Pure Mathematics

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This is the collection for the University of Waterloo's Department of Pure Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 6 of 6

Applications of Operator Systems in Dynamics, Correlation Sets, and Quantum Graphs
(University of Waterloo, 2020-07-24) Kim, Se Jin; Davidson, Kenneth; Kennedy, Matthew
The recent works of Kalantar-Kennedy, Katsoulis-Ramsey, Ozawa, and Dykema-Paulsen have demonstrated that many problems in the theory of operator algebras and quantum information can be approached by looking at various subspaces of bounded operators on a Hilbert space. This thesis is a compilation of papers written by the author with various coauthors that apply the theory of operator systems to expand on some of these results. This thesis is split into two parts. In Part I, we start by expanding on the theory of crossed product of operator algebras of Katsoulis and Ramsey. We first develop an analogous crossed product of operator systems. We then reduce two open problems on the uniqueness of universal crossed product operator algebras into one of operator systems and show that it has answers in the negative. In the final chapter of Part I, we generalize results of Kakariadis, Dor On-Salmon, and Katsoulis- Ramsey to characterize which tensor algebras of C*-correspondences admit hyperrigidity. In Part II, we look at synchronous correlation sets, introduced by Dykema-Paulsen as a symmetric form of Tsirelson’s quantum correlation sets. These sets have the distinct advantage that there is a nice C*-algebraic characterization that we present in Chapter 6. We show that the correlation sets coming from the tensor models on finite and infinite dimensional Hilbert spaces cannot be distinguished by synchronous correlation sets and that one can distinguish this set from the correlation sets which arise as limits of correlation sets arising from finite dimensional tensor models. Beyond this, we show that Tsirelson’s problem is equivalent its synchronous analogue, expanding on a result of Dykema-Paulsen. We end the thesis by looking at generalizations of graphs by the ways of operator subspaces of the space of matrices. We construct an analogue of the graph complement and show its robustness by deriving various generalizations of known graph inequalities.
Dilation methods in semigroup dynamics and noncommutative convexity
(University of Waterloo, 2022-08-25) Humeniuk, Adam; Davidson, Kenneth; Kennedy, Matthew
Since seminal work of Stinespring, Arveson, and others, dilation theory has been an indispensable tool for understanding operator algebras. Dilations are fundamental to the representation theory of operator systems and (non-selfadjoint) operator algebras. This thesis is a compilation of three research papers in operator algebras and noncommutative convexity linked by their use of dilations and operator systems. A semicrossed product is a non-selfadjoint operator algebra encoding the action of a semigroup on an operator or C*-algebra. In Chapter 2, we describe the C*-envelopes of a large class of semicrossed products. We prove that, when the positive cone of a discrete lattice ordered abelian group acts on a C*-algebra, the C*-envelope of the associated semicrossed product is a full corner of a crossed product by the whole group. After dilating the semigroup action to an automorphic action of the whole group using a direct product construction, we explicitly compute the Shilov ideal and therefore compute the C*-envelope. This generalizes a result of Davidson, Fuller, and Kakariadis from Z_+^n to the class of all discrete lattice ordered abelian groups. Chapters 3 and 4 present results in noncommutative (or ``matrix") convexity. By the noncommutative Kadison duality of Webster-Winkler and Davidson-Kennedy in the unital setting, and Kennedy-Kim-Manor in the nonunital setting, the category of compact noncommutative (nc) convex sets is dual to the category of operator systems. Thus nc convexity allows a new avenue to study operator systems geometrically. In Chapter 3, we prove a noncommutative generalization of the classical Jensen's Inequality for multivariable nc functions which are convex in each variable separately. The proof involves a sequence of dilations resembling a noncommutative analogue of Fubini's Theorem. This extends a single-variable nc version of Jensen's Inequality of Davidson and Kennedy. We demonstrate an application of the multivariable separate nc Jensen's Inequality to free semicircular systems in free probability. In Chapter 4, we discuss duals of operator systems. Recently, C.K. Ng obtained a nice duality theory for operator systems. Call a (possibly nonunital) operator system S dualizable if its dual S* embeds into B(H) via a complete order embedding and complete norm equivalence. Through the nonunital noncommutative Kadison duality of Kennedy, Kim, and Manor, we characterize dualizability of S using geometric conditions on its associated nc convex quasistate space K in two ways. Firstly, in terms of an nc affine embedding of K into the nc unit ball of a Hilbert space satisfying a certain extension property. Secondly, we show that Ng's characterization is dual to a normality condition between K and the nc cone R_+ K. As applications, we obtain some permanence properties for dualizability, and give a new nc convex-geometric proof of Choi's Theorem.
Nevanlinna-Pick Spaces and Dilations
(University of Waterloo, 2016-07-12) Hartz, Michael Peter; Davidson, Kenneth
The majority of this thesis is devoted to the study of Nevanlinna-Pick spaces and their multiplier algebras. These spaces are Hilbert function spaces in which a version of the Nevanlinna-Pick interpolation theorem from complex analysis holds. Their multiplier algebras occupy an important place at the interface between operator algebras, operator theory and complex analysis. Over the last few years, the classification problem for these algebras has attracted considerable attention. These investigations were pioneered by Davidson, Ramsey and Shalit, who used a theorem of Agler and McCarthy to identify a given multiplier algebra with the restriction of the multiplier algebra of the universal Nevanlinna-Pick space, namely the Drury-Arveson space, to an analytic variety in a complex ball. In this thesis, the classification problem is studied from three different angles. In Chapter 3, we investigate multiplier algebras associated to embedded discs in a complex ball. In particular, we exhibit uncountably many embedded discs which are biholomorphic in a strong sense, but whose multiplier algebras are not isomorphic. Motivated by these issues, we use in Chapter 4 a different approach to the classification problem. Thus, we study the spaces and their multiplier algebras directly without making use of the existence of a universal Nevanlinna-Pick space. This allows us to completely classify the multiplier algebras of a special class of spaces on homogeneous varieties. In Chapter 5, we investigate the complexity of this classification problem from the point of view of Borel complexity theory. In Chapter 6, we show that the Hardy space on the unit disc is essentially the only Nevanlinna-Pick space whose multiplication operators are all hyponormal. The last part of this thesis is concerned with dilations and von Neumann’s inequality. It has been known since the seventies that there are three commuting contractions which do not satisfy von Neumann’s inequality. In Chapter 7, we show that every tuple of commuting contractions which forms a multivariable weighted shift dilates to a tuple of commuting unitaries and hence satisfies von Neumann’s inequality, thereby providing a positive answer to a question of Shields and Lubin from 1974.
Regular Dilation on Semigroups
(University of Waterloo, 2018-08-07) Li, Boyu; Davidson, Kenneth
Dilation theory originated from Sz.Nagy's celebrated dilation theorem which states that every contractive operator has an isometric dilation. Regular dilation is one of many fruitful directions that aims to generalize Sz.Nagy's dilation theorem to the multi-variate setting. First studied by Brehmer in 1961, regular dilation has since been generalized to many other contexts in recent years. This thesis is a compilation of my recent study of regular dilation on various semigroups. We start from studying regular dilation on lattice ordered semigroups and shows that contractive Nica-covariant representations are regular. Then, we consider the connection between regular dilation on graph products of N, which uni es Brehmer's dilation theorem and the well-known Frazho-Bunce-Popescu's dilation theorem. Finally, we consider regular dilation on right LCM semigroups and study its connection to Nica-covariant dilation.
Strong Morita Equivalence and Imprimitivity Theorems
(University of Waterloo, 2016-09-15) Kim, Se-Jin; Davidson, Kenneth
The purpose of this thesis is to give an exposition of two topics, mostly following the books \cite{R & W} and \cite{Wil}. First, we wish to investigate crossed product $C^*$-algebras in its most general form. Crossed product $C^*$-algebras are $C^*$-algebras which encode information about the action of a locally compact Hausdorff group $G$ as automorphisms on a $C^*$-algebra $A$. One of the prettiest example of such a dynamical system that I have observed in the wild arises in the gauge-invariant uniqueness theorem \cite{Rae}, which assigns to every $C^*$-algebra $C^*(E)$ associated with a graph $E$ a \emph{gauge action} of the unit circle $\T$ to automorphisms on $C^*(E)$. Group $C^*$-algebras also arise as a crossed product of a dynamical system. I found crossed products in its most general form very abstract and much of its constructions motivated by phenomena in a simpler case. Because of this, much of the initial portion of this exposition is dedicated to the action of a discrete group on a unital $C^*$-algebra, where most of the examples are given. I must admit that I find calculations of crossed products when one has an indiscrete group $G$ acting on our $C^*$-algebra daunting except under very simple cases. This leads to our second topic, on imprimitivity theorems of crossed product $C^*$-algebras. Imprimitivity theorems are machines that output (strong) Morita equivalences between crossed products. Morita equivalence is an invariant on $C^*$-algebras which preserve properties like the ideal structure and the associated $K$-groups. For example, no two commutative $C^*$-algebras are Morita equivalent, but $C(X) \otimes M_n$ is Morita equivalent to $C(X)$ whenever $n$ is a positive integer and $X$ is a compact Hausdorff space. Notice that Morita equivalence can be used to prove that a given $C^*$-algebra is simple. All this leads to our concluding application: Takai duality. The set-up is as follows: we have an action $\alpha$ of an abelian group $G$ on a $C^*$-algebra $A$. On the associated crossed product $A \rtimes_\alpha G$, there is a dual action $\Hat{\alpha}$ from the Pontryagin dual $\Hat{G}$. Takai duality states that the iterated crossed product $(A \rtimes_\alpha G) \rtimes \Hat{G}$ is isomorphic to $A \otimes \calK(L^2(G))$ in a canonical way. This theorem is used to show for example that all graph $C^*$-algebras are nuclear or to establish theorems on the $K$-theory on crossed product $C^*$-algebras.
Techniques in operator algebras: classification, dilation and non-commutative boundary theory
(University of Waterloo, 2017-08-11) Dor On, Adam; Davidson, Kenneth
In this thesis we bring together several techniques in the theory of non-self-adjoint operator algebras and operator systems. We begin with classification of non-self-adjoint and self-adjoint operator algebras constructed from C*-correspondence and more specifically, from certain generalized Markov chains. We then transitions to the study of non-commutative boundaries in the sense of Arveson, and their use in the construction of dilations for families of operators arising from directed graphs. Finally, we discuss connections between operator systems and matrix convex sets and use dilation theory to obtain scaled inclusion results for matrix convex sets. We begin with classification of non-self-adjoint operator algebras. In Chapter 3 we solve isomorphism problems for tensor algebras arising from weighted partial dynamical systems. We show that the isometric isomorphism and algebraic / bounded isomorphism problems are two distinct problems, that require separate criteria to be solved. Our methods yield an alternative solution to Arveson's conjugacy problem, first solved by Davidson and Katsoulis. A natural bridge between operator algebras / systems and C*-algebras is the C*-envelope, which is a non-commutative generalization of the notion of \emph{Shilov boundary} from the theory of function algebras. In Chapter 4 we investigate C*-envelopes arising from the operator algebras of stochastic matrices via subproduct systems. We identify and classify these non-commutative boundaries in terms of the matrices, and exhibit new examples of C*-envelopes of non-self-adjoint operator algebras arising from a subproduct system construction. In Chapter 5 we apply Arveson's non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We also obtain a generalization of our dilation result to the context of colored directed graphs, which relies on the complete injectivity of amalgamated free products of operator algebras. The last part of this thesis is devoted to the interplay between matrix convex sets and operator systems, inspired by the work of Helton, Klep and McCullough. In Chapter 6 we establish a functorial duality between finite dimensional operator systems and matrix convex sets that recovers many interpolation results of completely positive maps in the literature. We proceed to investigate dual, minimal and maximal matrix convex sets, and relate them to dilation theory, scaled inclusion results and operator systems. By using dilation theory, we provide \emph{rank-independent} optimal scaled inclusion results for matrix convex sets satisfying a certain symmetry condition, and prove the existence of an essentially unique \emph{self-dual} matrix convex set.

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Browsing Pure Mathematics by Author "Davidson, Kenneth"