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

Browsing Pure Mathematics by Author "Forrest, Brian"

Filter results by typing the first few letters
Now showing 1 - 5 of 5
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    Exotic Group C*-algebras, Tensor Products, and Related Constructions
    (University of Waterloo, 2016-06-15) Wiersma, Matthew; Forrest, Brian; Spronk, Nicolaas
    Recently there has been a rejuvenated interest in exotic group C*-algebras, i.e., group C*-algebras which are "intermediate" to the full and reduced group C*-algebras. This resurgence began with the introduction of the class of group $L^p$-representations and their associated C*-algebras (a class of potentially exotic group C*-algebras) by Brown and Guentner. Unlike previous examples of exotic group C*-algebras, this class of examples is universally defined for all locally compact groups. In this thesis we compare this new class of exotic group C*-algebras to previously known examples of exotic group C*-algebras in several key examples and produces new examples of exotic group C*-algebras. Similar to the definition of exotic group C*-algebras, an exotic C*-tensor product is a C*-tensor product which is intermediate to the minimal and maximal C*-tensor products. Borrowing from the theory of $L^p$-representations, we construct many exotic C*-tensor products for group C*-algebras. We will also study the $L^p$-Fourier and Fourier-Stieltjes algebras of a locally compact group. These are ideals which of the Fourier-Stieltjes algebras containing the Fourier algebras and correspond to the class of $L^p$-representations. We study the structural properties of these algebras and classify the Fourier-Stieltjes spaces of SL(2,R) which are ideals in the Fourier-Stieltjes algebra. There are many different tensor products considered in the category of C*-algebras. In contrast, virtually the only tensor product ever considered for von Neumann algebras is the normal spatial tensor product. We propose a definition for what a generic tensor product in the category of von Neumann algebras should be and study properties of von Neumann algebras in relation to these tensor products.
  • Thumbnail Image
    Item
    Mind the GAP: Amenability Constants and Arens Regularity of Fourier Algebras
    (University of Waterloo, 2023-08-28) Sawatzky, John; Forrest, Brian; Wiersma, Matthew
    This thesis aims to investigate properties of algebras related to the Fourier algebra $A(G)$ and the Fourier-Stieltjes algebra $B(G)$, where $G$ is a locally compact group. For a Banach algebra $\cA$ there are two natural multiplication operations on the double dual $\cA^{**}$ introduced by Arens in 1971, and if these operations agree then the algebra $\cA$ is said to be Arens regular. We study Arens regularity of the closures of $A(G)$ in the multiplier and completely bounded multiplier norms, denoted $A_M(G)$ and $A_{cb}(G)$ respectively. We prove that if a nonzero closed ideal in $A_M(G)$ or $A_{cb}(G)$ is Arens regular then $G$ must be a discrete group. Amenable Banach algebras were first studied by B.E. Johnson in 1972. For an amenable Banach algebra $\cA$ we can consider its amenability constant $AM(\cA) \geq 1$. We are particularly interested in collections of amenable Banach algebras for which there exists a constant $\lambda > 1$ such that the values in the interval $(1,\lambda)$ cannot be attained as amenability constants. If $G$ is a compact group, then the central Fourier algebra is defined as $ZA(G) = ZL^1(G) \cap A(G)$ and endowed with the $A(G)$ norm. We study the amenability constant theory of $ZA(G)$ when $G$ is a finite group.
  • Thumbnail Image
    Item
    On Asynchronous Interference Channels
    (University of Waterloo, 2016-11-15) Moshksar, Kamyar; Forrest, Brian; Spronk, Nicolaas
    In the first part of the thesis, a decentralized wireless network of separate transmitter-receiver pairs is studied where there is no central controller to assign the resources to the users and users do not explicitly cooperate. For simplicity, we focus on a single-burst scenario where each transmitter sends a single codeword upon activation and remains silent afterwards. Users are block-asynchronous meaning there exists a mutual delay between their transmitted codewords. We show how the receivers learn about the number of active users, channel coefficients and activation times of the transmitters using piecewise linear regression. It is essential that each receiver fi nds the exact arrival time of the codeword sent by its corresponding transmitter. To achieve this goal, preamble sequences are embedded at the beginning of a transmitted codeword. As different users do not necessarily know each other's preamble sequences, there is no guarantee that a receiver can estimate the arrival times of interference bursts along its desired data with vanishingly small probability of error. Nevertheless, the estimates are reliable enough to guarantee successful decoding at each receiver. The second part of the thesis addresses a Gaussian interference channel with two transmitter-receiver (Tx-Rx) pairs under stochastic data arrival (GIC-SDA). Information bits arrive at the transmitters according to independent and asynchronous Bernoulli processes (Tx-Tx~asynchrony). Each information source turns off after generating a given total number of bits. The transmissions are asynchronous (Tx-Rx~asynchrony) in the sense that each Tx sends a codeword to its Rx immediately after there are enough bits available in its buffer. Such asynchronous style of transmission is shown to significantly reduce the transmission delay in comparison with the existing Tx-Rx synchronous transmission schemes. The receivers learn about the activity frames of both transmitters by employing sequential joint-typicality detection. As a consequence, the GIC-SDA under Tx-Rx asynchrony is represented by a standard GIC with state known at the receivers. The cardinality of the state space is $\binom{2N_1+2N_2}{2N_2}$ in which $N_1, N_2$ are the numbers of transmitted codewords by the two transmitters. Each realization of the state imposes two sets of constraints on $N_1, N_2$ referred to as the geometric and reliability constraints. In a scenario where the transmitters are only aware of the statistics of Tx-Tx~asynchrony, it is shown how one designs $N_1,N_2$ to achieve target transmission rates for both users and minimize the probability of unsuccessful decoding. An achievable region is characterized for the codebook rates in a two-user GIC-SDA under the requirements that the transmissions be immediate and the receivers treat interference as noise. This region is described as the union of uncountably many polyhedrons and is in general disconnected and non-convex due to infeasibility of time-sharing. Special attention is given to the symmetric case where closed-form expressions are developed for the achievable codebook rates.
  • Thumbnail Image
    Item
    On the structure of invertible elements in certain Fourier-Stieltjes algebras
    (University of Waterloo, 2021-08-27) Thamizhazhagan, Aasaimani; Forrest, Brian; Spronk, Nicolaas
    Let $G$ be a locally compact group. The Fourier-Stieltjes and Fourier algebras, $B(G)$ and $A(G)$ are defined by Eymard to act as dual objects of the measure and group algebras, $M(G)$ and $L^1(G)$, in a sense generalizing Pontryagin duality from the theory of abelian locally compact groups. Hence there is natural expectation that properties of the latter two algebras ought to be reflected in the former. Joseph L. Taylor wrote a series of ten papers within the span of 1965 - 1972 studying the structure of convolution measure algebras $\mathfrak{M}$ (important examples are $M(G)$, measure algebra and $L^1(G)$, group algebra for an locally compact abelian group $G$) and characterizes the invertible elements in a measure algebra of a locally compact abelian group. Several results indicated that the spectrum $\Delta$ of $M(G)$ is quite complicated, as is the problem of deciding when $\mu \in M(G)$ is invertible. He proved that the spectrum of any abelian measure algebra can be represented as the space $\hat{S}$ of semicharacters on some compact semigroup $S$. However, for $M(G)$ we have no concrete description of the corresponding $S$ or $\hat{S}$. Though this failed to give a really satisfactory description of the spectrum of $M(G)$, using this description and certain semigroup techniques, he proved a theorem which significantly simplifies the problem of deciding when a measure in $M(G)$ is invertible. Basically, this theorem asserts if $\mu^{-1}$ exists it must lie in a certain "small" subalgebra of $M(G)$. This reduces the invertibility problem in $M(G)$ to the same problem in an algebra which is far less complicated than $M(G)$. He does so by identifying the closed linear span of all maximal subalgebras which are isometrically isomorphic to a group algebras of some locally compact abelian group for his convolution measure algebras. He calls this the 𝘴𝘱𝘪𝘯𝘦 of $\mathfrak{M}$. In 2007, Nico Spronk and Monica Illie develop the non-commutative dual analogue of the spine of an abelian measure algebra, provide an explicit dual description of how the above could be realised in abelian Fourier-Stieltjes algebra and conjecture if the invertible elements in general $B(G)$ admit such a dual characterisation. In this thesis, we bring together a class of interesting examples within Lie groups, in particular, many totally minimal groups, classical motion groups and the $ax+b$-group that witness the desired characterisation of invertible elements in their Fourier-Stieltjes algebra.