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)