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Abraham C. S. Ng

Publications and source records attributed to Abraham C. S. Ng.

10 recordsLinked to original sources
arXiv2025

K-theory of C*-algebras arising from commuting Hilbert bimodules and invariant ideals

We study the K-theory of the Cuntz-Nica-Pimsner C*-algebra of a rank-two product system that is an extension determined by an invariant ideal of the coefficient algebra. We use a construction of Deaconu and Fletcher that describes the Cuntz-Nica-Pimsner C*-algebra of the product system in terms of two iterations of Pimsner's original construction of a C*-algebra from a right-Hilbert bimodule. We apply our results to the product system built from two commuting surjective local homeomorphisms of a totally disconnected space, where the Cuntz-Nica-Pimsner C*-algebra is isomorphic to the C*-algebra of the associated rank-two Deaconu--Renault groupoid. We then apply a theorem of Spielberg about stable finiteness of an extension to obtain sufficient conditions for stable finiteness of the C*-algebra of the Deaconu-Renault groupoid.

math.OA↗
arXiv2023

Reconstruction of topological graphs and their Hilbert bimodules

We show that the Hilbert bimodule associated to a compact topological graph can be recovered from the C*-algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson-Katsoulis and of Davidson-Roydor on local conjugacy of topological graphs and isomorphism of their tensor algebras. In particular, we give a direct proof that a compact topological graph can be recovered up to local conjugacy from its Hilbert bimodule, present an example of nonisomorphic locally conjugate compact topological graphs with isomorphic Hilbert bimodules. We also give an elementary proof that for compact topological graphs with totally disconnected vertex space the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of C*-algebraic triples, and isomorphism all coincide.

math.OA↗
arXiv2021

Stably finite extensions of rank-two graph C*-algebras

We study stable finiteness of extensions of 2-graph C*-algebras determined by saturated hereditary sets of vertices. We use two iterations of the Pimsner-Voiculescu sequence to calculate the map in K-theory induced by the inclusion of a hereditary subgraph into the larger 2-graph it lives in. We then apply a theorem of Spielberg about stable finiteness of extensions to provide a sufficient condition for the C*-algebra of the larger 2-graph to be stably finite. We illustrate our results with examples.

math.OA↗
arXiv2020

A categorical approach to operator semigroups

The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be applied to past operator theoretic results. In particular, we investigate the inverse-producing Arens extension for Banach algebras (Trans. Amer. Math. Soc. 88:536-548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Operator Theory 78(2):473-500, 2017) in this new framework, asking and answering questions using categorical language. We demonstrate that the Arens extension defines an extension functor in this setting and that it forms an adjunction with the suitably defined forgetful functor. As a by-product of this categorical framework, we also revisit the work on Banach direct sums by Lachowicz and Moszyński (Semigroup Forum 93(1):34-70, 2016). This paper can be considered as a brief exploration of the triple interface between operator semigroups, Banach algebras, and category theory.

math.FA↗
arXiv2020

Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces

The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator $T$ on a Banach space we have $||T^n(I-T)\|\to0$ if and only if $σ(T)\cap\mathbb{T}\subseteq\{1\}$. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms $\|R(e^{iθ},T)\|$ as $|θ|\to0$ satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case of polynomial resolvent growth. We further show that, under a natural additional assumption, our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold. By considering a suitable class of Toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators, for which we in addition obtain a more general result.

math.FA↗
arXiv2020

Direct integrals of strongly continuous operator semigroups

The goal of this article is to develop a theory for direct integrals of $C_0$-semigroups on Hilbert spaces parallel to the recent approach by Lachowicz and Moszyński for direct sums of Banach spaces, diagonal operators, and semigroups. In it we deal with the existence and characterisation of semigroups, asymptotic rates, and questions of decomposability.

math.FA↗
arXiv2019

Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

Using recent results in the theory of $C_0$-semigroups due to Batty, Chill and Tomilov (J. Eur. Math. Soc. 18(4):853-929, 2016) we study energy decay in a one-dimensional coupled wave-heat system with finite wave part and infinite heat part. Our main result provides a sharp estimate for the rate of energy decay of a certain class of classical solutions. The present paper can be thought of as a natural sequel to a recent work by Batty, Paunonen and Seifert (J. Evol. Equ. 16:649-664, 2016), which studied a similar wave-heat system with finite wave and heat parts using a celebrated result due to Borichev and Tomilov.

math.AP↗
arXiv2019

Optimal energy decay in a one-dimensional wave-heat-wave system

Harnessing the abstract power of the celebrated result due to Borichev and Tomilov (Math.\ Ann.\ 347:455--478, 2010, no.\ 2), we study the energy decay in a one-dimensional coupled wave-heat-wave system. We obtain a sharp estimate for the rate of energy decay of classical solutions by first proving a growth bound for the resolvent of the semigroup generator and then applying the asymptotic theory of $C_0$-semigroups. The present article can be naturally thought of as an extension of a recent paper by Batty, Paunonen, and Seifert (J.\ Evol.\ Equ.\ 16:649--664, 2016) which studied a similar wave-heat system via the same theoretical framework.

math.AP↗