Domination of semigroups on standard forms of von Neumann algebras

Consider (Tt)t≥0 and (St)t≥0 as real C -semigroups generated by closed and symmetric sesquilinear forms on a standard form of a von Neumann algebra. We provide a characterisation for the domination of the semigroup (Tt)t≥0 by (St)t≥0 , which means that - Stv≤ Ttu≤ Stv holds for all t≥ 0 and all real u and v that satisfy - v≤ u≤ v . This characterisation extends the Ouhabaz characterisation for semigroup domination to the non-commutative L2 -spaces. Additionally, we present a simpler characterisation when both semigroups are positive as well as consider the setting in which (Tt)t≥0 need not be real.