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Spheres, Euler classes and the K-theory of C²-algebras of subproduct systems

By Francesca Arici

Appears in collection : Group operator algebras and Non commutative geometry / Algèbres d'opérateurs de groupes et Geometrie non commutative

In this talk, we shall consider equivariant subproduct system of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras. We will provide results about their topological invariants through K(K)-theory. More specifically, we will show that the Toeplitz algebra of the subproduct system of an SU(2)-representation is equivariantly KKequivalent to the algebra of complex numbers so that the (K)K- theory groups of the Cuntz–Pimsner algebra can be effectively computed using a Gysin exact sequence involving an analogue of the Euler class of a sphere bundle. Finally, we will discuss why and how C²-algebras in this class satisfy KK-theoretic Poincaré duality.

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Citation data

  • DOI 10.24350/CIRM.V.20165403
  • Cite this video Arici, Francesca (23/04/2024). Spheres, Euler classes and the K-theory of C²-algebras of subproduct systems. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20165403
  • URL https://dx.doi.org/10.24350/CIRM.V.20165403

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Bibliography

  • ARICI, Francesca, GERONTOGIANNIS, Dimitris Michail, et NESHVEYEV, Sergey. KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems. arXiv preprint arXiv:2401.01725, 2024. - https://arxiv.org/abs/2401.01725
  • ARICI, Francesca et KAAD, Jens. Gysin sequences and SU (2)‐symmetries of C∗‐algebras. Transactions of the London Mathematical Society, 2021, vol. 8, no 1, p. 440-492. - https://doi.org/10.1112/tlm3.12038
  • ARICI, Francesca, KAAD, Jens, et LANDI, Giovanni. Pimsner algebras and Gysin sequences from principal circle actions. Journal of Noncommutative Geometry, 2016, vol. 10, no 1, p. 29-64. - https://doi.org/10.4171/jncg/228
  • HABBESTAD, Erik et NESHVEYEV, Sergey. Subproduct systems with quantum group symmetry. Journal of Noncommutative Geometry, 2023, vol. 18, no 1, p. 93-121. - https://arxiv.org/abs/2212.08512
  • HABBESTAD, Erik et NESHVEYEV, Sergey. Subproduct systems with quantum group symmetry. Journal of Noncommutative Geometry, 2023, vol. 18, no 1, p. 93-121. - https://doi.org/10.4171/JNCG/523
  • SHALIT, Orr et SOLEL, Baruch. Subproduct systems. Documenta Mathematica, 2009, vol. 14, p. 801-868. - https://doi.org/10.4171/DM/290

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