This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E0-semigroup. The gauge group of a spatial E0-semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E0-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 E0-semigroups of type II and index 1 which are not 2-fold transitive. These new examples also illustrate that an E0-semigroup of type IIk need not be a tensor product of an E0-semigroup of type II0 and another of type Ik.
ASJC Scopus subject areas