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KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems

Francesca Arici, Dimitris Michail Gerontogiannis, Sergey Neshveyev

Abstract

We prove that the Cuntz-Pimsner algebra of every Temperley-Lieb subproduct system is KK-self-dual. We show also that every such Cuntz-Pimsner algebra has a canonical KMS-state, which we use to construct a Fredholm module representative for the fundamental class of the duality. This allows us to describe the K-homology of the Cuntz-Pimsner algebras by explicit Fredholm modules. Both the construction of the dual class and the proof of duality rely in a crucial way on quantum symmetries of Temperley-Lieb subproduct systems. In the simplest case of Arveson's $2$-shift our work establishes $U(2)$-equivariant KK-self-duality of $S^3$.

KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems

Abstract

We prove that the Cuntz-Pimsner algebra of every Temperley-Lieb subproduct system is KK-self-dual. We show also that every such Cuntz-Pimsner algebra has a canonical KMS-state, which we use to construct a Fredholm module representative for the fundamental class of the duality. This allows us to describe the K-homology of the Cuntz-Pimsner algebras by explicit Fredholm modules. Both the construction of the dual class and the proof of duality rely in a crucial way on quantum symmetries of Temperley-Lieb subproduct systems. In the simplest case of Arveson's -shift our work establishes -equivariant KK-self-duality of .
Paper Structure (14 sections, 18 theorems, 112 equations)

This paper contains 14 sections, 18 theorems, 112 equations.

Table of Contents

  1. Preliminaries
  2. Temperley--Lieb subproduct systems
  3. KK-duality
  4. Duality classes for Temperley--Lieb subproduct systems
  5. The fundamental class
  6. The dual class
  7. An extension representing the Kasparov product
  8. Comparison with the Cuntz algebra case
  9. Proof of duality
  10. Quantum symmetries of Temperley--Lieb polynomials
  11. Reduction to m=2
  12. Computations for Uq(2)
  13. KMS-state
  14. A Fredholm module representative of the fundamental class

Key Result

Proposition 1.1

Let $H$ be an $m$-dimensional Hilbert space with an orthonormal basis $\lbrace \xi_i \rbrace_{i=1}^m$. Then there is a bijective inclusion-reversing correspondence between the proper homogeneous ideals $J \subset \mathbb{C} \langle X_1,\ldots,X_m \rangle$ and the standard subproduct systems $\lbrac

Theorems & Definitions (31)

  • Proposition 1.1: ShSo09
  • Definition 1.2: HaNe21
  • Theorem 1.3: HaNe22
  • Theorem 1.4: HaNe22
  • Corollary 1.5: HaNe22
  • Corollary 1.6
  • Definition 1.7
  • Lemma 2.1: HaNe21
  • Definition 2.2
  • Lemma 2.3
  • ...and 21 more