First, I would like to take this opportunity to thank my advisor, Jesse Peterson, for his guidance throughout my years at Vanderbilt. Third, I would like to thank the professors who taught me, whether in or out of the classroom: Dietmar Bisch, Vaughan Jones, Guoliang Yu, Alexander Olshanskiy, Alexander Powell, Gennadi Kasparov, Mike Mihalik, Denis Osin, Doug Hardin, etc. The concept of amenability is at the heart of the study of von Neumann algebras.
The second chapter introduces amenability, starting with amenable groups and then on to amenable von Neumann algebras. The treatment is brief and most of the proofs in this chapter are omitted, as they can be found in many standard textbooks in the field, [Dix81, KR97a, KR97b, Tak03].
Bounded operators on Hilbert spaces
Then every self-adjoint element in the closed unit sphere is SOT (equivalently, WOT) of the closure M, in the SOT closure of self-adjoint elements in the closed unit sphere M. Let (X,µ) be a standard gauge space and Γy(X,µ) be the conservation action measure class of the countable discrete group Γ. As a special case of the above case, if we take X to be the space of one point, the resulting von Neumann algebra is called the (left) group von Neumann algebra associated with Γ and denoted by L(Γ).
Type decompositions
Every von Neumann algebra can be uniquely decomposed into the direct sum of von Neumann algebras of type I, type II1, type II∞, and type III. All other examples of type II1, II∞, and III can be obtained via the set-mass-space construction.
Type II 1 factors
In previous sections we introduced the construction due to Murray and von Neumann, the group of von Neumann algebras. Moreover, each elementx∈L(Γ) can be formally written as an infinite linear combination of the canonical unitary elementsug,g∈Γ:x=∑g∈Γagug and we have. It is not difficult to see that for an infinite groupΓ,Γ ICC is if and only if L(Γ) is an II1 factor.
Hyperfinite factor II1: LetSn be the set of permutations in {1,· · ·,n} and letS∞ be the inductive limit of allSn. In other words, L(S∞) is the WOT closure of an increasing sequence of finite-dimensional algebras. The fundamental issue of II1 factor theory is to solve the factor isomorphism problem.
As the first two examples were introduced by the co-founders of the theory, it is known that the hyperfinite II1 factor and the free group factors are not isomorphic.
Ultraproducts of II 1 factors
Suitable groups were first introduced by von Neumann in his attempt to understand the Banach-Tarski paradox. Roughly speaking, an admissible group is a 'small' group comparable to the group of integers. The first examples of inadmissible groups are groups that contain non-abelian free groups as subgroups.
However, it took mathematicians many years to answer the question of whether every non-amenable group contains a free subgroup. As we mentioned in Chapter 1, the hyperfinite factor II1 is one of the first examples presented by Murray and von Neumann. Indeed, Connes' seminal work on the classification of injective von Neumann algebras [Con76] shows that they are equivalent.
Furthermore, Connes' result implies that for an ICC groupΓ, L(Γ)∼=Rif and only ifΓ is admissible. Therefore, to study non-admissible von Neumann algebras, it is natural to consider their admissible subalgebras.
Amenable groups
Amenable von Neumann algebras
Amenable extensions
Maximal amenable subalgebras
Given a suitable subalgebra within an inadmissible fuzzy factor II1, how many ways can we suitably expand it. In this regard, Jesse Peterson conjectures that there exists a unique maximal suitable extension for every suitable diffuse subalgebra in a free group factor. In this chapter we will present the main results in this thesis, that for the radial measure in a free group factor and for the cup subalgebra in a planar factor II1, unique results of appropriate extension can be obtained.
The main proof technique will also be explained and the proofs will occupy the last two chapters.
Main results
This is the first example of such a discontinuity for a suitable maximal subalgebra which is not known to be in a free position. BW16] The cup of the subalgebra is the maximally suitable unique extension of any diffuse subalgebra by itself.
Strong AOP and maximal amenable extensions
Both A and Qlie in the normalizer C, so that together they produce an amenable algebra containing A. Ozawa and Popa [OP10] showed that the factors of a free group are strongly robust and that singularities for subalgebras are known. Therefore, our main task is to prove s-AOP for those cases we mentioned in the last section.
Preliminaries
Recall that in a separable Hilbert space, a sequence of vectors {vn} forms a Riesz basis (for the basics of Riesz basis, see e.g. [Chr01]) if {vi} is the image of an orthonormal basis under a bounded invertible operator. In this case, each vector x in the Hilbert space has a unique decomposition x=∑cnvn, for some (cn)∈`2. Sometimes it will be convenient to use the following convention: we write ξn,mi for all n,m∈Z, where we defineξn,mi =0 whenever <0 orm<0.
The key calculation in [CFRW10] is that when considering the AOP in the radial mass case, the R˘adulescu basis plays the same role as the canonical basis for the mass generator case. The other side of the inequality is easy, since each of them appears only four times. L0k) is the left "projection" into the space of ξn,mi. i,n,m) with the first signature no more thanks.
However, one should be warned that they are simply idempotents, instead of projections, due to the presence of thosei's withl(i) =1. The definition of the ηn,mi's clearly implies that ηn,mi ωl ∈span{ηn,ki }k≥0, that is, multiplication of ωl on the right does not change the top or left index of ηn,mi, so L0k (ηn,mi ωl) =L0k(ηn,mi )ωland the proof is complete.
Proof of Theorem A
Moreover, one can construct {vk},{Sk} such that there is a sequence {Fk} of strictly increasing natural numbers such that L0N. Since span{ωn}n≥0is a *- dense weak subalgebra C, the Kaplansky Density Theorem implies that there exists a sequence{zk}kof elements in C whose Fourier expansions are finitely supported, such that||zk| | ≤3/2 and. Now sinceuk→0 weakly, there exists a natural number nk+1>nk, such that with respect to the basis {ωi}i≥0, the Fourier coefficient of znk+1 has absolute value less than 1.
The last statement can be obtained by letting the support of {vk}k mutually far away. Choose e.g. the distance between Si and Sj to be greater than 3N0, and let Fk be the collection of elements of N∪ {0}between minn∈Sk|n| −N0 and maxn∈Sk|n|+N0. Taking a subsequence if necessary, we can assume that {vk}is a sequence in C such that vk∈span{ωi:i∈Sk}for some finite subsetSk⊂N,||vk| | ≤2,||vk−uk||2≤ 1.
2k andviωk⊥vjωk, for alli,j≥1,i6= j and all 0≤k≤N0, and there is a sequence {Fk} of strictly increasing natural numbers such that L0N. The second line uses the assumption that (xk)k ∈B0∩Mω and the third line uses the fact that L0N. Since there are only finitely many such i's, we can restrict our attention to a single fixi.
In fact, the same conclusion as in Theorem 4.2.1 applies, if we replace the assumption "B⊂Cdiffuse" by "B⊂Cω diffuse".
Some remarks
The maximally unique injection extension for any diffuse subalgebra of generator measure was first shown by Houdayer [Hou15, Theorem 4.1]. Note that the disjoint result as in Theorem 4.2.6 is not true for the arbitrary maximum suitable measure of a factor II1. For example, if the inclusion A⊂M has a nice decomposition, then A does not have the uniqueness property as the measure of the generator in the above conclusion.
Let M=A1∗A0A2 be an amalgamated free product with Aiamenable and A0diffuse, A06=Ai, i=1,2, then A0 can be contained in different maximal amenable subalgebras. Let Λ<Γ be a singular subgroup in the sense of Boutonnet and Carderi ([BC15, Definition 1.2]) and assume that Γ acts on a finite scattered amenable von Neumann algebraQ.
Preliminaries
Planar algebras
The cup subalgebra
Proof of Theorem B
The same proof shows that there exists M>0 such that for anyn>M we havek(1⊗pη)ξnk26ε. Fix a subfactor plane algebra P with modulus δ >1 and denote by A⊂M its associated cup subalgebra. This implies that the A-bimodule L2(M) L2(A) is isomorphic to an infinite direct sum of the rough bimodule.
Consider x∈Mω Aω in the relative commutator of B and y∈M A, where ω is a free ultrafilter inN. Let us show that xy⊥yx. Note, GrP is a slightly dense subgebra∗ of M. Therefore, we can assume that y∈GrP by Kaplanski's density theorem. Let(xn) be a representative ixin ultrapowerMω. We can assume that for every>0 kemixn∈L2(M) L2(A). Let p∈B(L2(A)) be the orthogonal projection on Lm. Therefore, we can assume sexn∈L⊥m⊗V⊗L⊥m for every>0. Lemma 5.2.2 implies thatxny⊥yxn for every>0. This implies thatxy⊥xy.
BC13] R´emi Boutonnet and Alessandro Carderi, Maximal admissible subalgebras of von neumann algebras associated with hyperbolic groups, arXiv (2013), no Bro14] Arnaud Brothier, The beaker subalgebra of alII1 factor given by a subfactor planar algebra is maximally admissible, Pacific J. BW16] Arnaud Brothier and Chenxu Wen, the cup subalgebra has the absorbing tractability property, International J.
CJS14] Stephen Curran, Vaughan Jones and Dimitri Shlyakhtenko, On the symmetric envelope algebra of planar algebra subfactors, Transactions of the American Mathematical Society366(2014), No. GJS11] Alice Guionnet, V Jones and Dimitri Shlyakhtenko, A Semifinite Algebra associated with a subfactor planar algebra, J. Har13] Michael Hartglass, free product of neumann algebras associated with graphs, and guionnet, jones, shlyakhtenko subfactors in infinite depth, J.
HS10] Cyril Houdayer and Dimitri Shlyakhtenko, Strongly solid II1 factors with exotic MASA, International Notices of Research in Mathematics (2010), rnq117. JSW10] Vaughan Jones, Dimitri Shlyakhtenko and Kevin Walker, Orthogonal approach to the subfactor of a planar algebra, Pacific J. Pytlik, Radial functions on free groups and a decomposition of the regular representations into irreducible components, J.
