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Equivariant injectivity of crossed products

Joeri De Ro

TL;DR

This work develops a comprehensive framework for equivariant injectivity in the setting of operator spaces and operator systems under locally compact quantum groups. It defines $\mathbb{G}$-operator spaces and their crossed products, establishes core equivalences between $L^1(\mathbb{G})$-module injectivity, $\mathbb{G}$-injectivity, and amenability, and analyzes how these notions behave under crossed products and dual actions. The paper also links amenability and inner amenability of actions to dynamical properties of quantum groups, applies the results to non-commutative Poisson boundaries, and proves compatibility of equivariant injective envelopes with crossed products for discrete and compact quantum groups. Finally, it shows propagation of injectivity and amenability to closed quantum subgroups, thereby unifying and extending several prior results in the literature.

Abstract

We introduce the notion of a $\mathbb{G}$-operator space $(X, α)$, which consists of an action $α: X \curvearrowleft \mathbb{G}$ of a locally compact quantum group $\mathbb{G}$ on an operator space $X$, and we make a study of the notion of $\mathbb{G}$-equivariant injectivity for such an operator space. Given a $\mathbb{G}$-operator space $(X, α)$, we define a natural associated crossed product operator space $X\rtimes_α\mathbb{G}$, which has canonical actions $X\rtimes_α\mathbb{G} \curvearrowleft \mathbb{G}$ (the adjoint action) and $X\rtimes_α\mathbb{G}\curvearrowleft \check{\mathbb{G}}$ (the dual action) where $\check{\mathbb{G}}$ is the dual quantum group. We then show that if $X$ is a $\mathbb{G}$-operator system, then $X\rtimes_α\mathbb{G}$ is $\mathbb{G}$-injective if and only if $X\rtimes_α\mathbb{G}$ is injective and $\mathbb{G}$ is amenable, and that (under a mild assumption) $X\rtimes_α\mathbb{G}$ is $\check{\mathbb{G}}$-injective if and only if $X$ is $\mathbb{G}$-injective. We discuss how these results generalise and unify several recent results from the literature, and give new applications of these results.

Equivariant injectivity of crossed products

TL;DR

This work develops a comprehensive framework for equivariant injectivity in the setting of operator spaces and operator systems under locally compact quantum groups. It defines -operator spaces and their crossed products, establishes core equivalences between -module injectivity, -injectivity, and amenability, and analyzes how these notions behave under crossed products and dual actions. The paper also links amenability and inner amenability of actions to dynamical properties of quantum groups, applies the results to non-commutative Poisson boundaries, and proves compatibility of equivariant injective envelopes with crossed products for discrete and compact quantum groups. Finally, it shows propagation of injectivity and amenability to closed quantum subgroups, thereby unifying and extending several prior results in the literature.

Abstract

We introduce the notion of a -operator space , which consists of an action of a locally compact quantum group on an operator space , and we make a study of the notion of -equivariant injectivity for such an operator space. Given a -operator space , we define a natural associated crossed product operator space , which has canonical actions (the adjoint action) and (the dual action) where is the dual quantum group. We then show that if is a -operator system, then is -injective if and only if is injective and is amenable, and that (under a mild assumption) is -injective if and only if is -injective. We discuss how these results generalise and unify several recent results from the literature, and give new applications of these results.
Paper Structure (17 sections, 40 theorems, 165 equations)

This paper contains 17 sections, 40 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator spaces and operator systems
  4. Locally compact quantum groups.
  5. Actions of locally compact quantum groups and crossed products.
  6. Equivariant operator spaces, injectivity and amenability
  7. Crossed products
  8. Equivariant injectivity of crossed products
  9. Amenability and inner amenability of actions
  10. Inner actions
  11. Dynamical characterisation of (inner) amenability of a locally compact quantum group
  12. Application to the non-commutative Poisson boundary
  13. Automatic (inner) amenability of actions
  14. Equivariant injective envelopes of crossed products
  15. Discrete quantum groups
  16. ...and 2 more sections

Key Result

Proposition 2.1

Ham82 Assume that $X_j\subseteq B(\mathcal{H}_j), j=1,2$ are operator spaces and $\phi: X_1\to X_2$ is a completely bounded map. Fix an orthonormal basis $\{e_i\}_{i\in I}$ for $\mathcal{K}$ and let $\{E_{ij}\}_{i,j\in I}\subseteq B(\mathcal{K})$ be the associated matrix units. The assignment (with the sums converging in the strong topology) is a well-defined completely bounded map with $\|\phi\o

Theorems & Definitions (87)

  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 3.1
  • Example 3.2
  • ...and 77 more