Natural positive cone

Let $M\subseteq B(H)$ be a von Neumann algebra with cyclic separating vector $\Omega$ , and let $S=J\Delta^{1/2}$ be the polar decomposition of its Tomita operator. The natural positive cone is

P_\Omega^\natural =\overline{\{\Delta^{1/4}x\Omega:x\in M_+\}}\subseteq H,

where the closure is in the Hilbert-space norm, $\Delta$ is the modular operator, and $J$ is the modular conjugation. Thus $P_\Omega^\natural$ is a distinguished closed convex cone attached to the represented algebra and its modular data, not the operator-positive cone $M_+$ . It is the cone used to place this representation in standard form.

Self-duality and invariance

The cone is self-dual:

P_\Omega^\natural =\{\xi\in H:\langle\xi,\eta\rangle\geq0 \text{ for every }\eta\in P_\Omega^\natural\}.

It is fixed pointwise by $J$ , and $xJxJ$ maps it into itself for every $x\in M$ . These are structural theorems of modular theory, not extra clauses in the displayed construction.

Every normal positive functional on $M$ is represented by a unique vector $\xi\in P_\Omega^\natural$ through $\varphi(x)=\langle x\xi,\xi\rangle$ . This uniqueness is one reason the cone is more useful than an arbitrary positive-vector realization.

Relationship to standard form

The quadruple $(M,H,J,P_\Omega^\natural)$ is a standard form of $M$ . Different cyclic separating vectors may produce different Tomita operators, but the resulting standard forms are canonically unitarily equivalent. Consequently the natural cone expresses positivity in a representation-independent way once the standard-form identification is made.

References

H. Araki, “Some Properties of Modular Conjugation Operator of von Neumann Algebras and a Non-commutative Radon–Nikodym Theorem with a Chain Rule,” Pacific Journal of Mathematics 50 (1974), 309–354. DOI record. Relevant: construction, self-duality, and representation of normal positive functionals by cone vectors.
U. Haagerup, “The Standard Form of von Neumann Algebras,” Mathematica Scandinavica 37 (1975), 271–283. DOI record. Relevant: Theorem 1.6 and Definition 2.1 on the natural cone and standard form.