New York Journal of Mathematics
Volume 9 (2003) 165-269


Robert T. Powers

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

Published: November 19, 2003
Keywords: completely positive maps, *-endomorphisms, Eo-semigroups
Subject: Primary 46L57; Secondary 46L55

This paper concerns the structure of strongly continuous one parameter semigroups of completely positive contractions of B(H) = B(K ⊗ L2(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 ⊗ L2(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.

Author information

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104