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    An introduction to II1 factors

    Claire Anantharaman

    Sorin Popa

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    2010 Mathematics Subject Classification. Primary

    Abstract.

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    Contents

    Part 1. 1

    Chapter 1. A first approach: examples 3 1.1. Notation and preliminaries 3 1.2. Measure space von Neumann algebras 5 1.3. Group von Neumann algebras 5 1.4. Group measure space von Neumann algebras 12 1.5. Von Neumann algebras from equivalence relations 18 1.6. Infinite tensor product of matrix algebras 22 Exercises 24

    Chapter 2. Fundamentals on von Neumann algebras 29 2.1. Von Neumann’s bicommutant theorem 29 2.2. Bounded Borel functional calculus 31 2.3. The Kaplansky density theorem 34 2.4. Geometry of projections in a von Neumann algebra 35 2.5. Continuity and order 40 2.6. GNS representations 45 Exercises 48

    Chapter 3. Abelian von Neumann algebras 51 3.1. Maximal abelian von Neumann subalgebras of B(H) 51 3.2. Classification up to isomorphisms 53 3.3. Automorphisms of abelian von Neumann algebras 54 Exercises 56

    Chapter 4. II1 factors. Some basics 59 4.1. Uniqueness of the trace and simplicity 59 4.2. The fundamental group of a II1 factor 61

    Chapter 5. More examples 65 5.1. Tensor products 65 5.2. Crossed products 67 5.3. Free products 70 5.4. Ultraproducts 75 5.5. Beyond factors and abelian von Neumann algebras 77 Exercises 79

    iii

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    iv CONTENTS

    Chapter 6. Finite factors 83 6.1. Definitions and basic observations 83 6.2. Construction of the dimension function 85 6.3. Construction of a tracial state 89 6.4. Dixmier averaging theorem 92 Exercises 95

    Chapter 7. The standard representation 97 7.1. Definition and basic properties 97 7.2. The algebra of affiliated operators 101 7.3. Square integrable operators 106 7.4. Integrable operators. The predual 111 7.5. Unitary implementation of the automorphism group 117 Exercises 119

    Chapter 8. Modules over finite von Neumann algebras 121 8.1. Modules over abelian von Neumann algebras. 121 8.2. Modules over tracial von Neumann algebras 123 8.3. Semi-finite von Neumann algebras 124 8.4. The canonical trace on the commutant of a tracial von

    Neumann algebra representation 127 8.5. First results on finite modules 133 8.6. Modules over II1 factors 134 Exercises 135

    Chapter 9. Conditional expectations. The Jones’ basic construction 139

    9.1. Conditional expectations 139 9.2. Center-valued tracial weights 143 9.3. Back to the study of finite modules 145 9.4. Jones’ basic construction 146 Exercises 151

    Part 2. 157

    Chapter 10. Amenable von Neumann algebras 159 10.1. Amenable groups and their von Neumann algebras 159 10.2. Amenable von Neumann algebras 161 10.3. Connes’ Følner type condition 166 Exercises 169

    Chapter 11. Amenability and hyperfiniteness 173 11.1. Every amenable finite von Neumann algebra is AFD 173 11.2. Uniqueness of separable AFD II1 factors 185 Exercise 188

    Chapter 12. Cartan subalgebras 191

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    CONTENTS v

    12.1. Normalizers and Cartan subalgebras 191 12.2. Isomorphism of Cartan inclusions and orbit equivalence 198 12.3. Cartan subalgebras and full groups 201 12.4. Amenable and AFD Cartan inclusions 205 12.5. Amenable II1 equivalence relations are hyperfinite 208 Exercises 210

    Chapter 13. Bimodules 213 13.1. Bimodules, completely positive maps and representations 213 13.2. Composition (or tensor product) of bimodules 221 13.3. Weak containment 224 13.4. Back to amenable tracial von Neumann algebras 231 Exercises 236

    Chapter 14. Kazhdan property (T) 241 14.1. Kazhdan property (T) for groups 241 14.2. Relative property (T) for von Neumann algebras 242 14.3. Consequences of property (T) for II1 factors 247 14.4. Rigidity results from separability arguments 250 14.5. Some remarks about the definition of relative property (T) 252 Exercises 254

    Chapter 15. Spectral gap and Property Gamma 257 15.1. Actions with spectral gap 257 15.2. Spectral gap and Property Gamma 261 15.3. Spectral gap and full II1 factors 266 15.4. Property Gamma and inner amenability 267 Exercises 269

    Chapter 16. Haagerup property (H) 273 16.1. Haagerup property for groups 273 16.2. Haagerup property for von Neumann algebras 274 16.3. Relative property (H) 275 Exercise 278

    Chapter 17. Intertwining-by-bimodules technique 281 17.1. The intertwining theorem 281 17.2. Unitary conjugacy of Cartan subalgebras 285 17.3. II1 factors with two non-conjugate Cartan subalgebras 287 17.4. Cartan subalgebras of the hyperfinite factor R 288 Exercises 289

    Chapter 18. A II1 factor with trivial fundamental group 293 18.1. A deformation/rigidity result 293 18.2. Fundamental group and cost of an equivalence relation 295 18.3. A II1 factor with trivial fundamental group 297 Exercise 299

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    vi CONTENTS

    Chapter 19. Free group factors are prime 303 19.1. Preliminaries 303 19.2. Proof of the solidity of Fn 307 Exercises 308

    Appendix. 311 A. C∗-algebras 311 B. Standard Borel and measure spaces 315

    Bibliography 317

    Index 327

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    Part 1

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    CHAPTER 1

    A first approach: examples

    This chapter presents some basic constructions of von Neumann algebras arising from measure theory, group theory, group actions and equivalence relations. All these examples are naturally equipped with a faithful trace and are naturally represented on a Hilbert space. This provides a plentiful source of tracial von Neumann algebras to play with. More constructions will be given in Chapter 5.

    The most general von Neumann algebras are obtained from simpler building blocks, called factors. These are the von Neumann algebras with a trivial center. We will see that they appear frequently, under usual as- sumptions. Infinite dimensional tracial factors (II1 factors) are our main concern. We end this chapter with the most elementary example, the hy- perfinite II1 factor, which is constructed as an appropriate closure of an increasing sequence of matrix algebras.

    1.1. Notation and preliminaries

    Let H be a complex Hilbert space with inner-product 〈·, ·〉 (always as- sumed to be antilinear in the first variable), and let B(H) be the algebra of all bounded linear operators from H to H. Equipped with the involu- tion x 7→ x∗ (adjoint of x) and with the operator norm, B(H) is a Banach ∗-algebra with unit IdH. We will denote by ‖x‖, or sometimes ‖x‖∞, the operator norm of x ∈ B(H). Throughout this text, we will consider the two following weaker topologies on B(H):

    • the strong operator topology (s.o. topology), that is, the locally con- vex topology on B(H) generated by the seminorms

    pξ(x) = ‖xξ‖, ξ ∈ H, • the weak operator topology (w.o. topology), that is, the locally con-

    vex topology on B(H) generated by the seminorms pξ,η(x) = |ωξ,η(x)|, ξ, η ∈ H,

    where ωξ,η is the linear functional x 7→ 〈ξ, xη〉 on B(H). This latter topology is weaker than the s.o. topology. It is strictly weaker

    when H is infinite dimensional (see Exercise 1.1). An important observa- tion is that the unit ball of B(H) is w.o. compact. This is an immediate consequence of Tychonoff’s theorem.

    3

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    4 1. A FIRST APPROACH: EXAMPLES

    This unit ball, endowed with the uniform structure associated with the s.o. topology, is a complete space. In case H is separable, both w.o. and s.o. topologies on the unit ball are metrizable and second-countable. On the other hand, when H is infinite dimensional, this unit ball is not separable with respect to the operator norm (Exercise 1.2).

    A von Neumann algebra M on a Hilbert space H is a ∗-subalgebra of B(H) (i.e., a subalgebra invariant under the ∗-operation) which is closed in the s.o. topology and contains the identity operator IdH.

    1 We will sometimes write (M,H) to specify the Hilbert space on which M acts, but H will often be implicit in the definition of M . The unit IdH of M will also be denoted 1M or simply 1. We use the lettersH,K,L to denote complex Hilbert spaces, while the letters M,N,P,Q will typically denote von Neumann algebras.

    Given a subset S of B(H), we denote by S′ its commutant in B(H): S′ = {x ∈ B(H) : xy = yx for all y ∈ S}.

    The commutant (S′)′ of S′ is denoted S′′ and called the bicommutant of S. Note that S′ is a s.o. closed unital subalgebra of B(H); if S = S∗, then S′ = (S′)∗ and therefore S′ is a von Neumann algebra on H. We will see in the next chapter that every von Neumann algebra appears in this way (Theorem 2.1.3).

    The first example of von Neumann algebra coming to mind is of course M = B(H). Then, M ′ = C IdH. WhenH = Cn, we get the algebra Mn(C) of n×n matrices with complex entries, the simplest example of a von Neumann algebra.

    We recall that a C∗-algebra on H is a ∗-subalgebra of B(H) which is closed in the norm topology. Hence a von Neumann algebra is a C∗-algebra, but the converse is not true. For instance the C∗-algebra K(H) of compact operators on an infinite dimensional Hilbert space H is not a von Neumann algebra on H: its s.o. closure is B(H).

    We assume that the reader has a basic knowledge about C∗-algebras. We have gathered in the appendix, with references, the main facts that we will use. Note that for us, a homomorphism between two C∗-algebras preserves the algebraic operations and the involution2. We recall that it is automatically a contraction and a positive map, i.e., it preserves the positive con