 Tomita operator

For a Von Neumann algebra with a cyclic separating vector, this is a densely defined antilinear operator. It is closable: the commutant orbit lies in the domain of its antilinear adjoint and is dense. Write $S$ for its closure and $S=J{\Delta }^{1/2}$ for its polar decomposition. In finite dimension it is everywhere defined and invertible. For a tracial vector, $S_0$ is isometric and extends to an antiunitary involution.