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UCLA UCLA Electronic Theses and Dissertations 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 by 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 by 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 max- imal 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 iso- morphism, 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 approx- imately finite dimensional subfactor, or equivalently, whether every separable abelian von Neumann subalgebra of a II1 factor could be embedded into some approximately finite di- mensional 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 ! ! precise, he showed that for any free ultrafilter ! on N, any x 2 Ma \ (L(Fn) Ma ), and 2 ! any y1; y2 2 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 subalge- bras. 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 Zsid´o [SZ99], and Ge [Ge96].
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