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Negative definite functions for \(C^*\)-dynamical systems

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Abstract

Given an action \(\alpha \) of a discrete group G on a unital \(C^*\)-algebra A, we introduce a natural concept of \(\alpha \)-negative definiteness for functions from G to A, and examine some of the first consequences of such a notion. In particular, we prove analogs of theorems due to Delorme–Guichardet and Schoenberg in the classical case where A is trivial. We also give a characterization of the Haagerup property for the action \(\alpha \) when G is countable.

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Acknowledgements

Most of the present work was done during visits made by E.B. at the Sapienza University of Rome and by R.C. at the University of Oslo in 2015 and 2016. Both authors would like to thank these institutions for their kind hospitality.

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Authors and Affiliations

  1. Institute of Mathematics, University of Oslo, P.B. 1053, Blindern, 0316, Oslo, Norway

    Erik Bédos

  2. Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Università Sapienza di Roma, via A. Scarpa 16, 00166, Roma, Italy

    Roberto Conti

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  1. Erik Bédos
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  2. Roberto Conti
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Correspondence to Roberto Conti.

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Bédos, E., Conti, R. Negative definite functions for \(C^*\)-dynamical systems. Positivity 21, 1625–1646 (2017). https://doi.org/10.1007/s11117-017-0490-0

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  • DOI: https://doi.org/10.1007/s11117-017-0490-0

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