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.

将\((T_t)_{t\ge 0}\)和\((S_t)_{t\ge 0}\)视为实\(C_0\) -由标准形式上的封闭和对称倍半线性形式生成的半群冯诺依曼代数。我们通过\ ((S_t)_{t\ge 0}\) 提供半群 \( (T_t)_{t\ge 0}\)支配的表征，这意味着\(-S_t v\le T_t u\le S_t v\)对于所有\(t\ge 0\)以及满足\(-v\le u\le v\)的所有真实u和v成立。此表征将半群支配的 Ouhabaz 表征扩展到非交换\(L^2\)空间。此外，当两个半群均为正时，我们提出了更简单的表征，并考虑了\((T_t)_{t\ge 0}\)不必为实数的设置。