Domination of semigroups on standard forms of von Neumann algebras

Consider $(T_t{)}_{t\ge 0}$ and $(S_t{)}_{t\ge 0}$ as real $C_0$-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 $(T_t{)}_{t\ge 0}$ by $(S_t{)}_{t\ge 0}$, which means that $-S_{t}v\le T_{t}u\le S_{t}v$ holds for all $t\ge 0$ and all real $u$ and $v$ that satisfy $-v\le u\le v$. This characterisation extends the Ouhabaz characterisation for semigroup domination to the non-commutative $L^2$ spaces. Additionally, we present a simpler characterisation when both semigroups are positive as well as consider the setting in which $(T_t{)}_{t\ge 0}$ need not be real.