  Let $\mathcal{A}$ be the $C^*$-algebra associated with $SU_q(2)$, $\pi$ 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 $\pi(\cla)'$ 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 $SU_q(2)$. We also show that a minor
modification of this equivariant spectral triple gives a fundamental class and
thus implements Poincar\'e duality.
