Pure Mathematics

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

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Browsing Pure Mathematics by Author "Brannan, Michael"

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    On Amenability Properties and Coideals of Quantum Groups
    (University of Waterloo, 2022-07-27) Anderson-Sackaney, Benjamin; Spronk, Nico; Brannan, Michael
    We study amenability type properties of locally compact quantum groups and subobjects of quantum groups realized as submodules of their von Neumann algebras. An important class of such subobjects are the coideals, which offer a way of defining a “quasisubgroup” for locally compact quantum groups. Chapters 3, 4, and 5 are based on [3], [2], and [4] respectively. In Chapter 3, we establish the notion of a non-commutative hull of a left ideal of L1(Gb) for a discrete quantum group G. Non-commutative spectral synthesis is defined too, and is related to a certain Ditkin’s property at infinity, allowing for a description of the closed left ideals of L1(Gb) for many known compact quantum groups Gb from the literature. We apply this work to study weak∗ closed in ideals in the quantum measure algebra of coamenable compact quantum groups and certain closed ideals in L1(Gb) which admit bounded right approximate identities in relation to coamenability of Gb (Theorem 3.3.14). In Chapter 4, we study relative amenability and amenability of coideals of a discrete quantum group, and coamenability of coideals of a compact quantum group. Making progress towards answering a coideal version of a question of [65], we prove a duality result that generalizes Tomatsu’s theorem [122] (lemmas 4.4.14 and 4.1.9). Consequently, we characterize the reduced central idempotent states of a compact quantum group (Corollary 4.1.2). In Chapter 5, we study tracial and G-invariant states of discrete quantum groups. A key result here is that tracial idempotent states are equivalently G-invariant idempotent states (Proposition 5.3.12). A consequence is the resolution of an open problem in [96, 22] in the discrete case, namely that amenability of G is equivalent to nuclearity of and the existence of a tracial state on Cr(Gb) (Corollary 5.3.14). We also obtain that simplicity of Cr(Gb) implies no G-invariant states exist (Corollary 5.3.15). Finally, we prove existence and uniqueness results of traces in terms of the cokernel, HF , of the Furstenberg boundary and the canonical Kac quotient of Gb. In Chapter 6, we develop a notion of operator amenability and operator biflatness of the action of a completely contractive Banach algebra on another completely contractive Banach algebra. We study these concept on various actions defined for locally compact quantum groups and their quantum subgroups, and relate them to usual operator amenability and other related properties, including amenability, coamenability, and compactness.
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    Quantum superchannels on the space of quantum channels
    (University of Waterloo, 2023-06-20) Daly, Padraig Conor; Brannan, Michael; Kribs, David
    Quantum channels, defined as completely-positive and trace-preserving maps on matrix algebras, are an important object in quantum information theory. In this thesis we are concerned with the space of these channels. This is motivated by the study of quantum superchannels, which are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps, as has been done in the past, we motivate and define superchannels by considering them as transformations on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.