  If A is a weak C^*-Hopf algebra then the category of finite dimensional
unitary representations of A is a monoidal C^*-category with monoidal unit
being the GNS representation D_eps associated to the counit \eps. This category
has isomorphic left dual and right dual objects which leads, as usual, to the
notion of dimension function. However, if \eps is not pure the dimension
function is matrix valued with rows and columns labelled by the irreducibles
contained in D_eps. This happens precisely when the inclusions A^L < A and A^R
< A are not connected. Still there exists a trace on A which is the Markov
trace for both inclusions. We derive two numerical invariants for each C^*-WHA
of trivial hypercenter. These are the common indices I and \delta, of the Haar,
respectively Markov conditional expectations of either one of the inclusions
A^{L/R} < A and Adual^{L/R} < Adual. In generic cases I > \delta. In the
special case of weak Kac algebras we show that I=\delta is an integer.
