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An Elliott intertwining approach to classifying actions of C$^*$-tensor categories

Sergio Girón Pacheco, Robert Neagu

Abstract

We introduce a categorical approach to classifying actions of C$^*$-tensor categories $\mathcal{C}$ on C$^*$-algebras up to cocycle conjugacy. We show that, in this category, inductive limits exist and there is a natural notion of approximate unitary equivalence. Then, we generalise classical Elliott intertwining results to the $\mathcal{C}$-equivariant case, in the same fashion as done by Szabó for the group equivariant case in [39].

An Elliott intertwining approach to classifying actions of C$^*$-tensor categories

Abstract

We introduce a categorical approach to classifying actions of C-tensor categories on C-algebras up to cocycle conjugacy. We show that, in this category, inductive limits exist and there is a natural notion of approximate unitary equivalence. Then, we generalise classical Elliott intertwining results to the -equivariant case, in the same fashion as done by Szabó for the group equivariant case in [39].
Paper Structure (13 sections, 32 theorems, 189 equations)

This paper contains 13 sections, 32 theorems, 189 equations.

Table of Contents

  1. Preliminaries
  2. Hilbert bimodules
  3. Correspondences
  4. $\mathrm{C}^*$-tensor categories
  5. Szabó's cocycle category
  6. Actions of $\mathrm{C}^*$-tensor categories
  7. The generalised cocycle category
  8. Inductive limits
  9. Topology induced on cocycle representations
  10. Approximate unitary equivalence
  11. Two-sided Elliott intertwining
  12. Intertwining through reparametrisation
  13. One sided intertwining arguments

Key Result

Theorem D

Let $(F,J): \mathcal{C}\curvearrowright A$ and $(G,I): \mathcal{C}\curvearrowright B$ be actions on separable $\mathrm{C}^*$-algebras. Let be two extendible cocycle morphisms such that the compositions $(\psi,l)\circ (\phi, h)$ and $(\phi,h)\circ (\psi, l)$ are approximately inner. Then $(\phi, h)$ and $(\psi, l)$ are approximately unitarily equivalent to mutually inverse cocycle conjugacies.

Theorems & Definitions (113)

  • Definition A: Lemma \ref{['linearmapspicture']}
  • Definition B: Lemma \ref{['lemma: topologycocyclemor']}
  • Definition C: Lemma \ref{['lemma: approxunitconj']}
  • Theorem D: Corollary \ref{['intertidentity']}
  • Definition A
  • Definition B
  • Example C
  • Definition D
  • Definition E
  • Remark F
  • ...and 103 more