  This paper concerns the structure of the group of local unitary cocycles,
also called the gauge group, of an E_0-semigroup. The gauge group of a spatial
E_0-semigroup has a natural action on the set of units by operator
multiplication. Arveson has characterized completely the gauge group of
E_0-semigroups of type I, and as a consequence it is known that in this case
the gauge group action is transitive. In fact, if the semigroup has index k,
then the gauge group action is transitive on the set of k+1-tuples of
appropriately normalized independent units. An action of the gauge group having
this property is called k+1-fold transitive.
  We construct examples of E_0-semigroups of type II and index 1 which are not
2-fold transitive. These new examples also illustrate that an E_0-semigroup of
type II_k need not be a tensor product of an E_0-semigroup of type II_0 and
another of type I_k.
