On Maximal Amenable Subalgebras of Amalgamated Free Product von Neumann Algebras

49 Pages·2015·0.3 MB·English

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
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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
(cid:13)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 N , and B is a von Neumann subalgebra of N with the property that no
1 1
corner of P embeds into B inside N in the sense of Popa’s intertwining by bimodules, then
1
weconcludethatP isamaximalamenablesubalgebraoftheamalgamatedfreeproductofN
1
and N over B, where N is another finite von Neumann algebra containing B. To this end,
2 2
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 = N to be amenable.
1
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
favoritepeoplethatI’vemethere; andtoMiranda, Johann, Siddharth, Bryon, andStephanie
for their contributions to making my time in this city memorable. I am truly indebted to all
of you.
vi
Vita
2008–2009 TeachingAssistant, MathematicsDepartment, CarnegieMellonUniversity,
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 II factor R,
1
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 II . Moreover,
1
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 II factor can be embedded into some approx-
1
imately finite dimensional subfactor, or equivalently, whether every separable abelian von
Neumann subalgebra of a II factor could be embedded into some approximately finite di-
1
mensional subfactor. In 1983, Popa [Pop83a] provided a negative answer to the problem by
constructing an abelian subalgebra of a II factor that is a maximal amenable subalgebra,
1
and hence a maximal approximately finite dimensional subalgebra by Connes’ theorem. In
particular, he proved that the abelian von Neumann subalgebra M of L(F ) generated by a
a n
1