Equivariant spectral triples and Poincaré duality for $SU_q(2)$

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Let $\mathcal{A}$ be the $C^*$-algebra associated with $SU_q(2)$, $π$ be the representation by left multiplication on the $L_2$ space of the Haar state and let $D$ be the equivariant Dirac operator for this representation constructed by the authors earlier. We prove in this article that there is no operator other than the scalars in the commutant $π(\cla)'$ that has bounded commutator with $D$. This implies that the equivariant spectral triple under consideration does not admit a rational Poincaré dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a $K$-homology fundamental class for $SU_q(2)$. We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincaré duality.
v2: main result strengthened, a new section added, title changed; 21 pages, LaTeX v1: 9 pages, Latex2e

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Operator Algebras, Quantum Algebra, 58B34, 46L87, 19K33

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http://salesiana.dossiersoluciones.com/handle/123456789/180674

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0211367
 http://arxiv.org/abs/math/0211367

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arXiv