GYROGROUPS AND HOMOGENEOUS LOOPS

For a canonical unbounded CP-map (lambda)(.) from the vector space of densely defined closed operators on a Hilbert space H into the vector space of densely defined closed operators on the symmetric Fock space over H, we prove that (lambda)(A) is directly (lambda)(B)-relatively bounded provided A and B are positive and A is directly B-relatively bounded, that is, ||A(PSI)|| <= c||B(PSI)|| on dom(B). We consider formal generators on the symmetric Fock space corresponding to this map (lambda)(.) and show that they determine uniquely certain quantum dynamical semigroups.