KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems
- Resource Type
- Working Paper
- Authors
- Arici, Francesca; Gerontogiannis, Dimitris Michail; Neshveyev, Sergey
- Source
- Subject
- Mathematics - Operator Algebras
Mathematics - K-Theory and Homology
Mathematics - Quantum Algebra
46L52, 46L67, 46L85 (primary), 19K35 (secondary)
- Language
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 $S^3$.
Comment: 20 pages; v2: minor corrections