Tomita operator ($S_0(a\Omega )=a^{\ast }\Omega$)

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.

The positive operator in the polar decomposition $S=J{\Delta }^{1/2}$ of the Tomita operator. In a matrix-algebra standard representation with cyclic vector $D>0$, it is $\Delta (X)=D^{2}X{D}^{-2}$. For a tracial vector it is the identity.

The antiunitary factor in the polar decomposition of the Tomita operator. In the finite-dimensional and tracial settings it satisfies $J^2=I$ and $JMJ=M^{\prime }$. In a matrix-algebra standard representation by left multiplication, with positive invertible cyclic vector $D$, it acts by $J(X)=X^{\ast }$.