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UCLA Electronic Theses and Dissertations UCLA UCLA Electronic Theses and Dissertations Title Spectral Gap Rigidity and Unique Prime Decomposition Permalink https://escholarship.org/uc/item/1j63188h Author Winchester, Adam Jeremiah Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California University of California Los Angeles Spectral Gap Rigidity and Unique Prime Decomposition A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Adam Jeremiah Winchester 2012 c Copyright by Adam Jeremiah Winchester 2012 Abstract of the Dissertation Spectral Gap Rigidity and Unique Prime Decomposition by Adam Jeremiah Winchester Doctor of Philosophy in Mathematics University of California, Los Angeles, 2012 Professor Sorin Popa, Chair We use malleable deformations combined with spectral gap rigidity theory, in the framework of Popas deformation/rigidity theory to prove unique tensor product decomposition results for II1 factors arising as tensor product of wreath product factors and free group factors. We also obtain a similar result regarding measure equivalence decomposition of direct products of such groups. ii The dissertation of Adam Jeremiah Winchester is approved. Dimitri Shlyakhtenko Yehuda Shalom Jens Palsberg Sorin Popa, Committee Chair University of California, Los Angeles 2012 iii To All My Friends iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Preliminaries ::::::::::::::::::::::::::::::::::::: 8 2.1 Intro to von Neumann Algebras . 8 2.2 von Neumann Subalgebras . 13 2.3 GNS and Bimodules . 14 2.4 The Basic Construction . 17 2.5 S-Malleable Deformations . 18 2.6 Spectral Gap Rigidity . 20 2.7 Intertwining By Bimodules . 20 3 Relative Amenability :::::::::::::::::::::::::::::::: 22 3.1 Definition . 22 3.2 Locating Subalgebras . 23 4 Wreath Product Factors :::::::::::::::::::::::::::::: 25 4.1 Deformation of Wreath Products . 25 4.2 Spectral Gap of Wreath Products . 26 4.3 Locating Subalgebras . 28 4.4 Prime Factorization . 29 5 Free Group Factors ::::::::::::::::::::::::::::::::: 32 5.1 Deformation of Free Group Factors . 32 5.2 Spectral Gap in Free Group Factors . 33 v 5.3 Locating Subalgebras . 35 5.4 Prime Factorization . 36 References ::::::::::::::::::::::::::::::::::::::::: 38 vi Acknowledgments I want to thank my mother Sharon Winchester for all her love, support, and friendship and my father Mark Winchester who has taught me that you can overcome any difficulties as long as you never give up. My beautiful girlfriend Tiffany Davis deserves more thanks than I can give for always being there and giving me confidence. This dissertation would not be possible without the guidance and kindness of Sorin Popa. Without his direction and knowledge, I would never have been able to complete this project. I want to give special thanks to my friend Owen Sizemore because I am lucky to have worked with and learned from him. Chapters 3 and 4 come from joint work completed with Owen Sizemore preprinted in [SW11] on a problem suggested by Sorin Popa. The Functional Analysis Group at UCLA has provided a rich, welcoming, and inspirational environment. I could not have begun the process of research without the help and friendship of fellow graduate students like Jane, Brady, Mike, Hem, and Neel. I also want to thank the Renaldo's and the Avchen's for treating me like family and for all their help. Finally, I thank all my friends, family, and teachers. vii Vita 2004 B.S. (Mathematics), UNLV, Las Vegas, Nevada. 2006 M.A. (Mathematics), UCLA, Los Angeles, California. 2005{2009 VIGRE Fellow, Mathematics Department, UCLA. 2005{2011 Teaching / Research Assistant, Mathematics Department, UCLA. Publications and Presentations Adam Winchester. \Busemann Points of infinite Graphs." Paper presented at Mathfest 2003, Boulder, Colorado, 2003. Corran Webster and Adam Winchester, \Boundaries of Hyperbolic Metric Spaces" Pac. J. of Math. 221(1):149-158, 2005. Corran Webster and Adam Winchester. \Busemann Points of Infinite Graphs" Trans. Amer. Math. Soc. 358:4209-4224, 2006. viii CHAPTER 1 Introduction The elusive nature and structure of II1 factors make these algebras an interesting and com- pelling area of research. A von Neumann Algebra is simply a unital ∗-subalgebra of bounded operators on a Hilbert space closed in the weak operator topology. This apparently simple definition leads to a rich classification theory. For an example of an abelian von Neumann Algebra, consider the probability space (X; µ). The essentially bounded functions L1(X; µ) form an abelian von Neumann algebra acting on the Hilbert space L2(X; µ). Matrices over the complex numbers also form basic examples of von Neumann algebras, but they are highly non-commutative with a trivial center, the center being all the elements that commute with every element of the algebra. A factor is a von Neumann algebra with trivial center, meaning that its center consists only of scalars multiples of the identity. A basic example of a factor is the set of all bounded operators on an infinite dimensional Hilbert space, denoted B(H). II1 factors are infinite dimensional factors equipped with a finite normal faithful trace, usually denoted τ, and the structure and decomposability of these factors are the main focus of this thesis. Many von Neumann algebras are constructed from others. The most prominent source for these types of examples of von Neumann algebras are crossed products formed by a trace preserving action σ of a countable group Γ on another von Neumann algebra A,Γ yσ A. The resulting Algebra AoΓ is the von Neumann algebra acting on L2(A, τ)⊗l2(Γ) generated by A and Γ in a way that makes the action of σ inner. In the case that A =L 1(X) and µ is measure preserving, this is the Murray/von Neumann group measure space construction which is a II1 factor whenever the action is free and ergodic. When X is a one point space, 1 we obtain the group von Neumann algebra of Γ, denoted L(Γ). This is also the von Neumann algebra generated by the left regular representation of Γ which is a II1 factor whenever the group has infinite conjugacy classes (ICC). This construction is essential to the theory and classification of II1 factors. Given two von Neumann algebras M1 acting on H1 and M2 acting on H2 we can form their tensor product M1⊗M2 as the von Neumann algebra generated by their algebraic tensor acting on the tensor product of their Hilbert spaces. If M1 and M2 are II1 factors then M1⊗M2 will also be a II1 factor. We can also form the their free product M1 ∗ M2 as defined in [VDN92]. The question now becomes \how do we classify II1 factors?" This has been a long term goal in the study of these factors. A major landmark was the realization, due to Connes [Con76], that all amenable such algebras are isomorphic. For example, any two ICC amenable groups will have isomorphic group von Neumann algebras. Also, if two amenable groups act on a measure space, the resulting group von Neumann algebras will be isomorphic. N1 These are also isomorphic to the hyperfinite II1 factor R = i=1 M2(C), which is the direct limit of finite dimensional subalgebras. However, in the non-amenable realm, there is much more variety, and a striking classification theory has developed. Not all II1 factors are isomorphic, as Murray and von Neumann proved that the free group factors L(Fn) are not isomorphic to R. To show that this classification theory is indeed nontrivial, we mention that D. McDuff was able to show that there are actually uncountably many non-isomorphic II1 factors [McD69]. Now that we can construct II1 factors with crossed products, tensor products, and free products, an essential question in the classification of these factors is whether or not we can construct a von Neumann algebra in more than one way. Can we take a group von Neumann algebra and deconstruct it as a tensor product? Or, if we have a II1 factor that we know to be a free product of two II1 factors, is it also possible to be the tensor product of two (possibly different) II1 factors? For example, Voiculescu showed that free group factors have no Cartan subalgebras (maximal abelian, regular subalgebras) proving that they cannot be 2 1 realized as a group measure space construction,L (X) o Γ [Voi96]. In this vein, we study whether certain factors can be written as a tensor product in two distinct ways; such results go back to the study of prime factors, (i.e. factors which cannot be written as the tensor product of two other II1 factors). The first result was obtained by Sorin Popa in [Pop83], where he showed that the group von Neumann algebra of an uncountable free group is prime. Later, in [Ge98], Ge proves that all group factors coming from finitely generated free groups are prime. Using C∗ techniques, this was greatly generalized by Ozawa [Oza04] to show that all I.C.C Gromov hyperbolic groups give rise to prime factors. The structure of II1 factors has been enormously elucidated in the past ten years due to Popa's deformation/rigidity theory. This theory has led to powerful rigidity results that unravel the complexities of non-amenable factors. For an example of the potential of these techniques, Popa proved the existence of cocyle superrigid group actions [Pop07a] which shows that cocycles of certain actions of \rigid" groups Γ y X with values in an arbitrary discrete group Λ are equivalent to group morphisms. The ability to deform a subalgebra can be thought of as \softness," and the resistance to the deformation, it's rigidity, can be thought of as \hardness." The success of this defor- mation/rigidity theory comes the \softness" of the II1 factor fighting against its \hardness" and taking advantage of this \local" give and take of the subalgebras.
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