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The stable uniqueness theorem for unitary tensor category equivariant KK-theory

Sergio Girón Pacheco, Kan Kitamura, Robert Neagu

Abstract

We introduce the Cuntz-Thomsen picture of $\mathcal{C}$-equivariant Kasparov theory, denoted $\mathrm{KK}^\mathcal{C}$, for a unitary tensor category $\mathcal{C}$ with countably many isomorphism classes of simple objects. We use this description of $\mathrm{KK}^\mathcal{C}$ to prove the stable uniqueness theorem in this setting.

The stable uniqueness theorem for unitary tensor category equivariant KK-theory

Abstract

We introduce the Cuntz-Thomsen picture of -equivariant Kasparov theory, denoted , for a unitary tensor category with countably many isomorphism classes of simple objects. We use this description of to prove the stable uniqueness theorem in this setting.

Paper Structure

This paper contains 10 sections, 27 theorems, 165 equations.

Table of Contents

  1. Preliminaries
  2. Hilbert modules
  3. Actions of unitary tensor categories
  4. Cocycle representations revisited
  5. Paschke dual type algebras
  6. Cuntz--Thomsen pictures for KKC
  7. Stable operator homotopy
  8. Absorbing C-cocycle representations
  9. Obtaining asymptotic unitary equivalence
  10. The C-equivariant stable uniqueness theorem

Key Result

Theorem A

Let $\mathcal{C}$ be a unitary tensor category with countably many isomorphism classes of simple objects, $(A,\alpha,\mathfrak{u})$ be a separable $\mathcal{C}$-C$^*$-algebra, and $(B,\beta,\mathfrak{v})$ be a $\sigma$-unital $\mathcal{C}$-C$^*$-algebra. Let be two $\mathcal{C}$-cocycle representations that form a $\mathcal{C}$-Cuntz pair. Then the pair $(\phi,\psi)$ represents the zero element i

Theorems & Definitions (84)

  • Theorem A: Theorem \ref{['thm: StableUniq']}
  • Lemma A: cf. Hilbertmodules
  • Lemma B: cf. MEY00
  • Definition C
  • Remark D
  • Remark E
  • Remark F
  • Lemma G
  • proof
  • Remark H
  • ...and 74 more