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Title On Maximal Amenable Subalgebras of Amalgamated Free Product von Neumann Algebras

Permalink https://escholarship.org/uc/item/0xk9x52t

Author Leary, Brian Andrew

Publication Date 2015

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California University of California Los Angeles

On Maximal Amenable Subalgebras of Amalgamated Free Product von Neumann Algebras

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics

Brian Andrew Leary

2015 c Copyright by Brian Andrew Leary 2015 Abstract of the Dissertation

On Maximal Amenable Subalgebras of Amalgamated Free Product von Neumann Algebras

Brian Andrew Leary Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Sorin Popa, Chair

In this thesis, we establish a sufficient condition for an amenable von Neumann algebra to be a maximal amenable subalgebra of an amalgamated free product von Neumann algebra. In particular, if P is a diffuse maximal amenable von Neumann subalgebra of a finite von

Neumann algebra N1, and B is a von Neumann subalgebra of N1 with the property that no

corner of P embeds into B inside N1 in the sense of Popa’s intertwining by bimodules, then we conclude that P is a maximal amenable subalgebra of the amalgamated free product of N1

and N2 over B, where N2 is another ﬁnite von Neumann algebra containing B. To this end, we utilize Popa’s asymptotic orthogonality property. We also observe several special cases in which this intertwining condition holds, and we note a connection to the Pimsner-Popa

index in the case when we take P = N1 to be amenable.

ii The dissertation of Brian Andrew Leary is approved.

Michael Gutperle

Dimitri Y. Shlyakhtenko

Edward G. Eﬀros

Sorin Popa, Committee Chair

University of California, Los Angeles

2015

iii To my sister

iv Table of Contents

1 Introduction ...... 1

2 Preliminaries ...... 5

2.1 Von Neumann Algebra Basics ...... 5

2.1.1 Deﬁnitions ...... 5

2.1.2 Constructions ...... 9

2.2 Amenability and Property Gamma ...... 13

2.3 Subalgebra Structure and Classiﬁcation ...... 17

2.4 Asymptotic Orthogonality ...... 19

2.5 Jones’ Basic Construction ...... 20

2.6 Popa’s Intertwining by Bimodules ...... 20

3 Main Result ...... 24

3.1 Amalgamated Free Products ...... 24

3.2 Maximal Amenability ...... 25

4 Special Cases ...... 33

4.1 Pimsner-Popa index ...... 33

4.2 Group von Neumann algebra case ...... 34

4.3 Finite Factor case ...... 35

4.4 Crossed Product von Neumann Algebra case ...... 35

4.5 Abelian case ...... 35

References ...... 38

v Acknowledgments

There are many people I would like to thank for their help and support over the years. To begin, I am deeply thankful for the guidance of my advisor, Sorin Popa, in the completion of this project. In addition to introducing me to this problem, his advice and support over the years have been invaluable to me, and I am extremely grateful for all he has done for me and for my career. I would also like to thank Thomas Sinclair for his mentorship and aid throughout his years at UCLA and beyond. I would like to further thank Jesse Peterson for suggesting the generalization of the main result presented in this thesis, Adrian Ioana for giving me an opportunity to speak about my work, Edward Effros and Dimitri Shlyakhtenko for teaching my first courses in this subject, and all the operator algebras grad students at UCLA during my time here, including Owen, Adam, Dom, Paul, Brent, Andreas, Alin, and Ian. Their contributions to student-run seminars were vital to my education in this field. Among these, I want to single out Ben Hayes for his help, as he has been a selfless, endless fount of knowledge, and, most importantly, a great friend to me.

Non-operator-theoretic inﬂuences include the limitless support of my family. I thank my mother for her love and care, my father for sharing his advice and experiences on surviving grad school and academia, my sister for always talking to me about important things, and my brother for always talking to me about absolutely everything else. I would also like to oﬀer my thanks to the numerous people who made my life in Los Angeles better. Thank you to Tori, who was the only person I knew when I moved to LA, and who was responsible for more fun experiences in the city than I can count; to Rob and Bailey for everything they’ve done, and for being my closest friends for many years; to Lee for his friendship, competitive foosball skills, and for giving my sister a reason to stay in LA; to Julie for being among my favorite people that I’ve met here; and to Miranda, Johann, Siddharth, Bryon, and Stephanie for their contributions to making my time in this city memorable. I am truly indebted to all of you.

vi Vita

2008–2009 Teaching Assistant, Mathematics Department, Carnegie Mellon University, Pittsburgh, Pennsylvania. Taught sections of Math 127 (an introduction to rigorous mathematics).

2009 B.S. (Mathematics), Carnegie Mellon University.

2009 M.S. (Mathematics), Carnegie Mellon University.

2009–2013 Teaching Assistant, Mathematics Department, University of California - Los Angeles, California.

2011 M.A. (Mathematics), UCLA.

2013–2015 Research Assistant, Mathematics Department, UCLA.

2015 Graduate Student Instructor, Mathematics Department, UCLA. Lecturer for Math 3C (probability for life sciences).

Publications and Presentations

On maximal amenable subalgebras in amalgamated free products, in preparation.

Workshop on von Neumann algebras and ergodic theory, UCLA, September 2014, On maximal amenable subalgebras of amalgamated free product von Neumann algebras.

vii CHAPTER 1

Introduction

The goal of this work is to establish maximal amenability results in certain amalgamated free product von Neumann algebras by using Popa’s asymptotic orthogonality method.

In the origins of the subject in the 1930s and 1940s, Murray and von Neumann gave the basic definitions of factoriality, type decomposition, and other isomorphism invariant properties in what are now known as von Neumann algebras, and they constructed the first examples. One such construction was the approximately finite dimensional II1 factor R, which can be realized as the tensor product of countably infinitely many copies of the space of 2 × 2 matrices over the complex numbers. They were also able to prove that, up to isomorphism, R was the unique approximately finite dimensional factor of type II1. Moreover, they showed that every infinite dimensional factor contains a copy of R. Later, Connes [Con76] was able to show that the property of a factor M ⊂ B(H) being approximately finite dimensional is equivalent to amenability, which is the existence of an M-central state on B(H) that extends the trace on M, and is also equivalent to injectivity, which is the existence of a conditional expectation from B(H) onto M.

In Kadison’s 1967 list of problems on von Neumann algebras [Kad67], he asked whether every self-adjoint operator in an arbitrary II1 factor can be embedded into some approximately ﬁnite dimensional subfactor, or equivalently, whether every separable abelian von

Neumann subalgebra of a II1 factor could be embedded into some approximately ﬁnite dimensional subfactor. In 1983, Popa [Pop83a] provided a negative answer to the problem by constructing an abelian subalgebra of a II1 factor that is a maximal amenable subalgebra, and hence a maximal approximately ﬁnite dimensional subalgebra by Connes’ theorem. In particular, he proved that the abelian von Neumann subalgebra Ma of L(Fn) generated by a

1 single generator a of Fn is a maximal amenable subalgebra. His method involved the analysis of Ma-central sequences in L(Fn) through the “asymptotic orthogonality property.” To be 0 ω ω precise, he showed that for any free ultraﬁlter ω on N, any x ∈ Ma ∩ (L(Fn) Ma ), and 2 ω any y1, y2 ∈ L(Fn) Ma, we have that y1x and xy2 are perpendicular in L (L(Fn) ). Then 0 using the strong mixingness of Ma inside L(Fn), he showed that L(Fn) ∩ Ma had a non-zero atomic part, and from this he was able to conclude maximal amenability.

In the last decade, there have been many more results about maximal amenable subalgebras. In 2010, Cameron, Fang, Ravichandran, and White [CFR10] showed that the Laplacian masa in L(Fn) is maximal amenable by modifying R˘adulescu’s proof of singularity of the Laplacian masa to show that it also had the asymptotic orthogonality property. In 2010, Jolissaint [Jol10] gave conditions on a subgroup H of Γ that imply that L(H) is a maximal amenable subalgebra of L(Γ) by ﬁrst establishing the asymptotic orthogonality property.

A special case of this showed that L(H1) is maximal amenable in the group von Neumann algebra of the amalgamated free product group L(H1 ∗Z H2), where H1 is infinite and abelian and Z is a finite common subgroup. In 2013, Boutonnet and Carderi [BC13] expanded Jolis- saint’s work to show that for any infinite maximal amenable subgroup H of a hyperbolic group Γ, L(H) is maximal amenable inside L(Γ). Houdayer [Hou14b] proved in 2014, among other things, that any diffuse amenable von Neumann algebra can be realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III1 factor by defining a relative version of the asymptotic orthogonality property that we will use in this thesis. Finally, Boutonnet and Carderi [BC14] in 2014 gave an alternative method of establishing maximal amenability in the specific cases of von Neumann algebras arising from groups, and their method was the first result that did not use Popa’s asymptotic orthogonality. Other maximal amenable von Neumann algebra results have been given by Brothier [Bro14], Hou- dayer [Hou14a], Gao [Gao10], Hou [Hou08], Fang [Fan07], Shen [She06], Str˘atil˘aand Zsidó [SZ99], and Ge [Ge96].

In this paper, we will utilize Popa’s method to give a maximal amenability result for amalgamated free product von Neumann algebras. We let (N1, τ1) and (N2, τ2) be ﬁnite von

Neumann algebras with a common von Neumann subalgebra (B, τB), and we construct the 2 amalgamated free product von Neumann algebra, M = N1 ∗B N2. We let P be a diﬀuse, maximal amenable subalgebra of N1 such that no corner of P embeds into a corner of B inside

N1, in the sense of Popa’s intertwining by bimodules. Then we establish that P satisﬁes the relative version of the asymptotic orthogonality property in M, and that P satisﬁes a relative weak mixing condition in M. From this, we conclude maximal amenability, and this is the main theorem of this work.

Theorem 1.1. Suppose that (N1, τ1) and (N2, τ2) are ﬁnite von Neumann algebras with a common von Neumann subalgebra (B, τB). Suppose further that P ⊂ N1 is a diﬀuse, maximal amenable subalgebra with the property that P 6≺N1 B. Then P is maximal amenable in M = N1 ∗B N2.

Clearly, a special case of this is when N1 itself is diﬀuse and amenable, and such that

N1 6≺N1 B. We will show that this particular intertwining by bimodules condition can be translated into a condition on the Pimsner-Popa index of B inside N1, which gives the following corollary.

Corollary 1.2. Suppose that (N1, τ1) is a ﬁnite, diﬀuse, amenable von Neumann algebra and

(N2, τ2) is a ﬁnite von Neumann algebra with a common von Neumann subalgebra (B, τB).

0 Suppose further that for every non-zero projection p ∈ B ∩N1, we have that [pN1p : Bp] = ∞.

Then N1 is maximal amenable in M = N1 ∗B N2.

The structure of this thesis is as follows: in the second chapter, we will present the preliminary material needed for the proofs of the main results, beginning with the basics of von Neumann algebra theory and constructions of von Neumann algebras. Next, we give definitions and notes on amenability and property Gamma. Then we discuss the definitions of asymptotic orthogonality and relative asymptotic orthogonality, and finally, we give the basic material on intertwining by bimodules. The third chapter begins with the construction of the amalgamated free product von Neumann algebra. We then prove in Theorem 3.2.3 that under our hypotheses, we have the relative asymptotic orthogonality property. Then combined with the weak mixingness established in Lemma 3.2.2, we are able to derive maximal amenability

3 and prove Theorem 1.1. In the fourth chapter, we prove Corollary 1.2, and give examples of how these hypotheses can be applied in diﬀerent von Neumann algebra constructions.

4 CHAPTER 2

Preliminaries

2.1 Von Neumann Algebra Basics

2.1.1 Deﬁnitions

We begin by deﬁning the basic terms and facts that we will utilize through this thesis. First, let H be a Hilbert space, and consider B(H), the space of all bounded linear maps from H to H. We deﬁne the following topologies on B(H).

Deﬁnition 2.1.1. Suppose (Ti)i is a net of operators in B(H), and T ∈ B(H).

• We say that Ti → T in the norm topology if kTi − T k → 0.

• We say that Ti → T in the strong operator topology if kTiξ − T ξk → 0 for all ξ ∈ H.

• We say that Ti → T in the weak operator topology if hTiξ, ηi → hT ξ, ηi for all ξ, η ∈ H.

Then we deﬁne a von Neumann algebra to be a *-subalgebra M ⊂ B(H) (i.e. M is closed under taking adjoints) such that M contains the unit of B(H) and M is closed in the weak operator topology. Note that by the Hahn-Banach Separation Theorem, this is equivalent to M being closed in the strong operator topology. Although this is a topological condition, we will show next that it is equivalent to a purely algebraic condition. To this end, we make the following deﬁnition.

Deﬁnition 2.1.2. For any subset N ⊂ B(H), we deﬁne the commutant of N inside B(H), denoted N 0, to be the set of all operators in B(H) that commute with all elements of N. That is, N 0 = {T ∈ B(H): TS = ST for all S ∈ N}.

5 Now we can state the Bicommutant Theorem of von Neumann.

WOT Theorem 2.1.3. (Bicommutant) Let A ⊂ B(H) be a *-subalgebra. Then A = A00.

WOT Proof. Let M = A . Then M is a von Neumann algebra. By continuity of multiplication, we have that A0 = M 0. We want to show that M 00 = M. First, note that M ⊂ M 00 is clear by the deﬁnition of the commutant. Suppose that T ∈ M 00. As noted above, since M is a subspace of a vector space, it is convex, so by the Hahn-Banach separation theorem, it suﬃces to show that T is in the closure of M in the strong operator

topology. Then we want to show that for all ε > 0 and for all ξ1, ξ2, . . . , ξn ∈ H there exists

an operator S ∈ M such that kT ξj − Sξjk < ε for all j. We ﬁrst take n = 1. Let ε > 0 and deﬁne P to be the projection onto the subspace Mξ. Then P ∈ M 0 since M leaves Mξ invariant, so we have that T commutes with P . That is, P (T ξ) = T (P ξ). But since ξ ∈ Mξ, we have that T (P ξ) = T (ξ), so T (ξ) = P (T ξ), so T ξ ∈ Mξ. Then we may choose S ∈ M such that kT ξ − Sξk < ε. Similarly, to deal with

⊕n the general n case, we can consider (ξ1, ξ2, . . . , ξn) as a vector ξ ∈ H , and we deﬁne P to

⊕n be the projection from H onto the closure of the subspace {(xξ1, xξ2, . . . , xξn): x ∈ M}.

⊕n ⊕n Then P commutes with all diagonal elements in Mn×n(M), so the element T ∈ B(H ) consisting of copies of T on the diagonal commutes with P . Thus, as before, we may choose

⊕n ⊕n S ∈ M such that kT ξ − S ξk < ε, and hence kT ξj − Sξjk < ε for all j.

A strengthening of the Bicommutant Theorem is the Kaplansky Density Theorem, which states that the unit ball of a *-subalgebra A of B(H) is SOT-dense in the unit ball of the SOT-closure of A. Thus, an element x in the von Neumann algebra A00 is SOT-close to an element in A with norm bounded by the norm of x. Here we use the notation A1 to denote the unit ball of A.

Theorem 2.1.4. (Kaplansky Density) Suppose A ⊂ B(H) is a *-subalgebra that is closed in the norm topology. Let B = A00. Then

SOT (a) Bsa = Asa , i.e. the self-adjoint elements in A are SOT-dense in the self-adjoint elements of B. 6 SOT (b) (Bsa)1 = (Asa)1 , i.e. the unit ball of the self-adjoint elements in A are SOT-dense in the unit ball of the self-adjoint elements of B.

SOT (c) (B)1 = A1 .

Proof. We prove ﬁrst that for any bounded and continuous function f : R → C, we have that T 7→ f(T ) is SOT-continuous on self-adjoint operators T ∈ B(H) for any Hilbert space H. To see this, consider the set

E = {f : R → C : T 7→ f(T ) SOT-continuous on self-adjoints in any B(H)}.

Then E is an algebra, since (Ti) and (Si) uniformly bounded with Ti → T and Si → S in the strong operator topology implies that TiSi → TS in the strong operator topology.

Recall that C0(R) is the space of complex-valued functions on R that vanish at inﬁnity. We will show that C0(R) ⊂ E by showing that E0 = E ∩ C0(R) separates points of R, so that by the Stone-Weierstrass theorem, E0 = C0(R). To this end, we ﬁrst note that E0 is a *-subalgebra, as it is closed under complex conjugation since T is a self-adjoint operator.

2 2 Next, we deﬁne the functions h(x) = 1/(1 + x ) and g(x) = x/(1 + x ). Then h, g ∈ C0(R) with khk∞ , kgk∞ ≤ 1. Then h and g separate points of R, since if x 6= y, then either x = −y or x2 6= y2, so g separates x and y in the ﬁrst case and h separates x and y in the second case. Thus, we want to show that h and g are in E0. We have that

g(S) − g(T ) = (1 + S2)−1[S(1 + T 2) − (1 + S2)T ](1 + T 2)−1

= (1 + S2)−1[S − T + S(T − S)T ](1 + T 2)−1, so

2 −1 2 −1 kg(S)x − g(T )xk ≤ (1 + S ) (S − T )(1 + T ) x

2 −1 2 −1 + (1 + S ) S (T − S)T (1 + T ) x

2 −1 2 −1 ≤ (S − T )(1 + T ) x + (T − S)T (1 + T ) x ,

so S → T strongly implies that g(S) → g(T ) strongly, so g ∈ E0. Then we also have that h = 1 − xg is in E0. Thus, E0 = C0(R). Then for any bounded and continuous f, we have 7 that fh and fg are in C0(R) ⊂ E, so f = fh + xfg ∈ E.

Now the proof of the theorem will follow easily. For (a), we let T ∈ Bsa. Choose a net

∗ ∗ ∗ Ti → T strongly, where Ti ∈ A. Then Ti → T weakly and (Ti + Ti )/2 → T weakly, since T ∗ WOT is self-adjoint, so since (Ti + Ti ) ∈ Asa, we have that Bsa ⊂ Asa , but since Asa is convex, WOT SOT we have that Asa = Asa , so we have (a).

For (b), let T ∈ (Bsa)1, and choose a net Ti → T strong, where Ti ∈ Asa. Deﬁne a function

−1 f ∈ C0(R) by f(t) = t on [−1, 1] and f(t) = t elsewhere. Then we have shown that f(Ti) → f(T ) strongly. But since kT k < 1, we have that f(T ) = T , so f(Ti) → T . But we

also have that kf(Ti)k ≤ 1 for all i by the deﬁnition of f, so f(Ti) ∈ (Asa)1 for all i, which gives (b).

Finally for (c), let T ∈ B1. We consider the self-adjoint element in (M2×2(B))1 given by   0 T T˜ =   . T ∗ 0

˜ ˜ Then by (b), there exists a net Ti ∈ (M2×2(A)sa)1 converging to T in the strong operator ˜ topology of B(H ⊕ H). Then for each i, we can write Ti in the form   A T ˜ i i Ti =   . ∗ Ti Bi

Then Ti → T strongly, and kTik ≤ 1, which gives (c).

Next, we give some deﬁnitions of certain classes of von Neumann algebras.

Deﬁnition 2.1.5. A von Neumann algebra M ⊂ B(H) is said to be ﬁnite if there is a map

τ : M → C satisfying:

• τ(x∗x) ≥ 0 for every x ∈ M and τ(1) = 1 (that is, τ is a state)

• τ(xy) = τ(yx) for every x, y ∈ M (that is, τ is a trace)

• τ is weakly continuous on the unit ball of M (that is, τ is normal)

• If x ∈ M with τ(x∗x) = 0, then x = 0 (that is, τ is faithful) 8 A von Neumann algebra M is said to be a factor if it has trivial center, i.e. M ∩ M 0 = C.

An inﬁnite dimensional ﬁnite factor is called a II1 factor. This will be an important class of von Neumann algebras throughout this thesis.

2.1.2 Constructions

The two simplest examples of von Neumann algebras are abelian von Neumann algebras and matrix algebras. Indeed, the matrix algebra M = Mn×n(C) acts on the finite dimensional Hilbert space H = Cn, and we have that M 00 = M. In fact, we have that the only matrices that commute with all other matrices are the constant multiples of the identity, so we have that M ∩ M 0 = C, so M is an example of a finite dimensional factor. In fact, every n- dimensional factor is isomorphic to Mn×n(C). We also have that Mn×n(C) is a finite von Neumann algebra, with the normal faithful tracial state given by the normalized matrix trace.

On the other end of the commutivity spectrum are the abelian von Neumann algebras. Let (X, µ) be a σ-ﬁnite measure space, and consider the Hilbert space L2(X, µ) consisting of all square-µ-integrable functions from X to C. Then A = L∞(X, µ) is a von Neumann algebra acting on L2(X, µ) by translation. In fact, for every abelian von Neumann algebra A, there exists such a compact measure space (X, µ) with µ a probability measure on X such that A ∼= L∞(X, µ). In this case, we also have that L∞(X, µ) is a ﬁnite von Neumann algebra, with the normal faithful tracial state given by τ(f) = R fdµ.

Another example of a ﬁnite von Neumann algebra is the von Neumann algebra generated by a discrete countable group Γ. We deﬁne a unitary representation of Γ on U(l2(Γ)) by the map (λ(g)(f))(h) = f(g−1h) for all g, h ∈ Γ and f ∈ l2(Γ).

Then λ is called the left regular representation of Γ. We deﬁne the group von Neumann algebra of Γ, denoted L(Γ), to be the von Neumann subalgebra of B(l2(Γ)) generated by

00 λ(Γ). That is, L(Γ) = {λ(g): g ∈ Γ} . We will deﬁne ug = λ(g) to be the unitaries generating L(Γ), and then we have that L(Γ) has a dense set consisting of sums of the form

9 P g∈Γ agug, where ag ∈ C for each g ∈ Γ. The normal faithful tracial state on L(Γ) is given by τ(x) = hxδe, δei, where e is the identity element of the group. Another way to describe

2 L(Γ) is through convolution operators, Lξ : η 7→ ξ ∗ η, where ξ ∈ l (Γ). Then we have that

2 2 2 2 L(Γ) = {Lξ : ξ ∈ l (Γ) and ξ ∗ l (Γ) ⊂ l (Γ)} ⊂ B(l (Γ)).

While L(Γ) is a finite von Neumann algebra for any countable discrete group Γ, there are certain groups for which L(Γ) is actually a II1 factor. We say that Γ has infinite conjugacy classes, or is ICC, if for every g 6= e in Γ, the set {hgh−1 : h ∈ Γ} is infinite. An example of

such a group is the free group Fn.

Proposition 2.1.6. Let Γ be a discrete countable group. Then L(Γ) is a II1 factor if and only if Γ is ICC.

0 Proof. Suppose ﬁrst that Γ is ICC, and let x ∈ (L(Γ)) ∩ L(Γ). Then uhx = xuh for every P h ∈ Γ, so if x = agug, then we have that ahgh−1 = ag for all g, h ∈ Γ, so g 7→ ag is a constant function on conjugacy classes. But since x ∈ l2(Γ) and the non-identity conjugacy

classes are inﬁnite, this means that ag = 0 for all g 6= e. Hence, x = aeue ∈ C · 1. Next, if Γ is not ICC, then choose a non-identity element g with a ﬁnite conjugacy class,

Pn ∗ {g1, . . . gn}. Then we can take x = i=1 ugi . Then for any h ∈ Γ, we have that uhxuh = n P −1 i=1 uhgih . But the conjugation of h simply permutes the elements in the conjugacy class n n ∗ P −1 P of gi, so that i=1 uhgih = j=1 ugj , so uhxuh = x, and hence x is a non-trivial element in the center of (L(Γ))0 ∩ L(Γ).

We can also construct a von Neumann algebra through an action of a group on another von Neumann algebra. Let Γ be a discrete countable group, and suppose that N ⊂ B(H) is a von Neumann algebra. Let σ :Γ → Aut(N) be an action of Γ on N. We deﬁne the crossed product von Neumann algebra M = N o Γ acting on the Hilbert space l2(Γ, H) of square-summable functions from Γ to H as being generated by the operators

(λ(g)(f))(h) = f(g−1h) for all g, h ∈ Γ and f ∈ l2(Γ, H),

and

2 (x(f))(h) = σh−1 (x)(f(h)) for all h ∈ Γ, f ∈ l (Γ, H), and x ∈ N. 10 That is, M = {xλ(g): x ∈ N and g ∈ Γ}00, and as in the group von Neumann algebra, we let

∗ ug = λ(g), and we note that ugxug = σg−1 (x). Also as in the group von Neumann algebra, we P have that the crossed product has a dense set consisting of sums of the form g xgug, where 2 here xg ∈ N for each g ∈ Γ. Since B(l (Γ, H)) can be realized in multiple diﬀerent ways, there are alternative descriptions of the crossed product. For example, note that we may consider B(l2(Γ, H)) as a space of matrices of operators in B(H) indexed by the elements

P 2 of Γ, (Tg,h)g,h∈Γ, via the relation (T f)(g) = h∈Γ((Tg,h)(f))(h), for T ∈ B(l (Γ, H)) and f ∈ l2(Γ, H). Then by considering N and {λ(g)} in this space, the above relations translate as

λ(g)h1,h2 = δh1,gh2 for all g, h1, h2 ∈ Γ,

and

xh ,h = δh ,h σ −1 (x) for all h1, h2 ∈ Γ and x ∈ N. 1 2 1 2 h2

2 And then M can be considered as the subset {T ∈ B(l (Γ, H)) : Tg,h = σh−1 (Tgh−1,e)}.

Then we have that M is a ﬁnite von Neumann algebra if and only if N is ﬁnite and the

trace τ on N satisﬁes τ(σg(x)) = τ(x) for all x ∈ X. In this case, the trace on M is given

byτ ˜(x) = τ(xδe).

A special case of this is when the von Neumann algebra N is abelian, and therefore is of the form L∞(X, µ) for some probability space (X, µ). We note that if Γ acts on (X, µ) by σ, a probability measure preserving action, then σ extends to an actionσ ˜ on L∞(X, µ)

∞ byσ ˜g(f)(x) = f(σg−1 (x)), and thus the crossed product M = L (X, µ) o Γ is a ﬁnite von Neumann algebra, which is known as the group measure space construction. This von Neumann algebra can be classiﬁed depending on properties of the action σ. We recall that the action σ :Γ y (X, µ) is said to be free if σg(x) 6= x for every g 6= e and a.e. x ∈ X. The action is said to be ergodic if for every Y ⊂ X, we have that σ(Y ) = Y implies that µ(Y ) = 0 or µ(Y ) = 1. The connection between ergodicity and factoriality can be seen by the following lemma.

Lemma 2.1.7. If Γ y (X, µ), then the action is ergodic if and only if for every f ∈ L∞(X, µ) with f(g · x) = f(x) for all g ∈ Γ and a.e. x ∈ X, then f is constant a.e. 11 Proof. First, if the action is not ergodic, choose Y ⊂ X such that g(Y ) = Y and such that 0 < µ(Y ) < 1. Let f be the characteristic function of Y . Then f is non-constant. Conversely, assume the action is ergodic. Let f ∈ L∞(X, µ) with f(g · x) = f(x) for a.e. x. Assume for sake of contradiction that f is not constant almost everywhere. Then either the real part of f or the imaginary part of f is not constant, so without loss of generality, suppose f is real-valued and not constant almost everywhere. Let m be the essential inﬁmum

of f and let M be the essential supremum of f. Choose y ∈ R such that m < y < M. Let Y = {x ∈ X : f(x) ≤ y}. Then 0 < µ(Y ) < 1, so by ergodicity, µ(g(Y )\Y ) > 0 for some g ∈ Γ, so for all x ∈ (g(Y )\Y ), we have that f(g−1·x) ≤ y < f(x), giving a contradiction.

∞ Proposition 2.1.8. If σ is a free action, then M = L (X, µ) o Γ is a II1 factor if and only if the action σ is ergodic.

Proof. Write A = L∞(X, µ). We ﬁrst prove that since σ is a free action, we have that A0 ∩ M = A (i.e. A is a maximal abelian subalgebra of M). To this end, let b ∈ A0 ∩ M. P P P Then for every a ∈ A, we have that ab = ba, so abgug = bguga = bgσg(a)ug, so

abg = bgσg(a) for every g ∈ Γ. Then since the action is free, we have that bg = 0 for all

0 0 0 g 6= e, so b = be ∈ A. Thus, we have that M ∩ M ⊂ A ∩ M = A. Thus, letting y ∈ M ∩ M,

∗ we have that y ∈ A. As y commutes with ug for every g ∈ Γ, we have that ugyug = y. But ∗ ugyug = σg−1 (y), so y = σg−1 (y) for every g ∈ Γ. But then we have that y(x) = y(σg(x)) for a.e. x ∈ X. Hence, by the lemma, y ∈ C · 1 if and only if the action is ergodic.

Now we consider ﬁnite von Neumann algebras N1 and N2, and we suppose N1 ⊂ B(H1)

and N2 ⊂ B(H2). Then we consider the tensor product Hilbert space H1 ⊗H2, and we deﬁne

the tensor product von Neumann algebra N1⊗N2 ⊂ B(H1 ⊗ H2) to be the von Neumann algebra generated by N1 ⊗ 1 and 1 ⊗ N2. Then we have that N1⊗N2 is a ﬁnite von Neumann

0 algebra, and we have that if N1 is a factor, then (N1⊗1) ∩(N1⊗N2) = (1⊗N2). Consequently,

N1⊗N2 is a factor if and only if N1 and N2 are both factors.

Finally, the last construction we will consider is the ultraproduct von Neumann algebra,

which plays a significant role in proofs of maximal amenability, as we will show. We let βN be the Stone-Cechˇ compactification of N. Then βN can be considered as the set of all ultrafilters 12 on N, and we have that N is a dense subset of βN. Then an element ω ∈ βN\N is a free ultrafilter on N. Note that C(βN), the continuous functions on βN, is isomorphic to l∞(N). ∞ If (an)n∈N ∈ l (N), then we write ω((an)) = limn→ω(an). Thus, if an → a as n → ∞, we have that limn→ω(an) = a. Now let (Mn, τn)n∈N be finite von Neumann algebras. We have that Q Mn = {(an): an ∈ Mn for all n and sup kank < ∞}. Note that if Mn = M for all n, n∈N n Q ∞ Q then M = l ( ,M). We define a trace τω : Mn → by τω((an)) = limn→ω τn(an). n∈N N C Q Consider the set I = {(an) : limn→ω kank2 = 0}. Then I is a two-sided ideal of Mn, since k(a )(b )k ≤ ka k kb k . Then we define the ultraproduct Qω M to be the quotient n n τω n ∞ n τω n Q ω Qω ( n∈ω Mn)/I. If Mn = M for all n, we denote this by M . Then we have that Mn is a

ﬁnite von Neumann algebra with the trace τω. Furthermore, if Mn is a factor for all n, we Qω have that Mn is a factor.

2.2 Amenability and Property Gamma

We ﬁrst recall the deﬁnition of an amenable group.

Deﬁnition 2.2.1. A discrete countable group G is said to be amenable if there exists a state

ψ : l∞(G) → C that is invariant under left translation: ψ(g · f) = ψ(f) for all g ∈ G and f ∈ l∞(G), where by deﬁnition (g · f)(x) = f(g−1x) for x ∈ G. Then ψ is said to be a G-invariant state.

Then the notion of amenability for von Neumann algebras was first defined by Kadison and Ringrose in 1971 via derivations. We present a different definition here.

Deﬁnition 2.2.2. A von Neumann algebra M ⊂ B(H) is said to be amenable if there exists a state Φ on B(H) such that Φ(uxu∗) = Φ(x) for all x ∈ B(H) and all u ∈ M (that is, Φ is an M-central state on B(H)), and such that Φ M = τ. Then Φ is called a hypertrace.

The connection between the two deﬁnitions is clear from the following proposition.

Proposition 2.2.3. Let G be a discrete countable group. Then the von Neumann algebra L(G) is amenable if and only if G is amenable as a group. 13 Proof. First, assume that G is an amenable group. Let ψ be a G-invariant state, and let

P 2 eg denote the projection onto δg. Then φ : x 7→ g egxeg deﬁnes a map φ from B(l (G)) to l∞(G). Let Φ = ψ ◦ φ. We claim that Φ is an L(G)-central state on B(l2(G)). Indeed,

∗ ∗ Φ(ugxug) = ψ(ugφ(x)ug) = ψ(g · φ(x)) = ψ(φ(x)) = Φ(x). Finally, for x ∈ L(G), we have that Φ(x) = hxδe, δei = τ(x). Thus, Φ is a hypertrace. Conversely, assume that L(G) is an amenable von Neumann algebra, with hypertrace Φ on

2 ∗ ∞ B(l (G)). Deﬁne ψ = Φ l∞(G). Then ψ(g · f) = Φ(ugfug) = Φ(f) = ψ(f) for any f ∈ l (G) and any g ∈ G, so ψ is a G-invariant state.

The simplest amenable von Neumann algebras are the abelian von Neumann algebras,

∞ 2 R 2 L (X, µ), whose hypertrace Φ : B(L (X, µ)) → C is given by integration, Φ(f) = |f| dµ. The previous proposition gives us that every amenable ICC group generates an amenable

II1 factor. For example, S∞, the inﬁnite symmetric group, generates an amenable II1 factor

L(S∞).

Murray and von Neumann defined a finite von Neumann algebra M to be approximately finite dimensional, or AFD, if there exists a nested sequence of finite dimensional subalgebras

00 N Q1 ⊂ Q2 ⊂ · · · ⊂ M such that M = (∪Qi) . For example, let R = 2( ), the tensor i∈NM C

product of inﬁnitely many copies of the space of 2×2 matrices. Then R is an AFD II1 factor.

An early result of Murray and von Neumann proved that all AFD II1 factors are isomorphic,

so we call R the unique AFD II1 factor up to isomorphism.

Furthermore, it is not diﬃcult to show that R is also amenable. Indeed, if we deﬁne Rk be the subalgebra of R given by a tensor product of only k many copies of the space of 2×2

Nk 2 matrices, Rk = i=1(M2(C), τ2), then we can deﬁne a state φ on B(L (R)) by mapping 2 R an element T ∈ B(L (R)) to limk→∞ uT udu. We deﬁne Φ = τ ◦ φ. Then Φ is an U(Rk) R-central state on B(L2(R)).

The groundbreaking theorem of Connes in [Con76] proved the converse, in the following sense.

Theorem 2.2.4. Let M be a ﬁnite von Neumann algebra. Then the following are equivalent:

1. M is AFD. 14 2. M is amenable.

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

Thus, there is a unique amenable II1 factor, up to isomorphism, so in particular, we have ∼ that R = L(S∞). Other examples of amenable ﬁnite von Neumann algebras include the group measure space von Neumann algebra L∞(X, µ)oΓ when Γ is an amenable group, and the tensor product P ⊗Q when P and Q are both amenable.

Next, we recall the deﬁnition of property Gamma, originally given by Murray and von Neumann in [MN43].

Deﬁnition 2.2.5. A II1 factor M has property Gamma if for every ε > 0 and x1, . . . xn ∈ M there is a unitary u ∈ M such that τ(u) = 0 and k[u, xk]k2 < ε for all 1 ≤ k ≤ n.

Equivalently, we have that M has property Gamma if and only there exists a sequence of unitaries uj ∈ U(M) such that τ(uj) = 0 for all j and such that k[uj, x]k2 → 0 as j → ∞ for all x ∈ M. Then such a sequence of unitaries is called an asymptotic central sequence.

Then R has property Gamma, as any finite set of elements in R can be approximated by elements in a finite dimensional matrix subfactor Rk of R, and thus by choosing a unitary with zero trace in Rk ∩ R, we satisfy the conditions of property Gamma. For example, for any x ∈ R and any ε > 0, there exists y ∈ Rk for some k with kx − yk2 < ε/2. Consider the unitary uj in R defined by   0 1 uj = id ⊗ id ⊗ · · · ⊗ id ⊗   ⊗ id ⊗ · · · , 1 0

  0 1 where the matrix   appears in the jth copy of M2(C). Then uj is a unitary with 1 0

15 trace zero. Now we note that

kujx − xujk2 ≤ kuj(x − y) − (x − y)ujk2 + kujy − yujk2

≤ 2 kx − yk2 + kujy − yujk

= 2 kx − yk2 + 0 for j > k < ε.

Also note that property Gamma is an invariant property under *-isomorphism. Murray and ∼ von Neumann proved that L(Fn) does not have property Gamma, and thus L(Fn) 6= R. This was the ﬁrst result that showed the existence of non-isomorphic von Neumann algebras. Analysis of such approximately central sequences is the key behind Popa’s asymptotic orthogonality condition, and he uses this fact that every amenable II1 factor has property

Gamma in establishing maximal amenability of the generator subalgebras of L(Fn).

The converse of this result fails, in that there exist II1 factors that have property Gamma and are not amenable. We consider M = L(Fn)⊗R. Then since L(Fn) is not amenable, we have that M is not amenable. However, the fact that R has property Gamma is suﬃcient for M to also have property Gamma, by again considering an asymptotic central sequence of unitaries of the form     0 1 uj = 1L(Fn) ⊗ id ⊗ · · · ⊗ id ⊗   ⊗ id ⊗ · · ·  . 1 0

∼ ∼ It is clear that this works because R = R⊗R, and hence L(Fn)⊗R = (L(Fn)⊗R)⊗R. This leads us to the following deﬁnition inspired by the work of McDuﬀ in [McD70]

Deﬁnition 2.2.6. A factor M is a McDuﬀ factor if M ∼= M⊗R.

Then we have shown that every McDuﬀ factor has property Gamma, and that moreover, any tensor product M = N1⊗N2 of II1 factors has property Gamma if at least one of the factors has property Gamma. One might ask if, in fact, every property Gamma factor is a McDuﬀ factor, but this turned out to be false, as Dixmier and Lance gave a construction of

a II1 factor with property Gamma that is not McDuff [DL69]. 16 The difference between McDuff factors and property Gamma factors can be seen by the following theorems that tie these properties to the analysis of the ultraproduct von Neumann algebra. The first is due to McDuff, from [McD70].

Theorem 2.2.7. Let ω be a free ultrafilter on N. A separable II1 factor M satisfies the McDuff property M ∼= M⊗R if and only if M 0 ∩ M ω is a non-commutative algebra.

The next theorem was proven ﬁve years later, by Connes in [Con76].

Theorem 2.2.8. Let ω be a free ultraﬁlter on N. A separable II1 factor M has property Gamma if and only if M 0 ∩ M ω 6= C.

2.3 Subalgebra Structure and Classiﬁcation

In addition to considering the properties in the last section in the classiﬁcation of von Neu- mann algebras, it is important to also consider the structure of the lattice of subalgebras, and the properties that they have, since the subalgebra structure is also an isomorphism invariant.

For example, Dixmier deﬁned the following properties of subalgebras.

Deﬁnition 2.3.1. Let A ⊂ M be a von Neumann subalgebra. Then A is said to be singular if N (A) = {u ∈ U(M): uAu∗ = A} is contained in A, and A ⊂ M is said to be regular if (N (A))00 = M.

Additionally, we deﬁne a maximal abelian subalgebra, or MASA, in a von Neumann algebra M to be an abelian subalgebra of M that is maximal with respect to inclusion among all abelian subalgebras of M. Note that A ⊂ M is maximal abelian if and only if A0 ∩ M = A. Then if A ⊂ M is a maximal abelian subalgebra that is regular, we say that A ⊂ M is a Cartan subalgebra.

For example, consider the group measure space von Neumann algebra L∞(X, µ) o Γ arising from a free, ergodic, measure-preserving action σ of a discrete countable group Γ on a probability space (X, µ). Then L(Γ) is a singular subalgebra of L∞(X, µ) o Γ, since every 17 ∞ unitary that normalizes L(Γ) must be ug for some g ∈ Γ. However, we have that L (X, µ)

∞ ∗ ∞ is a regular subalgebra of L (X, µ)oΓ, since ugaug = σg(a) ∈ L (X, µ) for every g ∈ Γ and every a ∈ L∞(X, µ). In fact, since we showed that L∞(X, µ) is a maximal abelian subalgebra

of L∞(X, µ)oΓ, we have that L∞(X, µ) is a Cartan subalgebra. Similarly, we can consider a discrete group Γ with a subgroup H, and then L(H) is a von Neumann subalgebra of L(Γ). Then we see that L(H) is a regular subalgebra of L(Γ) if and only if H is a normal subgroup of Γ.

Now we deﬁne the class of subalgebras that is of most importance to this thesis. A von Neumann subalgebra N of M is said to be maximal amenable if N is amenable and N is maximal with respect to inclusion among all amenable subalgebras of M. The following proposition shows that maximal amenability is, in some sense, a strong type of singularity.

Proposition 2.3.2. Let (M, τ) be a ﬁnite von Neumann algebra, and suppose that A ⊂ M is a maximal amenable subalgebra. Then A is singular.

Proof. Let u ∈ N (A), so uAu∗ = A. Let N be the von Neumann algebra generated by A and u. Since A is amenable, by Connes’ theorem, the conditional expectation from M to A extends to a conditional expectation E : B(L2(M)) → A. Then we deﬁne

n−1 2 1 X j −j Φ: B(L (M)) → C by Φ(x) = lim τ(E(u xu )), n n j=0

and we claim that Φ is an N-central state on B(L2(M)). Note that

n−1 1 X Φ(uxu∗) = lim τ(E(uj+1xu−(j+1))) = Φ(x), n n j=0

and for any unitary v ∈ A, we have that Φ(vxv∗) = Φ(x) by the bimodularity of E. Thus, Φ is an N-central state on B(L2(M)). Furthermore, since u normalizes A, we have that Φ(x) = τ(x) for any x ∈ N. Thus, Φ is a hypertrace, so N is amenable. But A ⊂ N is maximal amenable, so we must have that A = N, and thus u ∈ A. Hence, N (A) ⊂ A.

Lastly, we say that a von Neumann subalgebra N of M is maximal Gamma if N has property Gamma and is maximal with respect to inclusion among all subalgebras of M with 18 property Gamma. Since amenability is a stronger condition than property Gamma, a subalgebra N being a maximal Gamma subalgebra is a stronger condition that N being maximal

amenable. We will see that Popa’s proof that Ma is maximal amenable in L(Fn) actually

shows that Ma is a maximal Gamma subalgebra. Recently, maximal Gamma subalgebras and Gamma stability have been studied by Houdayer in [Hou14b].

2.4 Asymptotic Orthogonality

Popa’s proof relied on the following property:

Deﬁnition 2.4.1. A von Neumann subalgebra N ⊂ M is said to have the asymptotic orthogonality property if for any free ultraﬁlter ω on N, for any y1, y2 ∈ M N, and for 0 ω ω 2 ω any x ∈ N ∩ (M N ), we have that xy1 ⊥ y2x in L (M ).

0 ω In particular, for any x ∈ N ∩ M , y1, y2 ∈ M N,

2 2 2 ky1x − xy2k2 ≥ ky1(x − EN ω (x))k2 + k(x − EN ω (x))y2k2 .

Popa proved that any strongly mixing masa satisfying the asymptotic orthogonality prop-

erty in a ﬁnite von Neumann algebra is necessarily maximal amenable: Let a ∈ Fn be a

generator, and let Ma denote the abelian subalgebra generated in L(Fn) by the unitary ua.

Popa established maximal amenability of Ma by ﬁrst showing that Ma ∈ L(Fn) has the asymptotic orthogonality property.

Next, he showed that for any intermediate subalgebra N, there is a partition of unity

0 ω {ek} of the center of N such that that (N ∩ Ma )ek has a nonzero atomic part for each k,

and claimed that Nek was non-Gamma for all k, for if Nek were to have property Gamma

0 ω 0 ω for some k, we would have (Nek) ∩ (Nek) has no atoms. But as (N ∩ Ma )ek has a nonzero 0 ω ω atomic part, there exists x ∈ (Nek) ∩ (L(Fn)ek) such that x 6∈ Ma . Then taking w ∈ Nek

a unitary orthogonal to Ma, we have that [x, w] = 0. But then

2 2 2 0 = kwx − xwk ≥ w(x − E ω (x)) = x − E ω (x) > 0. 2 Ma 2 Ma 2

This contradiction shows that Nen must be non-Gamma, and consequently, Nen must be non-amenable. Therefore, N is not amenable. Thus, in addition to proving the existence of 19 an abelian maximal amenable subalgebra of a II1 factor, we see that Popa actually proved the stronger result that the generator subalgebras are maximal Gamma.

In [Hou14c], Houdayer deﬁned the following relative notion of Popa’s asymptotic orthogonality property:

Deﬁnition 2.4.2. For von Neumann algebras P ⊂ N ⊂ M, we say that the inclusion N ⊂ M has the asymptotic orthogonality property relative to P if for any free ultraﬁlter ω

0 ω ω on N, for any y1, y2 ∈ M N, and for any x ∈ P ∩ (M N ), we have that xy1 ⊥ y2x in L2(M ω).

In the next chapter, we will prove that the relative asymptotic orthogonality property holds in our amalgamated free product situation.

2.5 Jones’ Basic Construction

If Q ⊂ (M, τ) is a von Neumann subalgebra of a ﬁnite von Neumann algebra, then we

2 2 deﬁne the Jones projection eQ ∈ B(L (M)) to be the orthogonal projection from L (M)

2 onto L (Q). If EQ : M → Q is the unique trace-preserving faithful normal conditional

expectation, then we have that EQ is implemented on M by eQ via eQxeQ = EQ(x)eQ for all x ∈ M. Then we deﬁne the basic construction hM, eQi to be the von Neumann subalgebra

2 of B(L (M)) generated by M and eQ. Then hM, eQi has a trace given by Tr(xeQy) = τ(xy) for any x, y ∈ M.

If we deﬁne J ∈ B(L2(M)) by x1ˆ 7→ x∗1ˆ to be the canonical anti-unitary operator in

2 0 2 B(L (M)), then we have that hM, eQi = JQ J ∩ B(L (M)).

2.6 Popa’s Intertwining by Bimodules

The following deﬁnition and theorem were given by Popa in [Pop06b] and [Pop06a].

Deﬁnition 2.6.1. Let M be a ﬁnite von Neumann algebra and let P,Q be von Neumann

subalgebras of M. We say that a corner of P embeds into Q inside M, and write P ≺M Q if 20 there exist non-zero projections p ∈ P and q ∈ Q, a unital ∗-homomorphism Ψ: pP p → qQq, and a partial isometry v ∈ M such that vv∗ ∈ (pP p)0 ∩ pMp, v∗v ∈ (Ψ(pP p))0 ∩ qMq, and such that xv = vΨ(x) for all x ∈ pP p, with x = 0 whenever xv = 0.

Theorem 2.6.2. (Popa) For M a ﬁnite von Neumann algebra with von Neumann subalgebras P and Q, the following are equivalent:

1. P ≺M Q.

2. There exists a Hilbert space H ⊂ L2(M) such that H is a P -Q-bimodule with ﬁnite dimension as a right Q-module.

0 3. There exists a nonzero projection f0 ∈ P ∩ hM, eQi with ﬁnite trace.

4. It is not true that: For any a1, a2, . . . , an ∈ M and any ε > 0, there exists a unitary u ∈ P such that E (a ua∗) < ε for each i, j. Q i j 2

Proof. (1 ⇒ 3) First, for a positive integer n, we choose a projection p0 ∈ pP p such that p0 enters n many times in a central projection of P . I.e., we consider CtrP (p) ∈ Z(P ), which

∗ can be cut into n many copies that are equivalent to p0. Then since [pP p, veQv ] = 0, we

∗ also have that [p0P p0, p0veQv p0] = 0, so without loss of generality, we may assume that

∗ p = p0. Then we choose partial isometries v1, v2, . . . , vn ∈ P such that vi vi = p and we Pn ∗ P ∗ ∗ deﬁne z = i=1 vivi ∈ P(Z(P )). But then f0 = viveQv vi commutes with P z and

Tr(f0) < nτ(z).

2 (3 ⇒ 2) Take H = f0(L (M)). Since [f0,LP ] = 0 and [f0,RQ] = 0, we have that

0 2 RQ ∩ B(f0L (M)) is a ﬁnite von Neumann algebra, so HQ has ﬁnite dimension.

0 (2 ⇒ 1) Since H is a P -Q bimodule and RQ is a ﬁnite von Neumann algebra, there exists a nonzero ξ ∈ H and a projection p ∈ P such that pξ = ξ and pP pξ ⊂ ξQ. By subtracting oﬀ the largest subprojection under p whose product with ξ is 0, we may assume that xξ = 0 implies that x = 0 for all x ∈ pP p.

∗ −1/2 2 We deﬁne ξ0 = ξEQ(ξ ξ) ∈ L (M), and we observe that if η is the right support of 21 ∗ 2 ∗ 1 ∗ EQ(ξ0 ξ0), then η ∈ L (M), so η η ∈ L (M), and therefore EQ(ξ0 x0) is a projection in Q. ∗ 2 Thus, we may assume that EQ(ξ ξ) = q is a projection in Q, and then pP pξ ⊂ ξL (Q).

∗ ∗ ∗ We deﬁne Ψ(x) = EQ(ξ xξ) for any x ∈ pP p. Note that EQ(ξ xξ) = EQ(ξ ξy) for some y ∈ qL2(Q), so we get that Ψ(x) = qy, so we have that xξ = ξΨ(x). Also, ξ∗x = Ψ(x)ξ∗, so the map is multiplicative, and hence is a ∗-isomorphism. Finally, we take v = (ξξ∗)−1/2ξ.

0 (3 ⇒ 4) Let f0 ∈ P ∩ hM, eQi be a nonzero projection with ﬁnite trace. Then we can Pn ∗ assume f0 = j=1 ajeQaj for a ﬁnite set {a1, a2, . . . , an} ⊂ M. But then for any unitary u ∈ U(P ), we have that

E (a ua∗) 2 = τ(E (a ua∗)∗E (a ua∗)) = Tr(e (a ua∗)∗e (a ua∗)e ), Q i j 2 Q i j Q i j Q i j Q i j Q

so that X 2 E (a ua∗) = Tr(f uf u∗) = Tr(f ), Q i j 2 0 0 0 i,j so (4) holds.

(4 ⇒ 3) Choose ε > 0 and a ﬁnite set {a1, . . . , an} ⊂ M such that for every unitary u ∈ U(P ), we have that X E (a ua∗) ≥ ε. Q i j 2 i,j Pn ∗ Pn ∗ Deﬁne b = j=1 ai eQai ∈ hM, eQi+, where Tr(b) = j=1 τ(ai ai) < ∞. We let c be the unique element of minimal L2 norm in the L2-closed convex hull of the set {ubu∗ : u ∈ U(P )}

∗ in hM, eQi. By uniqueness of c, we have that since kucu k2 = kck2 for every unitary u ∈ U(P ), ∗ 0 we have that c = ucu , and hence c commutes with P , so c ∈ P ∩ hM, eQi, and since Tr(b) < ∞, we have that Tr(c) < ∞. Lastly, note that

X 2 X ε2 ≤ E (a ua∗) = Tr(E (a ua∗)∗E (a ua∗)) Q i j 2 Q i j Q i j i,j i,j

X ∗ ∗ ∗ = Tr(eQaju ai eQaiuaj eQ) i,j

X ∗ ∗ = Tr(eQaj(u bu)aj eQ). j 22 P ∗ ∗ 2 Then by normality of Tr, we have that j Tr(eQaj(u cu)aj eQ) ≥ ε , from which we conclude 0 that c 6= 0. Thus, c is a nonzero projection in P ∩ hM, eQi with ﬁnite trace.

23 CHAPTER 3

Main Result

3.1 Amalgamated Free Products

We now recall the construction of amalgamated free products as seen in [Voi85] and [Pop93].

Suppose that (N1, τ1) and (N2, τ2) are ﬁnite von Neumann algebras with a common von

Neumann subalgebra (B, τB), where τj B = τB for each j, so there exists a trace preserving

conditional expectation Ej : Nj → B. We deﬁne the free product of N1 and N2 with

amalgamation over B, denoted (M,EB) = (N1,E1) ∗B (N2,E2) = N1 ∗B N2, as follows:

N1 ∗B N2 has a dense *-subalgebra

M M B ⊕ sp(Ni1 B)(Ni2 B) ··· (Nin B)

n≥1 ij ∈{1,2} i16=i2,i26=i3,...,in−16=in

with EB : M → B satisfying EB = Ej and EB(x) = 0 for any word x = xi1 xi2 . . . xin with Nj

xik ∈ Nik B, i1 6= i2 6= ... 6= in.

M has a trace given by τ = τB ◦EB, and the subspaces B and sp(Ni1 B) ··· (Nin B) are all mutually orthogonal in L2(M, τ) with respect to the inner product hx, yi = τ(y∗x), and the closures in L2(M, τ) of these subspaces give mutually orthogonal Hilbert B-B-bimodules that sum to L2(M, τ):

2 ∼ 0 0 0 L ((Ni1 B) ··· (Nin B)) = Hi1 ⊗B Hi2 ⊗B · · · ⊗B Hin ,

0 2 2 with Hi = L (Ni) L (B).

24 3.2 Maximal Amenability

In this section, we will establish the results about amalgamated free products needed to prove the main theorem.

First, as far back as [Pop83b], mixing properties were used to study subalgebras of II1 factors. We recall the following deﬁnition as stated in [PV08].

Deﬁnition 3.2.1. For ﬁnite von Neumann algebras P ⊂ N ⊂ M, we say that the inclusion

N ⊂ M is weakly mixing through P if there exists a sequence un ∈ U(P ) such that

kEN (xuny)k2 → 0 for all x, y ∈ M N.

Then amalgamated free products satisfy this weak mixing condition, which can be seen from the following result in [IPP08].

Lemma 3.2.2. Suppose (N1, τ1) and (N2, τ2) are ﬁnite von Neumann algebras with a common subalgebra B such that τ1 B = τ2 B. Let M = N1 ∗B N2, and suppose that P ⊂ N1 is a von Neumann subalgebra such that P 6≺N1 B. Then for each k = 1, 2, Nk ⊂ M is weakly mixing through P .

Proof. It suﬃces to show that for all x0, x1, . . . , xn ∈ M Nk and for all ε > 0, there exists a unitary u ∈ U(P ) such that

E (x ux∗) ≤ ε for all 0 ≤ i, j ≤ n. Nk i j 2

Also, it suﬃces to consider the following two cases: (i) xi ∈ M Nk is a word of the form aiyi, where ai is a word ending in a letter in N2 B and yi is either 1 or a letter in N1 B, or (ii) k = 2, and xi ∈ M N2 is a word of the form aiyi, where ai = 1, and yi ∈ N1 B.

(These cases are suﬃcient by Kaplansky density, and since every monomial in M Nk has such a form.)

By assumption, P 6≺N1 B, so we may choose u ∈ U(P ) such that ε E (y uy∗) ≤ for all 0 ≤ i, j ≤ n. B i j 2 kaik kajk 25 ∗ In both (i) and (ii), we have that yi ∈ N1, and since P ⊂ N1, we have yiuyj ∈ N1 for all i, j. ∗ ∗ ∗ Thus w = yiuyj − EB(yiuyj ) ∈ N1 B. We consider aiwaj in case (i) and (ii).

∗ If both ai and aj come from case (i), then since ai ends in N2 B and aj starts in N2 B, ∗ we have that aiwaj is perpendicular to both N1 and N2.

If one or both of ai and aj come from case (ii), then either ai = 1 or aj = 1 or both, and we

∗ have that k = 2. But since w ∈ N1 B, we have that aiwaj is perpendicular to N2.

∗ ∗ Thus, in all cases, aiwaj ∈ M Nk, so ENk (aiwaj ) = 0. Thus,

∗ ∗ ∗ ∗ ∗ ∗ ENk (xiuxj ) = ENk (aiwaj + aiEB(yiuyj )aj ) = ENk (aiEB(yiuyj )aj ) and hence

E (x ux∗) = E (a E (y uy∗)a∗) ≤ ka k ka k E (y uy∗) ≤ ε. Nk i j 2 Nk i B i j j 2 i j B i j 2

In order to get maximal amenability, we must ﬁrst show that N1 ⊂ M has the asymptotic orthogonality property relative to P .

Theorem 3.2.3. Suppose (N1, τ1) and (N2, τ2) are ﬁnite von Neumann algebras with a common subalgebra B such that τ1 B = τ2 B. Let M = N1 ∗B N2, and suppose that P ⊂ N1 is a von Neumann subalgebra such that P 6≺N1 B. Let ω be a free ultraﬁlter on N, y1, y2 ∈ M N1, 0 ω ω 2 ω and x ∈ P ∩ (M N1 ). Then xy1 ⊥ y2x in L (M , τω).

ω Proof. First, we note that if we let (xn)n be a sequence representing x ∈ M , it suﬃces to show that for every ε > 0 there is an integer N such that hxny1, y2xni < ε for every n ≥ N.

Secondly, we claim that it suﬃces to consider the case in which y1, y2, and xn are ﬁnite

0 0 alg sums of ﬁxed length. To see this, we ﬁrst choose y1, y2 ∈ N1 ∗B N2 such that we have 0 2 0 0 kyi − yik < ε/(6 sup kxnk ) and y1, y2 ∈ M N1. Then there are some integers m, l ∈ N

26 0 such that each yj is the sum of at most m products of length at most l. Next, we choose 0 alg xn ∈ N1 ∗B N2 such that

0 ε kxn − xnk < 2 2 6 max(ky1k , ky2k ) 0 and [P, xn] → 0. Then

0 2 0 2 0 0 hxny1, y2xni ≤ ky1 − y1k2 kxnk + ky2 − y2k2 kxnk + hxny1, y2xni ε < + ky0 k2 kx0 − x k + ky0 k2 kx0 − x k + hx0 y0 , y0 x0 i 3 1 n n 2 2 n n 2 n 1 2 n ε ε < + + hx0 y0 , y0 x0 i . 3 3 n 1 2 n

0 0 0 So, it suffices to consider the case in which xn, y1, y2 are all finite sums of fixed length.

Then there are ﬁnite symmetric subsets 1 ∈ Fi ⊂ Ni such that the letters in y1 and y2 all

2 come from F1 ∪ F2. So there is a subspace X1 of L (M) consisting of sums of products that

⊥ start and end with something in F1 = {w ∈ N1 B : hw, vi = 0 for all v ∈ F1}, and then ⊥ X0 = X1 is ﬁnite dimensional.

Let (xn)X denote the projection of xn onto a subspace X. Then we may decompose

xn = (xn)X0 + (xn)X1 . We claim that (xn)X1 satisﬁes the mutual orthogonality with y1 and y2, and we claim that k(xn)X0 k2 is small for suﬃciently large n.

For the latter claim, we use that P 6≺N1 B, so for any ε > 0 there exists u1 ∈ U(P ) ∗ ∗ such that hu1X0u1,X0i < ε/4. Then there exists u2 ∈ U(P ) such that hu2X0u2,X0i < ε/16, ∗ ∗ and hu2X0u2, u1X0u1i < ε/16. ∗ ε ∗ ∗ ε Inductively, we choose uk so that hukX0u ,X0i < k and ukX0u , ujX0u < k for each k 22 k j 22 j < k. Then 2 X 2 (x ) ∗ ≈ (x ) ∗ . n X0∪u1X0u1∪··· 2 ε n uiX0ui 2 i 2 But then for any ε > 0 there exists k such that (x ) ∗ < ε/4. Since [P, x ] → 0, we n ukX0uk 2 n ∗ 2 choose N such that kukxnuk − xnk2 < ε/4 for all n ≥ N. Hence, for this k, we have that 27 2 ∗ 2 k(xn)X0 k2 = kuk(xn)X0 ukk2

∗ 2 2 ≤ 2 u (x ) u − (x ) ∗ + 2 (x ) ∗ k n X0 k n ukX0uk 2 n ukX0uk 2 ∗ 2 2 ≤ 2 ku x u − x k + 2 (x ) ∗ < ε. k n k n 2 n ukX0uk 2

Thus, it suffices to show that (xn)X1 satisfies the mutual orthogonality property, so we may ⊥ assume that xn is a finite sum of products that start and end with something in F1 . By repeated use of the triangle inequality, we may consider the case of monomials. Let y = y(i) ··· y(i), x = x0 ··· x0 . Then i k1 kl n j1 jr

∗ ∗ hxny1, y2xni = τ(xny2xny1) = τ((x0 )∗ ··· (x0 )∗(y(2))∗ ··· (y(2))∗x0 ··· x0 y(1) ··· y(1)). jr j1 kl k1 j1 jr k1 kl

We claim that

τ((x0 )∗ ··· (x0 )∗(y(2))∗ ··· (y(2))∗x0 ··· x0 y(1) ··· y(1)) = 0. jr j1 kl k1 j1 jr k1 kl

First, if k = 2, then E (x0 y(1)) = E ((y(2))∗x0 ) = 0. 1 B jr k1 B k1 j1 Similarly, if k = 2, then E ((x0 )∗(y(2))∗) = 0. l B j1 kl Then if k = 1, we have that y(1) ∈ F , so since x0 ∈ F ⊥, we still have that E (x0 y(1)) = 0. 1 k1 1 jr 1 B jr k1

Thus in all cases, the claim holds, which completes the proof.

As we have established asymptotic orthogonality, as in Popa’s result, we use weak mixingness to get maximal amenability. To be precise, we use the following generalization of Popa’s result given by Houdayer, and we include his proof:

Theorem 3.2.4. (Theorem 8.1 in [Hou14c]) Suppose P ⊂ N ⊂ M are tracial von Neumann algebras. Assume the following:

1. P is amenable.

2. N ⊂ M is weakly mixing through P .

3. N ⊂ M has the asymptotic orthogonality property relative to P . 28 Then for any intermediate amenable von Neumann subalgebra P ⊂ Q ⊂ M, we have that Q ⊂ N.

Proof. Suppose that P ⊂ Q ⊂ M is an intermediate amenable von Neumann subalgebra. First, we will utilize the following lemma, which appeared in this form as Prop. 6.14 in [PV08], and is based oﬀ of Theorem 3.1 in [Pop06b].

Lemma 3.2.5. If N ⊂ M is weakly mixing through P , then any sub-P -N-bimodule of L2(M) which is of ﬁnite type as an N-module in contained in L2(N). In particular,

k X (i.) if x ∈ M satisﬁes P x ⊂ yiN for some ﬁnite subset {y1, y2, . . . , yk} ⊂ M, then i=1 x ∈ N, and

(ii.) P 0 ∩ M ⊂ N.

Thus, we have that P 0 ∩ M ⊂ N. But since P ⊂ Q, Q0 ∩ M ⊂ P 0 ∩ M, so Q0 ∩ M ⊂ N, and therefore Q0 ∩ M = Q0 ∩ N. We choose z ∈ Z(Q0 ∩ N) to be the maximal projection such that Qz ⊂ zNz. Then it suﬃces to show that z = 1. We let z⊥ = 1 − z and we assume for sake of contradiction that z⊥ 6= 0. Deﬁne Q˜ = Qz⊥.

˜ We claim that it suﬃces to show that Q ≺M N. Indeed, if this is true, then there exist ˜ k ≥ 1, a projection p ∈ Mk(N), a unital normal ∗-homomorphism Ψ : Q → pMk(N)p, and a ⊥ ∗ ˜0 ⊥ ⊥ ∗ ˜ 0 nonzero partial isometry v ∈ M1,k(z M)p such that vv ∈ Q ∩z Mz , v v ∈ (Ψ(Q)) ∩pMp, and av = vΨ(a) for all a ∈ Q˜.

⊥ ⊥ Then v can be written as v = [v1, v2, . . . , vk] ∈ M1,k(z M)p, with vj ∈ z M for each ˜ ˜ Pk 1 ≤ j ≤ k. Since av = vΨ(a) for all a ∈ Q, we have that Qvi ⊂ j=1 vjN for all 1 ≤ i ≤ k. Because N ⊂ M is weakly mixing through P , then N ⊂ M is weakly mixing through Q ˜ and hence through Q. Therefore, by part (i) of the lemma, we conclude that vi ∈ N for all 1 ≤ i ≤ k. Since vv∗ ∈ Q˜0 ∩ z⊥Mz⊥ and Q0 ∩ M = Q0 ∩ N, we have that vv∗ ∈ Q˜0 ∩ z⊥Nz⊥. ˜ ∗ ∗ ∗ Furthermore, we have that Qvv ⊂ vv Nvv . Thus, by letting z0 be the central support of ∗ ˜0 ⊥ 0 vv in Q ∩ z , we have that z0 ∈ Z(Q ∩ N) is a nonzero projection such that Qz0 ⊂ z0Nz0, 29 ⊥ ˜ with z0 ≤ z , which contradicts the maximality of z. Thus, if Q ≺M N, we have that Q ⊂ N.

˜ Therefore, we conclude the proof by showing that Q ≺M N. Since Q is amenable, we have that Q˜ is amenable. Therefore, by Connes, we have that Q˜ is AFD, so that Q˜ can be ˜ S 00 ˜ written as Q = ( j Qj) , where Qj is a unital finite-dimensional ∗-subalgebra of Q for each ˜ j ≥ 1, and such that Q1 ⊂ Q2 ⊂ · · · ⊂ Q. 0 ˜ ˜ Furthermore, we claim that we may choose Qj in such a fashion so that for each j, Qj ∩Q ⊂ Q 2 ˜ 0 ˜ is a finite index inclusion, i.e., L (Q) has finite dimension as a right Qj ∩ Q-module. Such a ˜ construction of Qj can be obtained as follows: First, we decompose Q into a direct summand

of type II1 and direct summands of type In by choosing pairwise orthogonal central projec- ˜ P∞ ⊥ ˜ ˜ tions qn ∈ Z(Q) such that n=0 qn = z , and such that Qq0 = Z0⊗R and Qqn = Zn⊗Mn(C) N∞ for all n ≥ 1, where R = i=1(M2(C), τ2) is the AFD II1 factor and Zn is an abelian von Neumann algebra for each integer n ≥ 0. Nk Then we define Rk = i=1(M2(C), τ2), and we choose an increasing sequence of unital finite (k) S (k) 00 dimensional ∗-subalgebras Zn ⊂ Zn such that Zn = ( k Zn ) . Then we define

(j) M (j) X Qj = (Z0 ⊗Rj) ⊕ (Zk ⊗Mk(C)) ⊕ C qj, 1≤k≤j k>j

˜ S 00 and then Q = ( j Qj) , with Qj ﬁnite dimensional for each j. Furthermore, we have that

0 ˜ M M Qj ∩ Q = (Z0⊗(Rj ∩ R)) ⊕ (Zk⊗CIdMk(C)) ⊕ (Zk⊗Mk(C)). 1≤k≤j k>j

0 ˜ ˜ Hence, we have that Qj ∩ Q has ﬁnite index in Q for each j. ˜ ˜ Then to show that Q ≺M N, we assume for sake of contradiction that Q 6≺M N. Then since 0 ˜ ˜ 0 ˜ Qj ∩ Q ⊂ Q has ﬁnite index, we have that Qj ∩ Q 6≺M N (using Lemma 3.9 in [Vae08], for example).

0 ˜ 1 ⊥ Then for each n ≥ 1, there exists a unitary un ∈ U(Qn ∩Q) such that kEN (un)k2 ≤ n z 2. ˜0 ˜ω Let ω be a free ultraﬁlter on N, and we consider u = (un)n≥1 as an element in U(Q ∩ Q ). ω ω ˜0 Then since kEN (un)k2 → 0, we have that u ∈ M N . Moreover, since u ∈ Q , we have that u ∈ Q0, so that u ∈ Q0 ∩ (M ω N ω) ⊂ P 0 ∩ (M ω N ω). 30 By assumption, N ⊂ M has the asymptotic orthogonality property relative to P , so since 0 ω ω ˜ u ∈ P ∩ (M N ) and (x − EN (x)) ∈ M N for any x ∈ Q, we have that (x − EN (x))u

2 ω is orthogonal to u(x − EN (x)) in L (M ), so

2 2 2 2 k(x − EN (x))u − u(x − EN (x))k2 = k(x − EN (x))uk2 +ku(x − EN (x))k2 = 2 kx − EN (x)k2 .

But since u ∈ Q˜0, we have that xu = ux, so

2 2 k(x − EN (x))u − u(x − EN (x))k2 = kxu − EN (x)u − ux + uEN (x)k2

2 = kEN (x)u − uEN (x)k2 ,

and hence 2 2 ˜ kEN (x)u − uEN (x)k2 = 2 kx − EN (x)k2 for all x ∈ Q.

1 ⊥ By our choice of (un), we have that for n ≥ 4 we have that kEN (un)k2 ≤ 4 z 2. Then

kEN (un)u − uEN (un)k2 ≤ kEN (un)uk2 + kuEN (un)k2

= 2 kEN (un)k2

1 ⊥ ≤ z , 2 2

2 1 ⊥ 2 so that kEN (un)u − uEN (un)k2 ≤ 4 z 2. However,

kun − EN (un)k2 ≥ kunk2 − kEN (uk)k2

⊥ 1 ⊥ ≥ z − z 2 4 2 3 ⊥ = z . 4 2

2 18 ⊥ 2 ˜ Hence, kEN (un)u − uEN (un)k2 ≥ 16 z 2, which is a contradiction. Thus, Q ≺M N.

Now our theorem follows easily, as we have established the weak mixing condition and the relative asymptotic orthogonality.

Theorem 1.1. Suppose that (N1, τ1) and (N2, τ2) are ﬁnite von Neumann algebras with

a common von Neumann subalgebra (B, τB). Suppose that P ⊂ N1 is a von Neumann

subalgebra with P diﬀuse and maximal amenable in N1, and that P 6≺N1 B. Then P is

maximal amenable in M = N1 ∗B N2. 31 Proof. Assume for sake of contradiction that there exists an intermediate amenable von

Neumann algebra, P ( Q ⊂ M. Then by Proposition 3.2.2, N1 ⊂ M is weakly mixing through P , and by Proposition 3.2.3, N1 ⊂ M has the asymptotic orthogonality property relative to P . Hence, by Theorem 3.2.4, we have that Q ⊂ N1. But P is a maximal amenable subalgebra of N1, so since Q is amenable and P ( Q ⊂ N1, we get a contradiction.

32 CHAPTER 4

Special Cases

In order to consider examples in which these hypotheses are satisﬁed, we ﬁrst recall some properties of the Pimsner-Popa index in order to restate our hypotheses in another way. Then we will consider how our result applies to the cases of group von Neumann algebras and abelian von Neumann algebras.

4.1 Pimsner-Popa index

We recall some definitions and facts from [PP86]. The Pimsner-Popa index for an inclusion of finite von Neumann algebras was first considered as a generalization of the Jones index for an inclusion of subfactors.

Definition 4.1.1. For an inclusion of finite von Neumann algebras P ⊂ N ⊂ M, we define

the constant λ(N,P ) = max{λ ≥ 0 : EP (x) ≥ λx for all x ∈ N with x > 0}. Then the index can be deﬁned by [N : P ]−1 = λ(N,P ).

We will need to consider the case when [N : P ] = ∞, and we will use the following lemma from [PP86].

Lemma 4.1.2. Suppose that N ⊂ (M, τ) is a von Neumann subalgebra such that N 0 ∩ M contains no ﬁnite trace projections of M. Then for all ε > 0 and all x ∈ M with τ(x) < ∞, Pn P there exist projections q1, q2, . . . , qn ∈ N such that i=1 q1 = 1 and k qixqik2 < ε kxk2.

We will use the Pimsner-Popa index to give examples in the special case in which P = N1, by the following lemma.

33 Lemma 4.1.3. Suppose that M is a diﬀuse amenable ﬁnite von Neumann algebra, and N ⊂ M is a von Neumann subalgebra. The following are equivalent:

1. M 6≺M N.

2. [pMp : Np] = ∞ for every non-zero projection p ∈ N 0 ∩ M.

3. There exist unitaries un ∈ U(M) such that kEN (unx)k2 → 0 for every x ∈ N.

Proof. (1 ⇒ 2) (c.f. Lemma 2.4 in [CIK13], Lemma 1.4 in [Ioa12]). Suppose that M 6≺M N

0 and let p ∈ N ∩ M be a non-zero projection. Then we have that pMp 6≺M Np. Then by

0 the intertwining by bimodules theorem, we have that (pMp) ∩ hM, eNpi does not contain a non-trivial projection of ﬁnite trace. Then by the above lemma from Pimsner-Popa, we Pn have that for any ε > 0, there exist projections q1, q2 . . . qn ∈ pMp such that i=1 qi = 1 Pn Pn 2 Pn 2 and k i=1 qieNpqik2,Tr < ε. But then k i=1 qieNpqik2,Tr = i=1 kENp(qi)k2, so there exists some q = qi with kENp(q)k2 ≤ ε kqk2. Thus, [pMp : Np] = ∞.

P ◦ 2 2 (2 ⇒ 3) Note that for x1, . . . xk ∈ N, we have that ξ = xj ⊗xj ∈ L (M)⊗N L (M). Thus, ∗ P 2 P 2 for any u ∈ U(M), we have hu ξu, 1 ⊗ 1i = j kEN (uxj)k2. If kEN (uxi)k2 ≥ ε, there 2 2 2 2 ∼ 2 exists ξ ∈ L (M) ⊗N L (M) that commutes with M. But L (M) ⊗N L (M) = L (hM, eN i),

0 0 0 and hM, eN i ∩ M = ρ(M ∩ N ), so we get a projection p ∈ N ∩ M with [pMp : Np] < ∞.

(3 ⇒ 1) This follows from the intertwining by bimodules theorem.

4.2 Group von Neumann algebra case

Consider the case where Λ1 is a discrete countable amenable group and H is a subgroup of

Λ1 with [Λ1 : H] = ∞. Let Λ2 be another group containing H. Deﬁne Γ to be the amalgamated free product group Γ = Λ1 ∗H Λ2. Then L(Λ1) is a maximal amenable von Neumann subalgebra of L(Γ) = L(Λ1) ∗L(H) L(Λ2). Similarly, if Λ1 y X is a free, ergodic, probability ∞ measure preserving action on a probability space X, then if we take N1 = L (X) o Λ1 and

34 ∞ B = L (X)oH, we also get that N1 is maximal amenable in N1 ∗B N2 for any N2 contained B. This had been previously proven by Jolissaint in [Jol10] in the case where H was assumed to be ﬁnite. The inﬁnite index case was proven independently by Boutonnet and Carderi in [BC14] as a corollary of a more general result.

4.3 Finite Factor case

Suppose B is a finite amenable factor and P is diffuse and amenable, and we define N1 to be

0 the tensor product B⊗Q. Then [pN1p : Bp] = ∞ for every non-zero projection p ∈ B ∩ N1, so for N2 any other ﬁnite von Neumann algebra containing B, we have that N1 is maximal amenable in N1 ∗B N2.

4.4 Crossed Product von Neumann Algebra case

Suppose a discrete countable amenable group Γ acts on a II1 factor B such that the action is properly outer, and let N1 = B o Γ. Let N2 be another ﬁnite von Neumann algebra 0 containing B. Then [pN1p : Bp] = ∞ for every non-zero projection p ∈ B ∩ N1, so N1 is maximal amenable in N1 ∗B N2.

4.5 Abelian case

∞ ∞ For the abelian case, we have B = L (Y, ν) ⊂ L (X, µ) = N1. This inclusion induces a surjective measure-preserving map φ : X → Y . We can disintegrate the measure space as a direct integral, Z ⊕ −1 (X, µ) = (φ ({y}), µy)dν(y), Y −1 where µy is a probability measure on φ ({y}) such that:

−1 • For every measurable E ⊂ X, y 7→ µy(E ∩ φ ({y})) is measurable

R −1 • µ(E) = Y µy(E ∩ φ ({y}))dν(y).

35 Then the inclusion having this condition is equivalent to the following measure theoretic condition.

Proposition 4.5.1. [L∞(X, µ)p : L∞(Y, ν)p] = ∞ for all p ∈ L∞(Y, ν)0 ∩ L∞(X, µ) if and only if µy is a diﬀuse probability measure for a.e. y ∈ Y .

Proof. Suppose a positive measure set in Y has ﬁbers with atoms. Then we deﬁne the

−1 set E = {y : µy has an atom} ⊂ Y . Then E is measurable and φ : φ (E) → E, so by

−1 measurable selection, there exists a map ψ : E → φ (E) such that φ ◦ ψ = idE and ψ(y) is

∞ ∞ an atom of µy for every y ∈ F . Let p = χψ(E). We claim that [L (X, µ)p : L (Y, ν)p] = 1.

∞ To see this, let f ∈ L (X, µ). Note that f(x)χψ(E)(x) 6= 0 implies that there exists y ∈ E

−1 such that ψ(y) = x, so x ∈ φ ({y}) is an atom of µy. Thus, f(x)χψ(E)(x) = f(ψ(y))χE(y).

We deﬁne g : E → C by g(y) = f(ψ(y)). Then for every x ∈ ψ(E), we have that x = ψ(y) for some y, so g(φ(x)) = g(y) = f(ψ(y)) = f(x).

∞ ∞ Thus, (g ◦ φ)χψ(E) = fχψ(E), so L (Y, ν)p = L (X, µ)p.

For the other direction, ﬁrst assume that [L∞(X, µ): L∞(Y, ν)] < ∞. As noted in the proof of Prop 1.2 in [Pop99], we have that

[L∞(X, µ): L∞(Y, ν)]−1 = inf{E(p)(φ(x)) : x ∈ X, p ∈ L∞(X, µ) a projection, p(x) 6= 0}, where E : L∞(X, µ) → L∞(Y, ν) is the conditional expectation. Thus, there exists ε > 0 such that E(p)(y) ≥ ε for every y ∈ Y and p ∈ L∞(X, µ) such that p(φ−1({y})) 6= 0. Using the disintegration of measure, we have that for every f ∈ L∞(X, µ), Z Z Z fdµ = fdµy dν(y). X Y φ−1({y}) R Hence, E is deﬁned by E(f)(y) = φ−1({y}) fdµy. Therefore, if E ⊂ X is any measurable −1 subset, we have that E(χE)(y) = µy(E ∩ φ ({y})). Therefore, for every y ∈ Y and for every measurable E ⊂ X such that E ∩ φ−1({y}) 6= ∅, we

−1 have that µy(E ∩ φ ({y})) ≥ ε. Hence, µy has atoms for a.e. y. 36 The argument remains the same if we assume instead that [L∞(X, µ)p : L∞(Y, ν)p] < ∞ for some p ∈ L∞(Y, ν)0 ∩ L∞(X, µ), so the proof is complete.

Thus, we conclude that if µy is a diﬀuse probability measure for a.e. y ∈ Y , then for any

∞ ∞ ﬁnite von Neumann algebra N2 containing B = L (Y, ν), we have that N1 = L (X, µ) is a maximal amenable subalgebra of N1 ∗B N2.

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