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A categorical approach to operator semigroups

Journal Article


Abstract


  • 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. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. 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.

Publication Date


  • 2021

Citation


  • Ng, A. C. S. (2021). A categorical approach to operator semigroups. Semigroup Forum, 102(2), 495-516. doi:10.1007/s00233-020-10158-7

Scopus Eid


  • 2-s2.0-85100739323

Start Page


  • 495

End Page


  • 516

Volume


  • 102

Issue


  • 2

Abstract


  • 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. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. 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.

Publication Date


  • 2021

Citation


  • Ng, A. C. S. (2021). A categorical approach to operator semigroups. Semigroup Forum, 102(2), 495-516. doi:10.1007/s00233-020-10158-7

Scopus Eid


  • 2-s2.0-85100739323

Start Page


  • 495

End Page


  • 516

Volume


  • 102

Issue


  • 2