Equivariant spectral triples and Poincare duality for SU_q(2)

Abstract

Let A be the C^∗-algebra associated with SU_q(2), let π be the representation by left multiplication on the L₂ 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(A)' that has bounded commutator with D. This implies that the equivariant spectral triple under consideration does not admit a rational Poincare 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 Poincare duality.