On the rational functions in non-commutative random variables Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaften der Fakult¨atf¨urMathematik und Informatik der Universit¨atdes Saarlandes Sheng Yin Saarbr¨ucken, 2019 Datum der Verteidigung: 27. Januar 2020 Dekan: Prof. Dr. Sebastian Hack Pr¨ufungsausschuss: Vorsitzender: Prof. Dr. Gabriela Weitze-Schmithsen Berichterstatter: Prof. Dr. Roland Speicher Prof. Dr. Moritz Weber Prof. Dr. Thomas Schick Akademischer Mitarbeiter Dr. Marwa Banna Abstract This thesis is devoted to some problems on non-commutative rational functions in non- commutative random variables that come from free probability theory and from random matrix theory. First, we will consider the non-commutative random variables in tracial W ∗-probability spaces, such as freely independent semicircular and Haar unitary random variables. A natural question on rational functions in these random variables is the well-definedness question. Namely, how large is the family of rational functions that have well-defined evaluations for a given tuple X of random variables? Note that for a fixed rational function r, the well-definedness of its evaluation r(X) depends on the interpretation of the invertibility of random variables. This is because the invertibility of a random variable in a tracial W ∗-probability space (or an operator in a finite von Neumann algebra) can be also considered in a larger algebra, i.e., the ∗-algebra of affiliated operators. One of our goals in this thesis to show some criteria that characterize the well-definedness of all rational functions in the framework of affiliated operators. In particular, one of these criteria is given by a homological-algebraic quantity on non-commutative random variables. We will also show that some notions provided by free probability are related to this quantity. So we can finally answer the well-definedness question via these related notions from free probability. Those criteria for the well-definedness of rational functions are actually intrinsic con- nected to the Atiyah conjecture or Atiyah property. We will explore these connections between the Atiyah property and our question on the well-definedness of rational func- tions. In particular, we will present a result to show a connection between the so-called strong Atiyah property and the invertibility of evaluations of rational functions. In this result, the evaluation of a rational function at a tuple of random variables may not be well-defined, but it is always invertible as an affiliated operator once it is well-defined. In the last part of this thesis, we will turn to the questions on rational functions in random matrices. Besides the well-definedness problem for rational functions in random matrices, we will also address the convergence problem for rational functions in random matrices. Due to the unstableness of the convergence in distribution, we will limit our ran- dom matrices to the ones that strongly converge in distribution and our rational functions to the ones that have bounded evaluations. We will show that both the well-defineness and the convergence problem have an affirmative answer under such conditions. i Abstrakt Diese Doktorarbeit widmet sich Problemen aus der freien Wahrscheinlichkeitstheorie und der Zufallsmatrizentheorie ¨uber nicht-kommutative rationale Funktionen in nicht- kommutativen Zufallsvariablen. Zun¨achst betrachten wir nicht-kommutative Zufallsvariable in endlichen (d.h. tra- cial) W ∗-Wahrscheinlichkeitsr¨aumen,wie z.B. freie Halbkreiselemente oder freie Haar unit¨areZufallsvariable. Eine nat¨urliche Frage ¨uber rationale Funktionen in solchen Zu- fallsvariablen ist die nach der Wohldefiniertheit. Genauer gesagt, wie groß ist die Familie von rationalen Funktionen, die eine wohldefinierte Auswertung f¨urein gegebenes Tu- pel X von Zufallsvariablen haben? Man muss beachten, dass f¨ureine feste rationale Funktion r die Wohldefiniertheit der Auswertung r(X) von der Interpretation der In- vertierbarkeit von Zufallsvariablen abh¨angt. Dies liegt daran, dass die Invertierbarkeit einer Zufallsvariablen in einem endlichen W ∗-Wahrscheinlichkeitsraum (oder eines Oper- ators in einer endlichen von Neumann Algebra) in einer gr¨oßerenAlgebra, n¨amlich der ∗-Algebra der affiliierten Operatoren, betrachtet werden kann. Eines der Ziele dieser Doktorarbeit ist es Kriterien zu finden, welche die Wohldefiniertheit von allen rationalen Funktionen im Rahmen von affiliierten Operatoren charakterisieren. Insbesondere wird eines dieser Kriterien durch eine homologisch-algebraische Gr¨oßevon nicht-kommutativen Zufallsvariablen gegeben sein. Wir werden auch zeigen, dass diese Gr¨oßemit verschiede- nen Gr¨oßenaus der freien Wahrscheinlichkeitstheorie zusammenh¨angt. So werden wir schließlich die Wohldefiniertheitsfrage durch diese Gr¨oßenaus der freien Wahrscheinlich- heitstheorie beschreiben. Diese Kriterien f¨urdie Wohldefiniertheit von rationalen Funktionen h¨angeninh¨arent mit der Atiyah Vermutung/Eigenschaft zusammen. Wir werden dieser Verbindung zwis- chen der Atiyah Eigenschaft und unserer Frage nach der Wohldefiniertheit von rationalen Funktionen auf den Grund gehen. Insbesondere werden wir den Zusammenhang zwis- chen der sogennanten starken Atiyah Eigenschaft und der Wohldefiniertheitsfrage kl¨aren. Dabei mag die Auswertung einer rationalen Funktion an einem Tupel von Zufallsvari- ablen nicht wohldefiniert sein, aber sofern sie wohldefiniert ist, ist sie immer invertierbar als affiliierter Operator. Im letzten Teil der Doktorarbeit wenden wir uns Fragen zu rationalen Funktionen in Zufallsmatrizen zu. Neben dem Wohldefiniertheitsproblem f¨urrationale Funktionen in Zufallsmatrizen werden wir auch das Konvergenzproblem in dem Rahmen ansprechen. Wegen der Instabilit¨atder Konvergenz in Verteilung schr¨anken wir uns dabei auf Zu- fallsmatrizen ein, welche stark in Verteilung konvergieren, und betrachen nur rationale Funktionen, welche beschr¨ankteAuswertungen besitzen. Wir werden zeigen, dass under solchen Voraussetzungen sowohl die Wohldefiniertheitsfrage als auch die Konvergenzfrage eine positive Antwort hat. iii Acknowledgments First and foremost, I would like to thank my supervisor, Prof. Dr. Roland Speicher, for his guidance and support during the last four years. I am deeply grateful to him for the opportunity to have him as my Ph.D. supervisor and to learn free probability theory from him. I learnt from him not only from the lectures and talks that he gave but also those discussions and chats that happened during the everyday lunchtime and coffee break. It was a great experience to be a member of his inspiring and supporting working group. Besides, a major part of this thesis is based on a joint project together with him and Dr. Tobias Mai. This project progressed with innumerous discussions that we had during the last two years. They contained the exciting ones as well as the fruitless but inspiring ones, which both I appreciate very much. It was a great collaboration experience and I enjoyed very much working with him. I would also like to express my sincere gratitude towards Dr. Tobias Mai. This thesis also could not be realized in its present form without him as a coauthor in the aforementioned joint project. Furthermore, he always answered my questions with great patience and excellent explanations since the very beginning of my Ph.D. study. My sincere thanks should also go to all present and former members of our research group. I would particularly like to thank Prof. Dr. Moritz Weber for all his help and advice that he offered me in the last four years (and also for exchanging with me his collections of coins). My thanks are also due to Miguel Rodriguez, a friend and an office buddy, for all the time we spent together, in particular those time in travelings. I also want to thank a former member, Dr. Guillaume C´ebron,for the inspiring discussions with him that helped a lot on my first project and also for his help and advice on my application for the postdoc position. I would also like to thank Prof. Dr. Uwe Franz and Prof. Dr. Christian Le Merdy for their valuable mentoring on my master projects and also for their excellent lectures on C∗-algebra theory. My heartfelt thanks also go to Prof. Dr. Quanhua Xu for always being there for helping and encouraging me since I was an undergraduate student at Wuhan University. It was his crystal clear and excellent lectures in my third year of bachelor's degree that inspired my interest in functional analysis and influenced my choices for my mathematics career. Furthermore, I want to thank Prof. Dr. Thomas Schick that he kindly agreed to review this thesis. My special thanks go to Mr. Bo Wang in my high school, who enlightened the sparkle of my passion for the research work. I would also like to thank Dr. Agnieszka Will for her earnest help on my English writing when I was writing this thesis. Besides, I want to thank Ziwei He and Han Du, my friends and my roommates, for all the fun (and also the cooking) we had together in the last years. Last but not least, I would like to thank my parents and my sister for always being there for me. Particularly, I thank for their understanding and support of my study aboard in the last years. Besides, I also want to thank my little nephew for his affection for me even though he barely saw me one week per year since he was born. v Contents Introduction . 1 Convergence problem for rational functions . 3 Atiyah property and zero divisors . 4 Chapter I. An introduction to free probability theory . 7 I.1. Non-commutative probability spaces . 8 I.2. Non-commutative distributions . 12 I.3. Atoms and zero divisors . 17 I.4. Free independence . 20 I.5. Unbounded random variables . 23 Chapter II. Asymptotic limits of Random Matrices . 29 II.1. Asymptotic behavior of random matrices . 29 II.2. Asymptotic freeness and strong asymptotic freeness . 34 Chapter III. Inner rank . 37 III.1. Some basics of the inner rank . 37 III.2. Sylvester domains . 41 III.3. Linear matrices . 44 III.4. Central eigenvalues . 45 Chapter IV. Free field . 49 IV.1. Non-commutative rational expressions . 50 IV.2. Non-commutative rational functions . 52 IV.3.