  We study various noncommutative geometric aspects of the compact quantum
group SU_q(2) for positive q (not equal to 1), following the suggestion of
Connes and his coauthors [CL, CD] for considering the so-called true Dirac
operator.
  However, it turns out that the method of the above references do not extend
to the case of positive (not equal to 1) values of q in the sense that the true
Dirac operator does not have bounded commutators with "smooth" algebra elements
in this case, in contrast to what happens for complex q of modulus 1.
Nevertheless, we show how to obtain the canonical volume form, i.e. the Haar
state, using the true Dirac operator.
