Tomita operator - Knowlpedia

Let $M\subseteq B(H)$ be a von Neumann algebra and let $\Omega\in H$ be both a cyclic vector and a separating vector for $M$ . On the dense subspace $M\Omega$ , define the antilinear operator

S_0(x\Omega)=x^*\Omega,\qquad x\in M.

Separatingness makes this formula well-defined, and cyclicity makes its domain dense. The operator $S_0$ is closable; its closure $S=\overline{S_0}$ is the Tomita operator of $(M,\Omega)$ . It is closed, densely defined, and satisfies $S^2\xi=\xi$ whenever both sides are defined.

Closability and the commutant

The companion operator $F_0(x'\Omega)=x'^*\Omega$ on $M'\Omega$ is densely defined because separatingness for $M$ is equivalent to cyclicity for the commutant $M'$ . The inclusions $S_0^*\supseteq F_0$ and $F_0^*\supseteq S_0$ prove that both operators are closable. In fact, their closures satisfy $S^*=F$ .

Polar decomposition and modular data

The polar decomposition of the closed antilinear operator is

S=J\Delta^{1/2}.

Here $J$ is the modular conjugation and $\Delta=S^*S$ is the modular operator. These objects depend on the pair $(M,\Omega)$ , not on $M$ alone. Their decisive properties are the content of the Tomita–Takesaki theorem.

Examples and scope

For $M=B(K)$ acting by left multiplication on the Hilbert–Schmidt operators $\operatorname{HS}(K)$ , the identity Hilbert–Schmidt vector is available only when $K$ is finite-dimensional; then $S(x)=x^*$ . More generally a faithful density operator supplies a cyclic separating vector in the standard Hilbert–Schmidt realization.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VI, §1 on the closable operators $S_0$ and $F_0$ .