Continuous spatial semigroups of completely positive maps of B(H)

This paper concerns the structure of strongly continuous one parameter semigroups of completely positive contractions of B(H) = B(K⊗L(0,∞)) which are intertwined by translation. These are called CP -flows over K. Using Bhat’s dilation result each unital CP -flow over K dilates to an Eo-semigroup of B(H1) where H1 can be considered to contain B(K⊗L(0,∞)). Every spatial Eo-semigroup is cocycle conjugate to one dilated from a CP -flow. Each CP -flow is determined by its associated boundary weight map which determines the generalized boundary representation. The index of the Eo-semigroup dilated from a CP -flow is calculated. Machinery for determining whether two CP -flow dilate to cocycle conjugate Eo-semigroups is developed. This paper is available via http://nyjm.albany.edu:8000/j/2003/9-13.html