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On hyperrigidity and non-degenerate C∗-correspondences

Lookup NU author(s): Joseph Dessi, Dr Evgenios Kakariadis

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Abstract

© Instytut Matematyczny PAN, 2026.We revisit the results of Kim, and of Katsoulis and Ramsey concerning hyperrigidity for non-degenerate C∗-correspondences. We show that the tensor algebra is hyperrigid, if and only if Katsura’s ideal acts non-degenerately, if and only if Katsura’s ideal acts non-degenerately under any representation. This gives a positive answer to the question of Katsoulis and Ramsey, showing that their necessary condition and their sufficient condition for hyperrigidity of the tensor algebra are equivalent. Non-degeneracy of the left action of Katsura’s ideal was also shown by Kim to be equivalent to hyperrigidity for the selfadjoint operator space associated with the C∗-correspondence, and our approach provides a simplified proof of this result as well. In the process we study unitisations of selfadjoint operator spaces in the sense of Werner, and revisit Arveson’s criterion connecting maximality with the unique extension property and hyperrigidity, in conjunction with the work of Salomon on generating sets.

Publication metadata

Author(s): Dessi JA, Kakariadis ETA, Paraskevas IA

Publication type: Article

Publication status: Published

Journal: Studia Mathematica

Year: 2026

Volume: 287

Issue: 3

Pages: 257-303

Online publication date: 10/03/2026

Acceptance date: 02/04/2018

ISSN (print): 0039-3223

ISSN (electronic): 1730-6337

Publisher: Institute of Mathematics. Polish Academy of Sciences

URL: https://doi.org/10.4064/sm250511-23-9

DOI: 10.4064/sm250511-23-9

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