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A note on Arveson's hyperrigidity and non-degenerate C*-correspondences

Joseph A. Dessi, Evgenios T. A. Kakariadis, Ioannis Apollon Paraskevas

TL;DR

The paper establishes a decisive link between hyperrigidity and the non-degeneracy of Katsura’s ideal $J_X$ for non-degenerate C*-correspondences, proving that hyperrigidity of the tensor algebra ${\mathcal T}_X$ is equivalent to $J_X$ acting non-degenerately under any representation. It unifies and extends results of Katsoulis–Ramsey and Kim, providing a streamlined proof that connects hyperrigidity for selfadjoint operator spaces to non-degeneracy, and clarifies the role of Arveson’s maximality criterion and Salomon’s generating-set framework. The authors also show that the C*-envelope of the selfadjoint operator space ${\mathfrak S}(A,X)$ is ${\mathcal O}_X$, and they reconcile various equivalent formulations of hyperrigidity (ucp/ccp, unique extension properties, and maximality) in both unital and separating contexts. Overall, the work offers a cohesive theory tying hyperrigidity to representation-theoretic and module-theoretic properties of C*-correspondences, with broad implications for tensor algebras and Cuntz–Pimsner-type constructions.

Abstract

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 revisit Arveson's criterion connecting maximality with the unique extension property and hyperrigidity, in conjunction with the work of Salomon on generating sets.

A note on Arveson's hyperrigidity and non-degenerate C*-correspondences

TL;DR

The paper establishes a decisive link between hyperrigidity and the non-degeneracy of Katsura’s ideal for non-degenerate C*-correspondences, proving that hyperrigidity of the tensor algebra is equivalent to acting non-degenerately under any representation. It unifies and extends results of Katsoulis–Ramsey and Kim, providing a streamlined proof that connects hyperrigidity for selfadjoint operator spaces to non-degeneracy, and clarifies the role of Arveson’s maximality criterion and Salomon’s generating-set framework. The authors also show that the C*-envelope of the selfadjoint operator space is , and they reconcile various equivalent formulations of hyperrigidity (ucp/ccp, unique extension properties, and maximality) in both unital and separating contexts. Overall, the work offers a cohesive theory tying hyperrigidity to representation-theoretic and module-theoretic properties of C*-correspondences, with broad implications for tensor algebras and Cuntz–Pimsner-type constructions.

Abstract

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 revisit Arveson's criterion connecting maximality with the unique extension property and hyperrigidity, in conjunction with the work of Salomon on generating sets.

Paper Structure

This paper contains 16 sections, 29 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. C*-envelope and hyperrigidity
  3. Non-degenerate C*-correspondences revisited
  4. Hyperrigidity and maximality revisited
  5. Contents
  6. Preliminaries
  7. Operator Spaces
  8. Selfadjoint Operator Spaces
  9. C*-algebras
  10. Hyperrigidity
  11. Hyperrigidity, UEP and maximality
  12. Unital generated C*-algebra
  13. Separating generating set
  14. Hyperrigidity for non-degenerate C*-correspondences
  15. Preliminaries on C*-correspondences
  16. ...and 1 more sections

Key Result

Lemma 2.2

Let ${\mathfrak{S}}$ and ${\mathfrak{T}}$ be selfadjoint operator spaces and let $\phi \colon {\mathfrak{S}} \to {\mathfrak{T}}$ be a completely isometric completely positive map and invertible onto its image with a completely positive inverse. Then the map $\phi$ is an embedding if and only if ever

Theorems & Definitions (69)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 59 more