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UNIVERSITY of CALIFORNIA Los Angeles on Bi-Free Probability And UNIVERSITY OF CALIFORNIA Los Angeles On bi-free probability and free entropy. A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Ian Lorne Charlesworth 2017 c Copyright by Ian Lorne Charlesworth 2017 ABSTRACT OF THE DISSERTATION On bi-free probability and free entropy. by Ian Lorne Charlesworth Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Dimitri Y. Shlyakhtenko, Chair Free probability is a non-commutative analogue of probability theory. Recently, Voiculescu has introduced bi-free probability, a theory which aims to study simultaneously \left" and \right" non-commutative random variables, such as those arising from the left and right regular representations of a countable group. We introduce combinatorial techniques to characterise bi-free independence, generalising results of Nica and Speicher from the free setting to the bi-free setting. In particular, we develop the lattice of bi-non-crossing partitions which is deeply tied to the action of bi-freely independent random variables on a free product space. We use these techniques to show that a conjecture of Mastnak and Nica holds, and bi-free independence is equivalent to the vanishing of mixed bi-free cumulants vanishing. Moreover, we extend the theory into the operator-valued setting, introducing operator-valued cumulants which correspond to bi-freeness with amalgamation in the same way. Finally, we investigate regularity problems in algebras of non-commuting random variables. Using operator theoretic techniques show that in an algebra generated by non-commutative random variables which admit a dual system, any self-adjoint element with spectral measure singular with respect to Lebesgue measure is a multiple of 1. We are also able to slightly improve on prior results in the literature and show that any non-constant self-adjoint polynomial evaluated at a set of non-commutative random variables which are free, algebraic, and have finite free entropy must produce a variable with finite free entropy. ii The dissertation of Ian Lorne Charlesworth is approved. Milos D. Ercegovac Sorin Popa Dimitri Y. Shlyakhtenko, Committee Chair University of California, Los Angeles 2017 iii For Sue, Dave, Laurel, and Kate. iv TABLE OF CONTENTS 1 Introduction. ::::::::::::::::::::::::::::::::::::: 1 1.1 Preliminaries. .1 1.1.1 Probability. .2 1.1.2 Free probability. .4 1.1.3 Free cumulants. .7 1.1.4 The incidence algebra of non-crossing partitions. .9 1.1.5 Bi-free probability. 11 1.1.6 Free convolutions. 14 1.1.7 Fock space. 16 1.1.8 Operator-valued free probability. 17 1.1.9 Free entropy and regularity. 18 2 Bi-free independence. :::::::::::::::::::::::::::::::: 23 2.1 Some combinatorial structures. 23 2.1.1 Bi-non-crossing partitions. 23 2.1.2 Shaded LR-diagrams. 24 2.2 Computing joint moments of bi-free variables. 28 2.3 The incident algebra on bi-non-crossing partitions. 32 2.3.1 Multiplicative functions. 33 2.4 Unifying bi-free independence. 35 2.4.1 Summation considerations. 35 2.4.2 Bi-free independence is equivalent to combinatorial bi-free independence. 40 2.4.3 Voiculescu's universal bi-free polynomials. 41 v 2.5 An alternating moment condition for bi-free independence. 43 2.5.1 The equivalence. 45 2.5.2 Conditional bi-freeness. 47 2.6 Bi-free multiplicative convolution. 50 2.6.1 The Kreweras complement. 51 2.6.2 Cumulants of products. 52 2.6.3 Multiplicative convolution. 53 2.7 An operator model for pairs of faces. 54 2.7.1 Nica's operator model. 55 2.7.2 Skeletons corresponding to bi-non-crossing partitions. 56 2.7.3 A construction. 57 2.7.4 The operator model for pairs of faces. 63 2.8 Additional cases of bi-free independence. 68 2.8.1 Conjugation by a Haar pair of unitaries. 68 2.8.2 Bi-free independence for bipartite pairs of faces. 70 3 Amalgamated bi-free probability. :::::::::::::::::::::::: 72 3.1 Bi-free families with amalgamation. 72 3.1.1 Concrete structures for bi-free probability with amalgamation. 72 3.1.2 Abstract structures for bi-free probability with amalgamation. 77 3.1.3 Bi-free families of pairs of B-faces. 81 3.2 Operator-valued bi-multiplicative functions. 82 3.2.1 Definition of bi-multiplicative functions. 82 3.3 Bi-free operator-valued moment function is bi-multiplicative. 86 3.3.1 Definition of the bi-free operator-valued moment function. 86 vi 3.3.2 Verification of Property (iii) from Definition 3.2.1 for E......... 88 3.3.3 Verification of Property (iv) from Definition 3.2.1 for E......... 92 3.4 Operator-valued bi-free cumulants. 99 3.4.1 Cumulants via convolution. 99 3.4.2 Convolution preserves bi-multiplicativity. 100 3.4.3 Bi-moment and bi-cumulant functions. 101 3.4.4 Vanishing of operator-valued bi-free cumulants. 104 3.5 Universal moment polynomials for bi-free families with amalgamation. 106 3.5.1 Bi-freeness with amalgamation through universal moment polynomials. 106 3.6 Additivity of operator-valued bi-free cumulants. 112 3.6.1 Equivalence of bi-freeness with vanishing of mixed cumulants. 112 3.6.2 Moment and cumulant series. 113 3.7 Amalgamated versions of results from Chapter 2. 115 3.7.1 Amalgamated vaccine. 115 3.7.2 Amalgamated multiplicative convolution. 117 4 An investigation into regularity and free entropy. :::::::::::::: 118 4.1 Regularity results. 118 4.1.1 Useful results from the literature. 118 4.2 Algebraicity and finite entropy. 119 4.2.1 Properties of spectral measures. 120 4.3 Bi-free unitary Brownian motion. 124 4.3.1 Free Brownian motion. 125 4.3.2 The free liberation derivation. 127 4.3.3 A bi-free analogue to the liberation derivation. 128 vii 4.3.4 Bi-free unitary Brownian motion. 131 viii ACKNOWLEDGMENTS I would like to thank my advisor, Dima Shlyakhtenko, for years of invaluable advice and support. Without his guidance and encouragement, I would never have attained the expertise and will necessary to complete this dissertation. I am in his debt for all the help he has given me, with respect to mathematics and to my future career. I could not have asked for a better advisor. I am grateful to Ben Hayes, Brent Nelson, and Paul Skoufranis, in whose footsteps I have followed. They have served as great role models for me in graduate school, providing me with friendship, mentoring, and collaboration. I wish to extend thanks as well to the many people who have provided me variously with advice, opportunities, suggestions, insight, and camaraderie: Andreas Aaserud, Michael Anshelevich, Ken Dykema, Todd Kemp, Brian Leary, Dave Penneys, Jesse Peterson, Sorin Popa, and Dan Voiculescu. Last but by no means least, I am thankful to the many people who have provided me love and support over the years which has helped me stay motivated and focused: my parents, Sue and Dave; my sisters Laurel and Kate; and Nadia and Rami, Sean and Becky, Fred and Steph, Brittany, Jason, Alec, and Danny. The content of [9] is included in Chapter 2; this paper was joint work with Brent Nelson and Paul Skoufranis and arose from useful discussions we shared while at the University of California, Los Angeles. Thus some parts of Chapter 2 were published in the Cana- dian Journal of Mathematics, http://dx.doi.org/10.4153/CJM-2015-002-6. [8] has been distributed between Chapters 2 and 3, and arose from continuations of those same dis- cussions. The final publication is available at Springer via http://dx.doi.org/10.1007/ s00220-015-2326-8. Some of the arguments from [10] appear in Chapter 4: the main re- sult of that paper was discovered by Dima Shlyakhtenko and has not been included; the included pieces were arguments I discovered with his advice which were also included in the paper. The full publication appears at http://dx.doi.org/10.1016/j.jfa.2016.05.001. Finally, the preprint [7] contains content from Chapters 2 and 4. I have received support from the Natural Sciences and Engineering Research Council of ix Canada in the form of scholarships, and the National Science Foundation through grants made to my advisor. I would also like to thank the Hausdorff Research Institute for Math- ematics at the Universit¨atBonn for providing an excellent research environment during the von Neumann Algebras trimester program in the summer of 2016. x VITA 2007-2012 B.Math. Honours Pure Mathematics & Honours Computer Science, Uni- versity of Waterloo, Waterloo, Ontario, Canada. 2012-2016 Natural Sciences and Engineering Research Council of Canada Postgradu- ate Scholarship. 2016-2017 U.C.L.A. Dissertation Year Fellowship. PUBLICATIONS A. Nica, I. Charlesworth, and M. Panju. Analyzing Query Optimization Process: Portraits of Join Enumeration Algorithms. IEEE 28th International Conference on Data Engineering (2012). I. Charlesworth, B. Nelson, and P. Skoufranis. On Two-faced Families of Non-commutative Random Variables, Canadian Journal of Mathematics 67 (2015), no. 6, 1290-1325. I. Charlesworth, B. Nelson, and P. Skoufranis. Combinatorics of Bi-Freeness with Amalga- mation, Communications in Mathematical Physics 338 (2015), 801-847. I. Charlesworth and D. Shlyakhtenko. Free Entropy Dimension and Regularity of Non- commutative Polynomials, Journal of Functional Analysis 271 (2016), no. 8, 2274-2292. I. Charlesworth. An alternating moment condition for bi-freeness. arXiv preprint 1611.01262 (2016). xi I. Charlesworth, K. Dykema, F. Sukochev, and D. Zanin. Joint spectral distributions and invariant subspaces for commuting operators in a finite von Neumann algebra. arXiv preprint 1703.05695 (2017). xii CHAPTER 1 Introduction. Free probability is a non-commutative probability theory, introduced by Voiculescu in the 1980's.
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