We show that the conjugacy classes of continuous semigroups of V-endomorphisms of B(H) that possess a pure, normal, absorbing state (i.e., standard E 0 -semigroups) are indexed by the orbits of the natural action of the group of local cocycles on the set of intertwining semigoups of isometries. We use this to show that, in passing from an arbitrary E 0 -semigroup : to a cocycle perturbation in standard form, the boundary representation is compressed to a hereditary subalgebra of the domain D($ : ) of :. 2001 Academic Press 1. INTRODUCTION An E 0 -semigroup of B(H) is a w*-continuous one-parameter semigroup :=[: t : t 0] of unit preserving V-endomorphisms of B(H). It is called spatial, if it has a strongly continuous intertwining semigroup of isometries, in which case the generator $ : of : is spatially implemented (see the next section for definitions). There are exotic examples of E 0 -semigroups that have no intertwining semigroups of isometries [7]; they are called of type III and they are rather poorly understood. At the other extreme, there is a sequence of canonical examples of spatial E 0 -semigroups constructed via the second quantization from semigroups of isometries, using irreducible representations of either the CARs or the CCRs. These have enough intertwining semigroups to be reconstructed from them. They are called completely spatial (or of type I n , n=1, 2, ..., ) and they are completely classified up to cocycle conjugacy by a certain numerical index, the Arveson index [2]. Finally, there are spatial E 0 -semigroups that are not completely spatial. There is, at the momemt, only one known example of these for each value n=1, 2, ..., of the index [8], and uncountably many examples of index 0 [10]. They are referred to as type II n semigroups.
In this paper we study E 0 -semigroups that are in standard form, in the sense that there exists a pure, normal state |, with the property that