It is known that
every semigroup of normal completely positive maps
P = \{P_t: t\geq 0\} of B(H) which  
preserves the identity has a minimal
dilation to an E_0-semigroup acting on B(K) for
some Hilbert space K containing H. 
The minimal dilation of P is
unique up to conjugacy.  In a previous paper 
("The index of quantum dynamical semigroups") a
numerical index was introduced for semigroups
of completely positive maps and it was shown that
the index of P agrees with the index of its
minimal dilation to an E_0-semigroup.  However, no examples
were discussed, and no computations were made.

In this paper we calculate the index
of a unital completely positive semigroup whose
generator is a {\it bounded} operator

L: B(H)--> B(H)

in terms of natrual structures associated with
the generator.  This includes all unital CP semigroups
acting on matrix algebras.  We also show that the
minimal dilation of the semigroup P to an 
E_0-semigroup is conjugate to a cocycle perturbation
of a CAR/CCR flow.

(To appear: International Journal of Mathematics)