Browsing by Author "McBride, Matt"

Now showing 1 - 7 of 7
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    Derivations and Spectral Triples on Quantum Domains I: Quantum Disk
    (2017) Klimek, Slawomir; McBride, Matt; Rathnayake, Sumedha; Sakai, Kaoru; Wang, Honglin; Mathematical Sciences, School of Science
    We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.
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    Dirac type operators on the quantum solid torus with global boundary conditions
    (Elsevier, 2020-04) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of Science
    We define a noncommutative space we call the quantum solid torus. It is an example of a noncommutative manifold with a noncommutative boundary. We study quantum Dirac type operators subject to Atiyah-Patodi-Singer like boundary conditions on the quantum solid torus. We show that such operators have compact inverse, which means that the corresponding boundary value problem is elliptic.
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    Noncommutative geometry of the quantum disk
    (Springer, 2022) Klimek, Slawomir; McBride, Matt; Peoples, J. Wilson; Mathematical Sciences, School of Science
    We discuss various aspects of the noncommutative geometry of a smooth subalgebra of the Toeplitz algebra. In particular, we study the structure of derivations on this subalgebra.
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    A Note on Dirac Operators on the Quantum Punctured Disk
    (National Academy of Science of Ukraine, 2010-07-16) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of Science
    We study quantum analogs of the Dirac type operator −2z¯¯¯∂∂z¯¯¯ on the punctured disk, subject to the Atiyah–Patodi–Singer boundary conditions. We construct a parametrix of the quantum operator and show that it is bounded outside of the zero mode.
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    A Note on Gluing Dirac Type Operators on a Mirror Quantum Two-Sphere
    (2014-03) Klimek, Slawomir; McBride, Matt
    The goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary condition. We show that the resulting operators have compact resolvents, and so they are elliptic operators.
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    A Note on Spectral Triples on the Quantum Disk
    (National Academy of Science of Ukraine, 2019) Klimek, Slawomir; McBride, Matt; Peoples, John Wilson; Mathematical Sciences, School of Science
    By modifying the ideas from our previous paper [SIGMA 13 (2017), 075, 26 pages, arXiv:1705.04005], we construct spectral triples from implementations of covariant derivations on the quantum disk.
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    Unbounded Derivations in Algebras Associated with Monothetic Groups
    (Cambridge, 2021-12) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of Science
    Given an infinite, compact, monothetic group G we study decompositions and structure of unbounded derivations in a crossed product C∗ -algebra C(G)⋊Z obtained from a translation on G by a generator of a dense cyclic subgroup. We also study derivations in a Toeplitz extension of the crossed product and the question whether unbounded derivations can be lifted from one algebra to the other.