the index [N : M ]must take values in { 4 cos2(π/k) | k = 3, 4, 5, . . . }∪[4,∞]. Jones also onstrutedexamples showing all these values an be obtained when N is the hyper�nite II1fator. When [N : M ] < 4, M is automatially irreduible, i.e. M ′ ∩N = C1, see[28, Corollary 2.2.4℄. All of Jones' initial examples with index greater than 4 arenot irreduible; and the full range of all possible values for the index of irreduiblesubfators is still unknown. In the sequel, we shall mainly be interested in the23Here we have the usual onvention that x∞ = ∞ for any x.21

inlusion of a regular subfator24 � in this ase Jones has shown the followingadditional restrition on the values of the index.Proposition 1.3.2. If 1 ∈ M ⊂ N is a unital inlusion of II1 fators with Mregular in N , then [N : M ] ∈ N ∪ {∞}.In the ase when M is also assumed to be irreduible, whih is all we shalluse later, this an be found in [24℄. The more general ase is dealt with by theobservationA more re�ned analysis based on [27℄ shows that all regular subfatorshave integer index.found on page 150 of Jones' book [21℄ with Goodman and de la Harpe.1.3.2 Inlusions of index 2Historially, the ase of index 2 inlusions was the �rst to be onsidered. In1960, Goldman haraterised index 2 subfators25 as being those oming fromross produts over Z2, ([20℄, see also [28, Corollary 3.4.3℄). More preisely, if1 ∈ M ⊂ N is a unital inlusion of II1 fators with [N : M ] = 2, then thereexists a non-inner automorphism θ of M of order 2 suh that N is ∗-isomorphito M ⋊θ Z2. Although we have not introdued ross produts,26 we shall brie�yuse this formulation. For our purposes, it is enough to regard M ⋊θ Z2 as a∗-subalgebra of the 2 × 2 matries over M , by

N ∼= M ⋊θ Z2 ={(

x yθ(y) θ(x)

) ∣∣∣∣ x, y ∈M}. (1.3.6)Here M is inluded in M ⋊θ Z2, by regarding M as isomorphi to the algebra ofall matries of the form (

x 00 θ(x)

).Four years later, in her work [70℄ on semi-regular masas of varying lengthswhih we examine in setion 2.3, Tauer distinguishes between subfators of Rusing the following property for a subfator M .The invariant is the fat that the produt of two operators in M⊥ isalways in M .24A pedagogial hoie has left us with the undesirable situation of this onept not beingspei�ed until De�nition 1.4.5 in the next setion!25Before the onept of index had been expliitly de�ned.26and have no intention of doing so! 22

We have hanged the notation slightly from the original statement, in the proofof Lemma 6.7 of [70℄. The set M⊥ onsists, as one would expet, of all operatorsx ∈ R orthogonal to M in the sense that EM (x) = 0.27 We an see that anyunital inlusion of subfators 1 ∈ M ⊂ N of index 2 satis�es Tauer's property.Indeed, given suh an inlusion, use Goldman's Theorem to write N in the form(1.3.6) for some θ ∈ Aut (M) of order 2. Two elements of N orthogonal to M arethen of the form (

0 y1θ(y1) 0

) and ( 0 y2θ(y2) 0

)for some y1, y2 ∈M . These elements have produt(y1θ(y2) 0

0 θ(y1)y2

),whih lies in M . In fat Tauer's property also essentially haraterises inlusionsof index 2 although, unlike Goldman who, as noted in [28℄, more or less expliitlyde�nes the index by means of the M-dimension of L2(N), she was most likelyunaware of this.Proposition 1.3.3. Let 1 ∈ M ⊂ N be a �nite index unital inlusion of II1fators. The following two statements are equivalent:1. [N : M ] = 2;2. If x, y ∈ N have EM (x) = EM (y) = 0 then xy∗ ∈M .We have hosen to use the expression essentially haraterises index two in-lusions as it has proved awkward to extend this Proposition to show that noin�nite index inlusion an satisfy ondition 2 above. Let us �rst quikly see why,under the assumption of �nite index, ondition 2 guarantees that [N : M ] = 2.The method we shall use here is the generi nature of the basi onstrution, [28,Corollary 3.1.9℄.Proposition 1.3.4. Let 1 ∈ M ⊂ N be a �nite index, unital inlusion of II1fators. Then there is a subfator P ⊂ M with [M : P ] = [N : M ] suh that

〈M, eP 〉 ∼= N .We apply this by onsidering the element x = 1− [N : M ]eP ∈ 〈M, eP 〉. SineEM (eP ) = [N : M ]

−11,28 we have EM (x) = 0. On the other handxx∗ = (1 − [N : M ]eP )2 = 1 +

([N : M ]2 − 2[N : M ]

)eP ,27Formally, we should perhaps speak of orthogonality to L2(M) but, as we have previouslynoted, EM (x) = eM (x) for x ∈ N .28We are regarding N as (isomorphi to) 〈M, eP 〉, so that eP is an element of N and EM isthe onditional expetation from N onto the subfator M .23

so that xx∗ ∈ M if and only if [N : M ]2 − 2[N : M ] = 0, from whih we andedue that [N : M ] = 2. This onludes the proof of Proposition 1.3.3.There is no hope of generalising the proof above to the ase of an in�niteindex inlusion M ⊂ N . It is possible to give another, more involved proof, ofthe impliation 2 ⇒ 1 using Pimsner-Popa bases for the inlusion M ⊂ N . Itis possible that this idea might generalise, for there is a notion of a generalisedorthogonal basis for an in�nite index inlusion ([46℄), but this will onsist ofunbounded operators a�liated to M . An intriate analysis of these generalisedorthogonal bases is needed, to determine whether these operators an be foundin L2(M). In this thesis, we have been assiduous in our poliy of only workingwith bounded operators - we leave the in�nite index ase as a onjeture.Question 1.3.5. How do we extend Proposition 1.3.3 to the in�nite index situ-ation?We shall brie�y need these orthogonal bases in setion 2.4. We will not developthe theory and just state exatly what we need. The result, whih is well known,an easily be established by manipulating fats, from [39℄, about these Pimsner-Popa orthogonal bases. We give a proof whih uses these fats impliitly.Proposition 1.3.6. Suppose that M ⊂ N is an inlusion of II1 fators, andthat there are n unitaries (ui)ni=1 in N with EM (uiu∗j) = δi,j1 for all i, j, then[N : M ] ≥ n.Proof. We assume that [N : M ] < ∞, otherwise the result is trivial. In 〈N, eM〉,eah eMui is a partial isometry with domain projetion u∗i eMui and range pro-jetion eM . The hypothesis ensures that these domain projetions are pairwiseorthogonal, so that

n∑

i=1

uieMu∗i ≤ 1.Take the trae of both sides to see that nTr(eM ) ≤ 1, from whih the resultfollows.1.3.3 Inlusions of �nite dimensional C∗-algebrasHere, we give a brief exposition of the theory of inlusions of �nite dimensional C∗-algebras, whih are also neessarily weakly losed and so von Neumann algebras.All of this material an be found a variety of soures, suh as [21, Setion 2.3℄and [26, Setion 3.2℄. Firstly, reall that every �nite dimensional C∗-algebra isisomorphi to a diret sum of matrix algebras, see for example [10, Theorem24

III.1.1℄. Suppose that we have the identi�ationM ∼= Mata1(C) ⊕ Mata2(C) ⊕ · · · ⊕ Matam(C),where Matk(C) denotes the algebra of k × k matries over C, then

a = (a1, . . . , am)T ∈ Nm,is alled the dimension vetor of M . Similarly, let N have diret sum deompo-sition

N ∼= Matb1(C) ⊕ Matb2(C) ⊕ · · · ⊕ Matbn(C),with orresponding dimension vetor b ∈ Nn. An inlusion 1 ∈ M ⊂ N or, morepreisely, a unital injetive ∗-homomorphism fromM into N , is determined (up tounitary onjugation in N) by an n×m inlusion matrix Λ over N0, with Λa = b.29The i, j-th entry λi,j of Λ is naively de�ned to be the number of times the ithomponent Matai(C) of M is repeated in the jth summand Matbj (C) of N .30The data for a �nite dimensional inlusion is also often ontained in a Brattelidiagram, an onept best explained by example. The Bratteli diagram2 5

2

OO @@�������1

KS

1

^^=======represents the unital inlusion of M , with dimension vetor (2, 1, 1)T, into N ,with dimension vetor (2, 5)T, by the inlusion matrixΛ =

(1 0 01 2 1

).Sine the dimension vetor of N is determined from that ofM and Λ, the dimen-sions of N are often omitted from the diagram.Sine there is a unique normalised trae on any matrix algebra (as with any�nite fator), a normalised trae tr on M is determined by a trae vetor

s = (s1, . . . , sm) ∈ Rm+ ,with sa = 1. The omponents are given by si = tr(pi), where pi is a minimalprojetion in the ith summand Matai(C) of M , and tr is then a faithful trae29There is no agreement in the literature about the orientation of the inlusion matrix. Theonvention we have hosen agrees with Davidson in [10℄ and Jones in [21℄, whereas Wenzl ([75℄)and Jones (again!) in [28℄ and [26℄ prefer to work with row vetors and the transpose of thismatrix.30Formally we should examine the representation struture of M and N , de�ning λi,j to bethe number of times the ith irreduible representation of M o

urs in the restrition to N ofthe jth irreduible representation of N but, as noted in [26℄, this does not provide additionalillumination. 25

if eah si > 0. If the trae on N is given by the trae vetor t ∈ Rn+, then therestrition of tr to M has trae vetor s = tΛ.In this �nite dimensional situation, the basi onstrution, 〈N, eM〉 deom-poses into m fators ⊕mi=1 〈N, eM〉i. A minimal projetion for 〈N, eM〉i an befound of the form eMpi, where pi is a minimal projetion in the ith summand ofM . With this ordering of the deomposition of 〈N, eM〉, the inlusion matrix of1 ∈ N ⊂ 〈N, eM〉 is given by ΛT. Sine the inlusion matrix for a hain of �nitedimensional inlusions is obtained by matrix multipliation31 (e.g. [26, Corollary2.3.2℄), the inlusion matrix for 1 ∈M ⊂ 〈N, eM〉 is ΛTΛ.1.3.4 The Wenzl index formulaIt is possible to onstrut an inlusion of fators by a hain of inlusions of �nitedimensional C∗-algebras. The Wenzl index formula, Theorem 1.5 of [75℄, allows usto ompute the resulting index for suitably periodi inlusions whih will appearlater. In fat we shall need a very slight extension of Wenzl's work whih weformulate in this setion. We begin with a de�nition of ommuting squares [26,De�nition 5.1.7℄, an idea originally due to Popa, who began the examinination ofommutating onditional expetations in [42℄.De�nition 1.3.7. Let N2 be a �nite von Neumann algebra with �xed faithfultrae tr. Consider the square of inlusions of von Neumann subalgebras of N2

N1� // N2

M1� //

?�

OO

M2?�

OOwith onditional expetations EM1 from N1 onto M1, and EM2 from N2 onto M2obtained from the trae tr on N2 in the usual way. The square is said to ommutewhen EM1 (x) = EM2 (x) for all x ∈ N1 or, more formally, whenN1

� //

EM1

��

N2

EM2

��M1

� //M2is a ommutative diagram of maps in the usual way. A diagram onsisting ofmultiple (and possibly in�nitely many) squares is a ommutative diagram if eahonstituent square ommutes.We shall examine in�nite ommutative diagrams of inlusions of �nite di-mensional C∗-algebras Mn and Nn of the form of Figure 1.1, where Λn is the31With our hoie of notation, this happens in a ontravariant way.26

N1� // N2

� // . . . � // Nn� // Nn+1

� // . . . � // N

M1� //

?�

Λ1

OO

M2� //

?�

Λ2

OO

. . . � //Mn� //

?�

Λn

OO

Mn+1� //

?�

Λn+1

OO

. . . � //M?�

OO

Figure 1.1: The setup for the Wenzl index formulainlusion matrix of 1 ∈ Mn ⊂ Nn. We shall insist that the ∗-algebras ⋃∞n=1Mnand ⋃∞n=1Nn are in�nite dimensional. Suppose that there is a unique normalisedtrae on the ∗-algebra ⋃∞n=1Nn, then N , de�ned to be the weak losure of the im-age of ⋃∞n=1Nn under the GNS representation orresponding to tr, is a II1 fator,whih is hyper�nite by onstrution. We shall also require that the restrition oftr to ⋃∞n=1Mn is the unique normalised trae on ⋃∞n=1Mn, so that M , de�ned tobe the von Neumann subalgebra of N generated by ⋃∞n=1Mn, is also a hyper�niteII1 fator. Atually, we only need to establish the uniqueness of these traes inthe very limited situation overed by the next proposition.Proposition 1.3.8. Suppose that we have a hain of inlusions of �nite dimen-sional C∗-algebras, with inlusion matries Γn as indiated below.

N1� Γ1 // N2

� // . . . � // Nn� Γn // Nn+1

� // . . .If, for in�nitely many n, every entry in the inlusion matrix Γn is idential, thenthere is at most one normalised trae on ⋃∞m=1Nm.Proof. Take some n with the desribed property, so Γn has idential entries - sayCn. Consider a normalised trae tr on ⋃∞m=1Nm with trae vetor t on Nn+1. Thetrae vetor s of Nn given by s = tΓn has si = Cn∑j tj, for eah i. Hene, therestrition of tr to Nn is unique by normalisation. As there are in�nitely many nfor whih the restrition of tr to Nn is unique, we see that there is at most onenormalised trae on ⋃∞m=1Nm.If in addition we had supposed that all the inlusion matries Γn in Propo-sition 1.3.8 had the desribed property, then an easy argument would also yieldthe existene of a normalised trae on ⋃∞n=1Nn. Despite this being exatly thesituation o

urring later, this is unneessary as all of the algebras Nn andMn willin fat lie in a larger hyper�nite II1 fator, giving us the existene of a normalisedtrae by restrition.The Wenzl index formula, Theorem 1.5 of [75℄, deals with diagrams of inlu-sions of the form of Figure 1.1 where we have additional periodiity requirementson the inlusions. We state the version of the formula from [21, Theorem 4.3.3℄,27

realling that a matrix Λ (with entries in R+) is primitive when there is somel ∈ N suh that all the entries of Λl are stritly positive.Theorem 1.3.9 (Wenzl index formula). Suppose we have �nite dimensional C∗-algebras Mn and Nn as in the ommutative diagram Figure 1.1. Suppose thatthere exists n0 ≥ 1 and p ≥ 1 suh that (with an appropriate ordering of thematrix algebras in the deompositions of Mn and Nn), we have for eah n ≥ n0:1. The inlusion matrix for Nn ⊂ Nn+1 is the same as that for Nn+p ⊂ Nn+p+1,and the inlusion matrix for Mn ⊂ Mn+1 is the same as that for Mn+p ⊂

Mn+p+1;2. The inlusion matries for Nn ⊂ Nn+p and Mn ⊂Mn+p are primitive;3. Λn = Λn+p.Then M ⊂ N is an inlusion of II1 fators with[N : M ] = ‖Λn‖2 ,for every n ≥ n0.As noted in [21℄ the periodiity data for the inlusions Nn ⊂ Nn+1 is onlyrequired to establish that N is a fator. While the index formula as stated willsu�e to ompute the index in one of the situations we shall require, in anotherase the size of the inlusion matries Λn will inrease with n, although they willretain the same struture so some periodiity will remain. We shall establish aversion of the Wenzl index formula in this situation as a orollary of the nextTheorem, in whih Wenzl examines the struture of general extensions of thebasi onstrution in �nite dimensions.Theorem 1.3.10 (Wenzl - [75, Theorem 1.1℄). Let 1 ∈ M ⊂ N be an inlusionof �nite dimensional C∗-algebras ating on the Hilbert spae H, with dimensionvetor a = (a1, . . . , am) for M . Fix a normalised faithful trae tr on N , whoserestrition to M has trae vetor s. Suppose that e is a projetion in B (H) suhthat

• exe = eEM (x) = EM (x) e for all x ∈ N• eM ∼= M ∼= Me,and let 〈N, e〉 be the (�nite dimensional) C∗-algebra generated by N and e.1. 〈N, e〉 ∼= 〈N, eM〉 ⊕K where K is isomorphi to a subalgebra of N .28

2. The entral projetion z in 〈N, e〉 onto 〈N, eM〉 (under the isomorphism of1) oinides with the entral support of e in 〈N, e〉.3. Let Tr be a trae on 〈N, e〉 extending tr. Then Tr(e) ≥ dTr(z), whered = mini=1,...,m ai/(Λ

TΛa)i.4. Let t be the trae vetor of Tr|〈N,eM 〉 (under the isomorphism in 1), thentΛTΛ ≤ s pointwise.We now ome to our well trailed orollary designed for appliation in setion2.4. It should be noted that the proof is a ombination of Lemma 1.4 and Theorem1.5(i) of [75℄ - we inlude it for ompleteness.Corollary 1.3.11. Suppose that in the situation of the ommutative diagram ofFigure 1.1 there is a unique faithful normalised trae on ⋃∞n=1Nn and on⋃∞n=1Mn.Suppose also that the dimension vetors a(n) for Mn are of the form

a(n) = An(

1 1 . . . 1)T,for some onstants An, and that there exists an integer λ ≥ 2 suh that eah ofthe inlusion matries Λn takes the form

Λn =

λ︷ ︸︸ ︷(I I . . . I

),where I is some identity matrix (whose size may vary with n).32 Then M ⊂ Nis an inlusion of II1 fators with

[N : M ] = λ.Proof. By hypothesis 1 ∈ M ⊂ N is an inlusion of II1 fators. Let Tr bea (possibly semi�nite) faithful normal trae on the basi onstrution 〈N, eM〉.Sine eM 〈M, eN 〉 eM = MeM ∼= M is �nite, Tr(eM)

Sine ∪∞n=1 〈Nn, eM〉 is dense in 〈N, eM〉, these zn onverge to 1, the entral supportof eM in 〈N, eM〉. In partiular Tr(1) ≤ λTr(eM)

masas have been used as tools in various areas of von Neumann algebra theory;Sinlair and Smith ([58℄) showed that a separable II1 fator ontaining a Cartanmasa has vanishing ontinuous Hohshild ohomology - a onept we have nointention of allowing further disussion of here!33 Sorin Popa, in [48℄, uses theuniqueness of a ertain Cartan masa34 to demonstrate that L (SL(2,Z) ⋊ Z2) hasa trivial fundamental group. Over the last ouple of years this amazing tehniqueof Popa's has been used to resolve many major problems in the lassi�ation ofII1 fators, [49, 40, 23℄.I would like to end these introdutory remarks with a plug for Allan Sinlairand Roger Smiths' forthoming book, [56℄, whih will over in full detail almostall the material in this setion. Indeed, the onept of strong-singularity, see sub-setion 1.4.3, was initially developed to produe a leaner proof of the singularityof ertain masas for this work.1.4.1 Basi properties of masasAs von Neumann algebras, masas inside separable II1 fators are all the same.This result dates bak to Murray and von Neumann, a proof an also be foundin [57, Lemma 5.3.4℄.Proposition 1.4.1. Let A be a masa in a separable II1 fator N . There is a∗-isomorphism from A onto L∞[0, 1] whih indues an isometry between L2(A, tr)and L2[0, 1].We should study the inlusion A ⊂ N of a masa inside a II1 fator, ratherthan just the masa itself. The terminology in the literature is that of onjugay.De�nition 1.4.2. Two masas A and B in a II1 fator N are said to be onjugatevia an automorphism of N , or sometimes lazily just onjugate in N , if there existsan automorphism θ of N with θ(A) = B. They are unitarily onjugate in N if θis an inner automorphism of N .We wait until the next subsetion to see some non-onjugate masas. In theremains of this subsetion we ollate some well known basi properties of masas.It is immediate that a von Neumann subalgebra A of a II1 fator N is a masa ifand only if it is its own relative ommutant, that is A = A′ ∩N . By thinking ofA as L∞[0, 1], we an �nd a hain, A1 ⊂ A2 ⊂ . . . , of �nite dimensional abelianC∗-algebras generating A as a von Neumann algebra. Popa onneted these twoobservations to give the following riterion for determining when suh a hain33See [57℄ for an a

ount of this ohomology theory.34a so alled HT masa. 31

generates a masa, whih he used to good e�et in [44℄. A proof an also be foundin [57, Lemma 5.3.2℄.Proposition 1.4.3. Given a hain (An)∞n=1 of �nite-dimensional abelian C∗-algebras in a II1 fator N , let A = (⋃∞n=1An)′′ be the abelian von Neumannalgebra it generates. Then A is a masa in N if and only iflimn→∞

∥∥EAn (x) − EA′n∩N (x)∥∥

2= 0, (1.4.1)for all x ∈ N .Both the onditional expetations appearing in (1.4.1) an easily be alulated.Suppose B is a �nite dimensional abelian C∗-algebra with minimal projetions

(ei)ni=1, ontained in a II1 fator N . Then (ei/ ‖ei‖2)ni=1 is an orthonormal basisfor L2(B) so that

EB (x) =

n∑

i=1

tr(xei)ei‖ei‖22

,for all x ∈ N . It is also well known that the onditional expetation onto therelative ommutant is given byEB′∩N (x) =

n∑

i=1

eixei, (1.4.2)for all x ∈ N . This an be seen by heking that (1.4.2) de�nes a onditionalexpetation, then appealing to uniqueness.In the ase of von Neumann algebras oming from groups, we have a wellknown riterion for an inlusion of groups to give rise to a masa, whih datesbak to Diximer, [11℄.Proposition 1.4.4. Let H be an abelian subgroup of the ountable disrete groupG. Then L (H) is a masa in L (G) if and only if the set

{hgh−1

∣∣ h ∈ H},is in�nite, whenever g ∈ G \H.1.4.2 Normalisers of masasWhen M ⊂ N is an inlusion of von Neumann algebras, we onsider the group

N (M) of all unitary normalisers of M in N given byN (M) = { u ∈ U(N) | uMu∗ = M } .When the larger von Neumann algebra is not obvious, we write NN (M) for

N (M). These groups where introdued by Dixmier ([11℄) to lassify masas byexamining how many normalisers there are, with the Cartan masas at one extremeand the singular masas at the other. 32

De�nition 1.4.5 (Dixmier). A von Neumann subalgebra M of a von Neumannalgebra N is said to be Cartan or regular when N (M)′′ = N ,35 and singularwhen N (M)′′ = M . A masa A in N is alled semi-regular when N (A)′′ is aproper subfator of N .These onepts gave the �rst examples, also in [11℄, of non-onjugate masasin a II1 fator. One of the prinipal su

esses of this lassi�ation program is theremarkable result of Connes, Feldman and Weiss, [8℄ (see also [45℄), Theorem 1.4.6below, on the uniqueness of the Cartan masa in R. Examples of non-onjugateCartan masas in a non-injetive II1 fator were later given in [9℄. In [44℄, Popashowed that singular masas an always be found in any separable II1 fator, onthe other hand, Cartan masas are not always present; Voiulesu (in [73℄) hasshown that no Cartan masa exists in a free group fators, L (Fk).Theorem 1.4.6 (Connes, Feldman and Weiss). Any two Cartan masas in thehyper�nite II1 fator R are onjugate via an automorphism of R.Pukánszky's examples ([51℄) gave ountably many pairwise non-onjugate sin-gular masas in the hyper�nite II1 fator R. His method, for showing this non-onjugay is now known as the Pukánszky invariant for a masa, an idea dis-ussed at length in hapter 3, and whih has reently been used by Størmerand Neshveyev ([38℄) to give unountably many pairwise non-onjugate singularmasas in R. In setion 3.3 we will give an alternative method of obtaining un-ountably many pairwise distint singular masas in R. It will not be possible touse Pukánszky's invariant to distinguish between these masas.When a masa is Cartan, one an normally verify this be writing down aolletion of normalisers whih generate the underlying II1 fator. Similarly inthe ase when A is a semi-regular masa in N with N (A)′′ = M for some givensubfator M of N , the inlusion M ⊂ N (A)′′ should be easy to verify simplyby exhibiting enough normalisers. To do this, it is often helpful to look at thegroupoid normaliser of A, GN (A), onsisting of all partial isometries v ∈ Nwith initial and range projetions in A and whih normalise A, in the sense thatvAv∗ = Avv∗. This groupoid normaliser generates the same von Neumann algebraas the normalisers do, as (see [25, 45℄) elements of GN (A) are preisely of theform ue for some u ∈ N (A) and projetion e ∈ A.However, the reverse inlusion, N (A)′′ ⊂ M is in general muh harder toestablish. Proving that a given masa is singular is of a similar level of di�ulty,35Cartan is used in the ontext of masas, while other von Neumann algebras are alled regular.Indeed, urrently the expression Cartan subalgebra is used in the literature to mean a Cartanmasa. 33

as the same style of inlusion has to be shown. Currently, the preferred methodof establishing singularity is to use strong singularity , as the name suggests, astronger onept than singularity whih is often easier to verify in pratie. Wewill disuss this idea further in subsetions 1.4.3 and 1.4.4. Here we give a briefdisussion of other methods for establishing upper bounds for N (A) that areeither historially interesting or appear later, onentrating mainly on the singularase.We have already mentioned, in subsetion 1.2.3, that Popa used orthogonalityto give a method for ontrolling the loation of normalising unitaries. The maintehnial result is Corollary 2.6 of [43℄, whih we state for ompleteness. Weremind the reader that a di�use von Neumann algebra is one with no minimalprojetions.Proposition 1.4.7 (Popa). Let M be a von Neumann subalgebra of the �nitevon Neumann algebra N and u be a unitary in N . If there exists a di�use vonNeumann subalgebra M0 of M suh that uM0u∗ is orthogonal to M , then u isorthogonal to N (M)′′.This proposition was partiularly useful in the ontext of group von Neumannalgebras, where it gives Proposition 4.1 of [43℄ below.Corollary 1.4.8 (Popa). Let H ⊂ H1 ⊂ G be an inlusion of in�nite disretegroups. If gHg−1 ∩ H = {1} for all g ∈ G \ H1, then whenever M is a di�usevon Neumann subalgebra of L (H), we have N (M)′′ ⊂ L (H1).If we take H = H1 in the above then we obtain the group theoreti ondition ofmalnormality, namely that for every g ∈ G\H we have gHg−1∩H = {1}. Supposein addition that H is abelian, then L (H) is a masa in L (G) by Proposition 1.4.436and so is di�use. In this instane Corollary 1.4.8 gives the singularity of L (H) inL (G).Popa is also responsible for two other methods of demonstrating the singularityof a masa. In [45℄, he gave a onnetion between the Pukánszky invariant of amasa and its normalisers - this is disussed in setion 3.1, where the Pukánszkyinvariant is de�ned. The tool he used in [44℄, to show that singular masas exist inany separable II1 fator was a δ-invariant. Formally when A is a masa in N , andv ∈ N is a partial isometry in N suh that v∗v and vv∗ are mutually orthogonalprojetions in A de�ne

δ(v) = supx∈vAv∗,‖x‖≤1

‖x− EA (x)‖2‖v∗v‖2

,36It is easily heked that the malnormality of H in G implies that { hgh−1 ∣∣ h ∈ H } isin�nite for every g ∈ G \ H . 34

whih takes values in [0, 1], and measures the distane between vAv∗ and A.37When v is a groupoid normaliser of A (i.e. vAv∗ ⊂ A), then δ(v) = 0. Popa'sδ-invariant is de�ned to be the in�mum of these distanes

δ(A) = inf { δ(v) | v∗v, vv∗ are mutually orthogonal projetions in A } .If δ(A) > 0, then A is neessarily singular, and indeed in [44℄ Popa obtainssingular masas by showing that a masa with δ(A) > 10−4 an be found in anyseparable II1 fator. More reently, in [47℄, Popa has gone on to show that anysingular masa in a separable II1 fator has δ(A) = 1, so the δ-invariant only takesthe values 0 or 1. We an view this result as a starting point for the perturbationwork of Popa, Sinlair and Smith, [50℄, whih we disuss in the next subsetion.As noted in [44℄, if A is a masa in N with δ(A) > 0, then Aω is a singular masa inNω, whenever ω ∈ βN \N.38 In partiular, any singular masa A in N , must thengive rise to a singular masa Aω in Nω. On the other hand, when D is the Cartanmasa in R, Dω is not Cartan in Rω. In [43℄, Popa showed that if N is a separableII1 fator then Nω ontains no Cartan masas. Reently, in [18, Corollary 6.2℄, itwas observed that the same method an be used to see that there are no Cartanmasas in Rω ∩R′.We end this setion by noting that in matrix algebras, a.k.a. �nite type Ifators, these onepts are moot. Here, by the elementary proess of simultaneousdiagonalisation, all masas are unitarily onjugate and Cartan.Proposition 1.4.9. Let N be a �nite type I fator. Any two masas A and B inN are unitarily onjugate and

{ u ∈ U(N) | uAu∗ = B }′′ = N.1.4.3 A metri on masas and strong singularityWe begin by introduing a norm for bounded linear maps between II1 fators,whih naturally gives a metri on the set of all von Neumann subalgebras of a II1fator.De�nition 1.4.10 (Sinlair and Smith). Let M and N be II1 fators. Given abounded linear map Φ : M → N , we de�ne ‖Φ‖∞,2 to be the norm of Φ regardedas an operator from M into L2(N). Formally, we have‖Φ‖∞,2 = sup

x∈M,‖x‖≤1

‖Φ(x)‖2 .37More a

urately, the distane between vAv∗ and Avv∗.38That a masa A in N gives rise to a masa Aω in Nω, in this situation is an easy alulationthat an be found in [43℄, where Popa also shows that there are no Cartan masas in Nω.35

This norm gives rise to a metri d∞,2 on the von Neumann subalgebras M of aII1 fator by

d∞,2(M1,M2) = ‖EM1 − EM2‖∞,2 .As noted in [50℄, the metri d∞,2 is equivalent to an older metri on this spaede�ned by Erik Christensen in [4℄. Although we do not de�ne Christensen's metrihere, preferring to work with the d∞,2-metri throughout, we shall ollate variousproperties whih appear in his work.Proposition 1.4.11 (Christensen). The d∞,2-metri makes the set of all vonNeumann subalgebras of a II1 fator N into a omplete metri spae. The maptaking M to its relative ommutant M ′ ∩N is d∞,2-ontinuous, and the followingsets are d∞,2-losed:1. The set of all masas in N ;2. The set of all singular masas in N ;3. The set of all subfators of N ;4. The set of all subfators with trivial relative ommutant in N .In [59℄ and [54℄ the onept of a strongly singular von Neumann subalgebraof a II1 fator was introdued. The idea, based on Popa's δ-invariant (whih weould have ouhed in terms of the d∞,2-metri) is to ontrol the distane of aunitary u to the subalgebra M by the distane between M and uMu∗.De�nition 1.4.12 (Sinlair and Smith). Let M ⊂ N be a von Neumann subal-gebra of the II1 fator N . For α ∈ (0, 1], M is said to be α-strongly singular iffor every unitary u ∈ N we haveα ‖u− EM (u)‖2 ≤ ‖EM − EuMu∗‖∞,2 = d∞,2(M,uMu∗). (1.4.3)We write α(M) for the supremum of all suh α for whih (1.4.3) holds for everyunitary u ∈ N , if suh α exist, otherwise we take α(M) = 0. If α(M) = 1, thenwe say that M is strongly singular.Indeed it is immediate that any α-strongly singular von Neumann subalgebrais singular, and any abelian α-strongly singular von Neumann subalgebra is a sin-gular masa. Every singular masa A for whih the α-invariant has been omputedhas turned out to be strongly singular: see [59℄ for generator masas oming fromprime elements of hyperboli groups; [60℄ for the Laplaian masa in a free groupfator; and, in appendix B, we use these methods to give the analogous alulation36

for ertain radial masas oming from free produts of �nite groups. In the spiritof Proposition 1.4.11, the appropriate losure result holds for strongly-singularmasas.Proposition 1.4.13 ([50, Corollary 6.7℄). For eah α > 0, the set of α-stronglysingular masas in a separable II1 fator N is d∞,2-losed.Motivated at least in part by Popa's work ([47℄) on the distane between masasin II1 fators, showing that the δ-invariant of a singular masa is 1, Popa, Sinlairand Smith have reently examined exatly how two von Neumann subalgebras ofa II1 fator an be lose in the d∞,2-metri, [50℄. This work applies in the generalsetting of all von Neumann subalgebras, but was originally done in the masa asewhih is all we shall give here. The starting point is to note that two masasare lose in d∞,2-metri when there are large utdowns of these masas whih areunitarily onjugate via a unitary u lose to 1. This is an easy estimate, versionsof whih an also be found in [59℄.Proposition 1.4.14 ([50, Theorem 6.5(i)℄). Let A and B be masas in the sepa-rably ating II1 fator N , suh that there are projetions p ∈ A and q ∈ B and aunitary in N with u(Ap)u∗ = Bq. Thend∞,2(A,B) = ‖EA − EB‖∞,2 ≤ 4 ‖u− EB (u)‖2 + ‖1 − p‖2 + ‖1 − q‖2 .In their perturbation work ([50℄) Popa, Sinlair and Smith, have shown thatthis is essentially the only way two masas an be lose in the d∞,2-metri. Morepreisely, they show that if two masas are su�iently lose in the d∞,2-metri,then there are large utdowns of these masas whih are unitarily onjugate.Theorem 1.4.15 ([50, Theorem 6.5(ii)℄). There are onstants 0 < δ < 1 and

K1, K2 suh that, whenever A and B are masas in a separable II1 fator N with‖EA − EB‖∞,2 = ǫ < δ, there exist projetions p ∈ A and q ∈ B and a unitaryu ∈ N suh that:

• u(Ap)u∗ = Bq;• ‖1 − p‖2 = ‖1 − q‖2 ≤ K1ǫ;• ‖u− EB (u)‖2 ≤ K2ǫ.We shall use this deep theorem to establish the d∞,2-ontinuity of the Pukán-szky invariant and of a new Γ-invariant for masas in hapter 3. To this end, westate exatly what we need as a orollary.37

Corollary 1.4.16. There exists onstants 0 < δ < 1 and K suh that, wheneverA and B are masas in a separable II1 fator N with ‖EA − EB‖∞,2 = ǫ < δ, thena masa B1 in N an be found with:

• B1 = uAu∗ for some unitary u in N ;• B1p = Bp for a projetion p ∈ B1 ∩ B with ‖1 − p‖2 ≤ Kǫ.Proof. Take δ and K to be the onstants δ and K1 of Theorem 1.4.15. Let u bethe unitary resulting from Theorem 1.4.15, p be the projetion q appearing there,and take B1 = uAu∗.Popa, Sinlair and Smith's perturbation work also gives a partial onverse tothe observation that every strongly singular masa is singular. They are able toshow that every singular masa is α-strongly singular for some α > 0, and in fatthat α ≥ α0 for some absolute onstant α0 > 0, whih does not even depend onthe underlying separable II1 fator in whih the masas live. The method usedto obtain this result, is to bound from below the d∞,2 distane between A anda unitary perturbation uAu∗ of A, by the distane between u and N (A). Thebest value of α0 is not yet known, although it seems reasonable to hope, basedon Popa's result [47℄ for the δ-invariant, that every singular masa will turn outto be strongly singular - we reord this onjeture formally as Question 1.4.18 forlater referene. The version of this result appearing as Theorem 6.4 of [50℄ gives

α0 ≥ 1/90, although we should observe that this is ertainly not the best valueof this onstant. It is noted in [50℄, that using methods spei� to masas39 leadsto the estimate α0 ≥ 1/31. For the purposes of this thesis, the exat value of α0will not be important.Theorem 1.4.17 (Popa, Sinlair, Smith). There exists a onstant 0 < α0 ≤ 1suh that whenever A is a masa in a separably ating type II1 fator N and u isa unitary in N , we haveα0d2(u,N (A)) ≤ ‖(I − EuAu∗)EA‖∞,2 ≤ d∞,2(A, uAu∗) ≤ 4d2(u,N (A)).In partiular any singular masa A is α-strongly singular for some α ≥ α0.Question 1.4.18. Is every singular masa in a separable II1 fator neessarilystrongly singular?4039Reall that [50℄ treats the more general ase of perturbations of von Neumann subalgebrasof II1 fators throughout.40Sine this thesis was written, this question has been answered. See Addendum 1.1.1 forfurther omments. 38

1.4.4 Establishing strong singularity: asymptoti homomor-phism propertiesTo demonstrate the strong singularity of ertain masas, the onept of an asymp-toti homomorphism was introdued in [59℄ and [54℄. The de�nition we give isthe hypothesis of Lemma 2.1 of [54℄.De�nition 1.4.19. Let M be a von Neumann subalgebra of the II1 fator N .We say that M has the weak asymptoti homomorphism property if and only if,for all ǫ > 0 and x1, . . . , xm ∈ N , we an �nd a unitary v ∈M with∥∥EM

(xivx

∗j

)− EM (xi) vEM

(x∗j)∥∥

2< ǫ, (1.4.4)for every i, j.The original asymptoti homomorphism property,41 de�ned in [59, De�nition4.1℄, required that we ould �nd a unitary v in M suh that

limn→∞

‖EM (xvny∗) − EM (x) vnEM (y∗)‖2 = 0,for every x, y ∈ N . Both these onepts were introdued as they imply strongsingularity, ([59, Theorem 4.7℄ and [54, Lemma 2.1℄) the proof of whih we useto obtain Lemma 1.4.24. Not all strongly singular masas have the asymptotihomomorphism property, [54, Remark 3.3℄. At present though, no singular masais known for whih the weak asymptoti homomorphism property fails, and soa

ordingly it is this property we shall fous on in relation to attempting todetermine the singularity of a masa heneforth.Question 1.4.20. Does every singular masa A in a separable II1 fator have theweak asymptoti homomorphism property?40We an use these ideas to give a riterion for bounding the normalising algebraN (M) from above in the more general non-singular situation.De�nition 1.4.21. Let M ⊂ B be von Neumann subalgebras of the II1 fatorN . We say that M has the weak asymptoti homomorphism property away fromB if and only if, for all ǫ > 0 and x1, . . . , xm ∈ N with EB (xi) = 0 for all i, wean �nd a unitary v ∈ M with

∥∥EM(xivx

∗j

)∥∥2< ǫ, (1.4.5)for every i and j.41In whih, by ontrast with the weak version, something does appear to be asymptotiallya homomorphism! 39

Observe that, with the notation of the de�nition above, if we take x, y ∈ Nand v ∈M , we haveEM (xvy

∗) = EM

((x− EB (x)) v (y∗ − EB (y∗))

)+ EM

(EB (x) v (y

∗ − EB (y∗)))

+ EM

((x− EB (x)) vEB (y∗)

)+ EM

(EB (x) vEB (y

∗))

= EM

((x− EB (x)) v (y∗ − EB (y∗))

)+ EM

(EB (x) vEB (y

∗)),as

EM

(EB (x) v (y

∗ − EB (y∗)))

= EM

((x− EB (x)) vEB (y∗)

)= 0,sine M ⊂ B, so EM = EMEB. This equality links the two notions of weakasymptoti homomorphism property, and will be repeatedly used in the sequel.Proposition 1.4.22. Let M ⊂ B be von Neumann subalgebras of the II1 fator

N . For x, y ∈ N and v ∈M we haveEM (xvy

∗) − EM(EB (x) vEB (y

∗))

= EM

((x− EB (x)) v (y∗ − EB (y∗))

).(1.4.6)Hene, M has the weak asymptoti homomorphism property away from B if andonly if, for all ǫ > 0 and x1, . . . , xm ∈M we an �nd a unitary v ∈M with

∥∥∥EM(xivx

∗j

)− EM

(EB (xi) vEB

(x∗j) )∥∥∥

2< ǫ, (1.4.7)for every i and j. In partiular, M has the weak asymptoti homomorphismproperty away from M preisely whenM has the weak asymptoti homomorphismproperty as given in De�nition 1.4.19.In setion 2, we shall use the weak asymptoti homomorphism riterion forsingularity repeatedly in the ontext of diret limits of ommuting squares, (seeDe�nition 1.3.7). In this situation a density argument makes things slightly easier.We state this here for later use.Lemma 1.4.23. Let M ⊂ B be von Neumann subalgebras of the II1 fator N .Suppose that for eah n we have von Neumann subalgebras Bn ⊂ Nn with eah

Bn ⊂ Bn+1 and Nn ⊂ Nn+1. Suppose further that B and N are the diret limits ofthe Bn and Nn respetively and that Figure 1.2 is made of ommutating squares.If, for eah n ≥ 1, ǫ > 0 and x1, . . . , xm ∈ Nn with EBn (xi) = 0, we an �nda unitary v ∈M with ∥∥EM (xivx∗j)∥∥2 < ǫ,for all i, j, then M has the weak asymptoti homomorphism property away fromB. 40

N1� // N2

� // . . . � // Nn� // Nn+1

� // . . . � // N

B1� //

?�

OO

B2� //

?�

OO

. . . � // Bn� //

?�

OO

Bn+1� //

?�

OO

. . . � // B?�

OO ,

Figure 1.2: Commutative diagram for Lemma 1.4.23.Proof. Sine (∪∞n=1Nn)′′ = N , the elements of ∪∞n=1Nn are 2-norm dense in N . Bythe preeding proposition, it is su�ient for the weak asymptoti homomorphismproperty away from B, to show that for given operators y1, . . . , ym in some Nnand ǫ > 0, we an �nd a unitary v ∈M with∥∥EM

(xivx

∗j

)∥∥2< ǫ,for all i and j, where we have taken xi = yi−EB (yi). Now note that, sine Figure1.2 ommutes, xi = yi−EBn (yi) ∈ Nn, so the ondition we are required to hekredues preisely to the hypothesis of the lemma.We are now in a position to state the tehnial result, generalising Lemma2.1 of [54℄, whih allows us to give an upper bound for the algebra generated bynormalisers. It should be noted that very few modi�ations to the proof in [54℄are required, although we give the details for ompleteness.Lemma 1.4.24. Let M ⊂ B be von Neumann subalgebras of the II1 fator N . If

M has the weak asymptoti homomorphism property away from B, then we have‖u− EB (u)‖2 ≤ ‖(I − EuMu∗)