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Invariant means on Boolean inverse monoids

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Abstract

The classical theory of invariant means, which plays an important rôle in the theory of paradoxical decompositions, is based upon what are usually termed ‘pseudogroups’. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to étale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when just such a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods.

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Notes

  1. There is a nice essay by Tyler Bryson, available as arXiv:1306.2985, that also recognises the importance of inverse semigroups in this context. As an aside, the second author observes that the co-attribution of [16] given in this essay is incorrect.

  2. UHF inverse monoids can be classified using supernatural numbers just as in the \(C^{*}\)-algebra case [26]. Thus we can define \(I_{n}\) where now n is a supernatural number. A concrete representation of \(I_{2^{\infty }}\) is constructed in [20].

  3. This has been confirmed by a counter-example in [35].

  4. There is clearly more work to be done here as one referee observed. The parallels between AF inverse monoids and AF \(C^{*}\)-algebras are just examples of a number of parallels between inverse semigroups and \(C^{*}\)-algebras.

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Acknowledgments

This research was carried out in July 2014 under the auspices of the Research in groups programme of the International Centre for Mathematical Sciences (ICMS), Edinburgh. We are grateful to the Scientific Director, Prof Keith Ball, and the Centre Manager, Ms Jane Walker, and all the staff at ICMS for their help and hospitality during our stay. We would also like to thank Alistair Wallis for some fruitful discussions. In addition, Kudryavtseva was also partially funded by the EU project TOPOSYS (FP7-ICT-318493-STREP) and by ARRS Grant P1-0288; Lawson was also partially supported by an EPSRC Grant (EP/I033203/1); and Resende was also partially supported by FCT/Portugal through projects EXCL/MAT-GEO/0222/2012 and PEst-OE/EEI/LA0009/2013. Finally, the authors would like to thank the two referees on their comments and suggestions.

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Authors and Affiliations

  1. Faculty of Civil and Geodetic Engineering, Jamova cesta 39, 1000, Ljubljana, Slovenia

    Ganna Kudryavtseva

  2. Jožef Štefan Institute, Jamova cesta 39, 1000, Ljubljana, Slovenia

    Ganna Kudryavtseva

  3. Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, 1000, Ljubljana, Slovenia

    Ganna Kudryavtseva

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

    Mark V. Lawson

  5. Mathematisches Institut, Friedrich-Schiller Universität Jena, Ernst-Abbé Platz 2, 07743, Jena, Germany

    Daniel H. Lenz

  6. Departamento de Matemática and the Centre for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Pedro Resende

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  1. Ganna Kudryavtseva
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  2. Mark V. Lawson
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  4. Pedro Resende
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Correspondence to Mark V. Lawson.

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Communicated by Jimmie D. Lawson.

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Kudryavtseva, G., Lawson, M.V., Lenz, D.H. et al. Invariant means on Boolean inverse monoids. Semigroup Forum 92, 77–101 (2016). https://doi.org/10.1007/s00233-015-9768-3

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