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.
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In the Hilbert space setting the author has elsewhere  generalised the notion of direct sums of \(C_0\)-semigroups, here treated as UB-coproducts, to direct integrals of \(C_0\)-semigroups.
The author actually first got the idea of thinking this way from watching https://www.youtube.com/watch?v=qHuUazkUcnU&ab_channel=mlbaker, where the notion of tensor products is explained as a standardising tool for turning multilinear maps into linear maps.
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The author is grateful to many people for their help during the production of this article. To Charles Batty and David Seifert for insightful discussions on the subject of this article, to Joshua Ciappara for making some crucial suggestions from an algebraist’s perspective (such as suggesting that I look at adjunctions), and to the reviewer for numerous useful comments as a result of which this paper is much improved. This work was partially supported by the University of Sydney through the Barker Graduate Scholarship while the author was a doctoral student at the University of Oxford. The revisions were completed when the author was funded by ARC Grant DP180100595.
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Communicated by Markus Haase.
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Ng, A.C.S. A categorical approach to operator semigroups. Semigroup Forum 102, 495–516 (2021). https://doi.org/10.1007/s00233-020-10158-7
- Category theory
- Operator theory