Equivariant spectral triples and Poincar\'e duality for $S{U}_q(2)$

Let $\mathcal{A}$ be the $C^{\ast }$-algebra associated with $S{U}_q(2)$, $\pi$ 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 $\pi (\text{cla}{)}^{\mathrm{\prime }}$ that has bounded commutator with $D$. This implies that the equivariant spectral triple under consideration does not admit a rational Poincar'e dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a $K$-homology fundamental class for $S{U}_q(2)$. We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincar'e duality.