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On Fock covariance for product systems and the reduced Hao-Ng isomorphism problem by discrete actions

Evgenios T. A. Kakariadis, Ioannis Apollon Paraskevas

TL;DR

The paper develops a unified framework for Fock covariance of product systems, giving a precise core-based criterion that generalizes and unifies Nica covariance and Toeplitz representations in key settings. By tying these covariance notions to coaction and Fell-bundle structures, it then applies the framework to the reduced Hao–Ng isomorphism problem, proving that for discrete generalized gauge actions the reduced crossed-product construction commutes with the tensor-algebra envelope. The approach leverages the fixed-point inductive limit of $K$-cores and the strong covariant bundle $A imes_X P$ to obtain canonical $*$-isomorphisms, thereby clarifying the universal properties of tensor algebras of product systems under covariant actions. Overall, the results connect Fock-space models, coactions, and crossed-product theory to yield concrete resolution of Hao–Ng-type questions in the discrete setting, with broad implications for the structure and envelopes of tensor algebras of product systems.

Abstract

We provide a characterisation of equivariant Fock covariant injective representations for product systems. We show that this characterisation coincides with Nica covariance for compactly aligned product systems over right LCM semigroups of Kwaśniewski and Larsen, and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao-Ng isomorphism problem for generalised gauge actions by discrete groups.

On Fock covariance for product systems and the reduced Hao-Ng isomorphism problem by discrete actions

TL;DR

The paper develops a unified framework for Fock covariance of product systems, giving a precise core-based criterion that generalizes and unifies Nica covariance and Toeplitz representations in key settings. By tying these covariance notions to coaction and Fell-bundle structures, it then applies the framework to the reduced Hao–Ng isomorphism problem, proving that for discrete generalized gauge actions the reduced crossed-product construction commutes with the tensor-algebra envelope. The approach leverages the fixed-point inductive limit of -cores and the strong covariant bundle to obtain canonical -isomorphisms, thereby clarifying the universal properties of tensor algebras of product systems under covariant actions. Overall, the results connect Fock-space models, coactions, and crossed-product theory to yield concrete resolution of Hao–Ng-type questions in the discrete setting, with broad implications for the structure and envelopes of tensor algebras of product systems.

Abstract

We provide a characterisation of equivariant Fock covariant injective representations for product systems. We show that this characterisation coincides with Nica covariance for compactly aligned product systems over right LCM semigroups of Kwaśniewski and Larsen, and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao-Ng isomorphism problem for generalised gauge actions by discrete groups.
Paper Structure (18 sections, 21 theorems, 291 equations)

This paper contains 18 sections, 21 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Fock covariance
  3. The Hao--Ng isomorphism problem
  4. Preliminaries
  5. Coactions on C*-algebras
  6. Semigroups
  7. C*-correspondences
  8. Product systems
  9. Fowler's product systems
  10. Fock covariant representations
  11. Fock covariant representations
  12. Compactly aligned product systems over right LCM semigroups
  13. The product system of a semigroup
  14. The reduced Hao--Ng isomorphism problem
  15. Operator algebras
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let ${\mathcal{B}}$ be a Fell bundle over a discrete group $G$, and $\Psi_1$ and $\Psi_2$ be equivariant representations of ${\mathcal{B}}$. Then if and only if If any of the above holds, and $\Psi_1({\mathcal{B}})$ is the induced Fell bundle in $\mathrm{C}^*(\Psi_1)$, then $\Psi_2$ defines a representation of $\Psi_1({\mathcal{B}})$, and thus there is a commutative diagram \xymatrix{ \mathrm{C}

Theorems & Definitions (44)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 34 more