E-semigroups Subordinate to CCR Flows

Abstract

The subordinate E-semigroups of a fixed E-semigroup are in one-to-one correspondence with local projection-valued cocycles of that semigroup. For the CCR flow we characterise these cocycles in terms of their stochastic generators, that is, in terms of the coefficient driving the quantum stochastic differential equation of Hudson-Parthasarathy type that such cocycles necessarily satisfy. In addition various equivalence relations and order-type relations on E-semigroups are considered, and shown to work especially well in the case of those semigroups subordinate to the CCR flows by exploiting our characterisation.