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MORPHISMS OF CUNTZ–PIMSNER ALGEBRAS FROM COMPLETELY POSITIVE MAPS

Part of: Selfadjoint operator algebras Ergodic theory

Published online by Cambridge University Press:  31 October 2025

KEVIN AGUYAR BRIX Open the ORCID record for KEVIN AGUYAR BRIX [Opens in a new window] ,
ALEXANDER MUNDEY Open the ORCID record for ALEXANDER MUNDEY [Opens in a new window]  and
ADAM RENNIE Open the ORCID record for ADAM RENNIE [Opens in a new window]
KEVIN AGUYAR BRIX
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK e-mail: kabrix.math@fastmail.com
ALEXANDER MUNDEY
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales, Australia e-mail: amundey@uow.edu.au
ADAM RENNIE*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales, Australia
*
e-mail: renniea@uow.edu.au
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Abstract

We introduce positive correspondences as right $C^*$-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the $C^*$-correspondence (Enchilada) category as a ‘retract’. Kasparov’s KSGNS construction provides a semi-functor from this semi-category onto the $C^*$-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz–Pimsner algebras, and we describe classes of examples arising from projections on $C^*$-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index.

Keywords

Cuntz–Pimsner algebrasHilbert modulespositive correspondencesbi-Hilbertian bimodulescorrespondence projections

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 37A55: Relations with the theory of $C^*$-algebras 46L08: $C^*$-modules

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 40
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by a Carlsberg Foundation Internationalisation Fellowship and a DFF-International postdoc (Case number 1025–00004B). The second author was supported by University of Wollongong AEGiS CONNECT grant 141765.

Communicated by Aidan Sims

In memory of Iain Raeburn, with gratitude for all of his contributions to mathematics and our community

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