Pure Mathematics

Permanent URI for this collectionhttps://uwspace.uwaterloo.ca/handle/10012/9932

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).

Waterloo faculty, students, and staff can contact us or visit the UWSpace guide to learn more about depositing their research.

Browse

Now showing 1 - 7 of 7

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.
Classification of Finitely Generated Operator Systems
(University of Waterloo, 2018-01-22) Hamzo, Chadi; Kennedy, Matthew
For the past few decades operator systems and their C*-envelopes have provided an invaluable tool for studying the theory of C*-algebras and positive maps. They provide the natural context in which to study the theory of completely positive maps. Furthermore, many of the important open problems in quantum information theory have found equivalent formulations in terms of operator systems. The question of the classification of operator systems and computing their C*-envelopes have been the center of much interest. Borrowing from the theory of representations of commutative C*-algebras by affine maps, we construct a new tool for classifying certain types of finitely generated operator systems. Using this tool, we show that all the information regarding such operator systems is usually encoded in the joint spectra of their generating operators. Using this tool we completely classify operator systems generated by finitely many normal operators. We also provide a different proof for the classification theorem of operator systems generated by a unitary with spectrum size different that 4. Furthermore, we settle the classification problem for operator systems generated by a single unitary with four points in its spectrum. In addition, we compute the C*-envelopes of such operator systems. Furthermore, we apply this tool to the classification problem of those operator systems generated by a unilateral shift with arbitrary multiplicity or by an isometry and we compute their C*-envelopes.
Coefficient spaces arising from locally compact groups
(University of Waterloo, 2020-08-17) Tanko, Zsolt; Forrest, Brian; Kennedy, Matthew
This thesis studies two disjoint topics involving coefficient spaces and algebras associated to locally compact groups. First, Chapter 3 investigates the connection between amenability and compactness conditions on locally compact groups and the homology of the Fourier algebra when viewed as a completely contractive Banach algebra. We provide characterizations of relative 1-projectivity, 1-flatness, and 1-biflatness of the Fourier algebra. These allow us to deduce a new hereditary property for an amenability condition, namely that inner amenability passes to closed subgroups. Our techniques also allow us to show that inner amenability coincides with Property (W) and to settle a conjecture regarding the cb-multiplier completion of the Fourier algebra. Our second theme is coefficient spaces arising from $L^p$-representations of locally compact groups. Chapter 4 is motivated by a question of Kaliszewski, Landstad, and Quigg regarding whether two coefficient space constructions coincide. We are able to provide a positive answer in special cases, in particular for the group $SL(2,\mathbb{R})$. We establish several results regarding the non-separability of algebras related to the $L^p$-Fourier algebras, and characterize when these algebras have a bounded approximate identity.
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.
Exactness and Noncommutative Convexity
(University of Waterloo, 2022-09-02) Manor, Nicholas; Kennedy, Matthew; Spronk, Nico
This thesis studies two separate topics in connection to operator systems theory: the dynamics of locally compact groups, and noncommutative convex geometry. In Chapter 1 we study exactness of locally compact groups as it relates to C*-exactness, i.e., the exactness of the reduced C*-algebra. It is known that these two properties coincide for discrete groups. The problem of whether this equivalence holds for general locally compact groups has recently been reduced by Cave and Zacharias to the case of totally disconnected unimodular groups. We prove that the equivalence does hold for the class of locally compact groups whose reduced C*-algebra admits a tracial state. In Chapter 2 we establish the dual equivalence of the category of generalized (i.e. potentially non-unital) operator systems and the category of pointed compact noncommutative (nc) convex sets, extending a result of Davidson and Kennedy. We then apply this dual equivalence to establish a number of results about generalized operator systems, some of which are new even in the unital setting. We develop a theory of quotients of generalized operator systems that extends the theory of quotients of unital operator systems. In addition, we extend results of Kennedy and Shamovich relating to nc Choquet simplices. We show that a generalized operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan's property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra. In Chapter 3 we expand on recent work of C.K. Ng about duals of operator systems. Call a nonunital operator system S dualizable if its dual S* embeds into B(H) via a complete order embedding and complete norm equivalence. Through the categorical duality of nonunital operator systems to pointed noncommutative convex sets discussed in Chapter 3, we characterize dualizability of S using geometric conditions on the nc 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 bounded positive extension property for nc affine functions. Secondly, we show that Ng's characterization is dual to a normality condition between K and its real nc cone. As applications, we obtain some permanence properties for dualizability and duality of mapping cones in quantum information, and give a new nc convex-geometric proof of Choi's Theorem.
Graph C*-algebras and the Abelian Core
(University of Waterloo, 2016-08-26) Eifler, Kari; Marcoux, Laurent; Kennedy, Matthew
We may associate to a C*-algebra to the directed graph E by associating edges to partial isometries and vertices to pairwise orthogonal Hilbert spaces which satisfy some additional conditions. Such graph algebras were first studied by Cuntz and Krieger in 1980. Because the structure theory of the C*-algebras is related to the combinatorial and geometrical properties of the underlying graph, graph algebras have gained a lot of attention. Our report will cover the basic terminology and properties, as well as look at multiple examples whose C*-algebras generated by the graph will be familiar to the reader. We introduce two well known uniqueness theorems, the Gauge-Invariant Uniqueness Theorem and the CK-Uniqueness Theorem. We conclude with an investigation of the 2012 paper by Nagy and Reznikoff on the abelian core. Their paper gives an additional uniqueness theorem which considers the spectrum of elements in the graph C*-algebra.
Tracial and ideal structure of crossed products and related constructions
(University of Waterloo, 2022-08-17) Ursu, Dan; Kennedy, Matthew
In this thesis, we concern ourselves with asking questions about the basic structure of group C*-algebras, crossed products, and groupoid C*-algebras. Specifically, we are concerned with two main topics. One is the simplicity of these algebras, and we either extend work that was already done in the case of group C*-algebras and crossed products, or characterize simplicity altogether in the case of groupoid C*-algebras. The other is the structure of traces on these algebras, in particular in the case of crossed products. In the third chapter, we give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra A and a discrete group G. In particular, we also answer the question of when the tracial states on the crossed products are in canonical bijection with the G-invariant tracial states on A. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group G are all finite, and in other cases gives previously known results, for example when the original C*-algebra A is commutative. We also obtain results and examples in the case of abelian groups that contradict existing results in the literature of Bédos and Thomsen. Specifically, we give a finite-dimensional counterexample, and provide a correction to the result of Thomsen. The fourth chapter is a short note on results in the von Neumann crossed product case that were never submitted for publication, and the author suspects might potentially be folklore, but cannot actually find anywhere. We extend the results on C*-crossed products from the third chapter to the case of von Neumann crossed products. In particular, we obtain results that characterize when a G-invariant normal tracial state on M has a unique normal tracial extension to the crossed product. As a consequence, we also characterize when such crossed products are finite factors. In the fifth chapter, we consider the notion of a plump subgroup that was recently introduced by Amrutam. This is a relativized version of Powers' averaging property, and it is known that Powers' averaging property for G is equivalent to C*-simplicity. With this in mind, we introduce a relativized notion of C*-simplicity, and show that for normal subgroups it is equivalent to plumpness, along with several other characterizations. For the sixth chapter, we prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products of a commutative unital C*-algebra C(X) and a discrete group G. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of the crossed product and to Kawabe's generalized space of amenable subgroups. This further lets us generalize a result of the first coauthor of the original publication of this chapter and Kalantar on simplicity of intermediate C*-algebras. For the seventh chapter, we characterise, in several complementary ways, étale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property, assuming that the groupoid is either Hausdorff or sigma-compact. This leads directly to a characterisation of the simplicity of this C*-algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C*-algebra. Specifically, we prove that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that this is equivalent to the uniqueness of equivariant pseudo-expectations. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras including the essential groupoid C*-algebra itself. For minimal groupoids, we further obtain a relative version of Powers averaging property. Examples arise from suitable group representations into simple groupoid C*-algebras. This is illustrated by the example of the quasi-regular representation of Thompson's group T with respect to Thompson's group F, which satisfies the relative Powers averaging property in the Cuntz algebra O2.

Browse

Browsing Pure Mathematics by Author "Kennedy, Matthew"