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Non-commutative Stone Duality

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Semigroups, Algebras and Operator Theory (ICSAOT 2022)

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Abstract

I show explicitly that Boolean inverse semigroups are in duality with Boolean groupoids, a class of étale topological groupoids; this is what we mean by the term ‘non-commutative Stone duality’. This generalizes classical Stone duality, which we refer to as ‘commutative Stone duality’.

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Notes

  1. 1.

    Throughout this chapter, filters will be assumed non-empty.

  2. 2.

    The fairy godmother of mathematics.

  3. 3.

    Not an established term.

  4. 4.

    Strictly speaking, ‘reverse definite’.

  5. 5.

    I made this word up. It comes from the Greek word ‘kallos’ meaning beauty. I simply wanted to indicate that these maps were sufficiently ‘nice’.

  6. 6.

    A term taken from ring theory.

  7. 7.

    The definition we have given here looks different from the one given in [30] but is equivalent by using [47].

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Acknowledgements

Some of the work for this chapter was carried out at LaBRI, Université de Bordeaux, during April 2018 while visiting David Janin. I am also grateful to Phil Scott for alerting me to typos. None of this work would have been possible without my collaboration with Daniel Lenz, and some very timely conversations with Pedro Resende. This chapter is dedicated to the memory of Iain Currie, colleague and friend.

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Authors and Affiliations

  1. Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK

    Mark V. Lawson

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  1. Mark V. Lawson
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Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala, India

    A. A. Ambily

  2. Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala, India

    V. B. Kiran Kumar

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Lawson, M.V. (2023). Non-commutative Stone Duality. In: Ambily, A.A., Kiran Kumar, V.B. (eds) Semigroups, Algebras and Operator Theory. ICSAOT 2022. Springer Proceedings in Mathematics & Statistics, vol 436. Springer, Singapore. https://doi.org/10.1007/978-981-99-6349-2_2

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