Equivariant $\mathcal{D}$-stability for Actions of Tensor Categories

TL;DR

The paper develops a framework for equivariant $\mathcal{D}$-stability for actions of unitary tensor categories on C$^*$-algebras, tying stability to embeddings of $\mathcal{D}$ into a fixed-point subalgebra of Kirchberg's central sequence algebra $F(A)^G$ when $\mathcal{D}$ is strongly self-absorbing. It proves a C-equivariant McDuff-type theorem equating several formulations of equivariant $\mathcal{D}$-stability, including cocycle conjugacy and approximate unitary equivalence of cocycle morphisms. The authors introduce a concrete subalgebra framework for central sequences that remains well-behaved under category actions and show how to verify $\mathcal{D}$-stability via embeddings into $F(A)^G$. As an application to stationary AF-actions, they show $\mathcal{Z}$-stability holds in broad cases, providing a practical criterion for $\mathcal{D}$-stability in a rich class of nongroup symmetries.

Abstract

We introduce a notion of equivariant $\mathcal{D}$-stability for actions of unitary tensor categories on C$^*$-algebras. We show that, when $\mathcal{D}$ is strongly self-absorbing, equivariant $\mathcal{D}$-stability of an action is equivalent to a unital embedding of $\mathcal{D}$ into a certain subalgebra of Kirchberg's central sequence algebra. We use this to show $\mathcal{Z}$-stability for a large class of AF-actions.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Example 3.3
• Definition 4.1