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Zappa–Szép product groupoids and \(C^*\)-blends

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Abstract

We study the external and internal Zappa–Szép product of topological groupoids. We show that under natural continuity assumptions the Zappa–Szép product groupoid is étale if and only if the individual groupoids are étale. In our main result we show that the \(C^*\)-algebra of a locally compact Hausdorff étale Zappa–Szép product groupoid is a \(C^*\)-blend, in the sense of Exel, of the individual groupoid \(C^*\)-algebras. We finish with some examples, including groupoids built from \(*\)-commuting endomorphisms, and skew product groupoids.

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Notes

  1. This unusual choice of labelling the range and source maps by b and t is explained in Sect. 3.

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Acknowledgments

This research was supported by the Australian Research Council and the University of Wollongong through a University Research Committee Grant to the fourth author. We would also like to take the opportunity to thank Ruy Exel and Charles Starling for many interesting discussions around the topics in this article.

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Authors and Affiliations

  1. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia

    Nathan Brownlowe, David Pask & David Robertson

  2. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Jacqui Ramagge

  3. School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow, Q12 8QW, Scotland

    Michael F. Whittaker

Authors
  1. Nathan Brownlowe
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  2. David Pask
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  3. Jacqui Ramagge
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  4. David Robertson
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  5. Michael F. Whittaker
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Corresponding author

Correspondence to Nathan Brownlowe.

Additional information

Communicated by Mark V. Lawson.

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Brownlowe, N., Pask, D., Ramagge, J. et al. Zappa–Szép product groupoids and \(C^*\)-blends. Semigroup Forum 94, 500–519 (2017). https://doi.org/10.1007/s00233-016-9775-z

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  • DOI: https://doi.org/10.1007/s00233-016-9775-z

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