Positive cone of the predual

Let $M$ be a von Neumann algebra with predual $M_*$ . The positive cone of the predual is

(M_*)_+=\{\varphi\in M_*:\varphi(x)\geq0\text{ for every }x\in M_+\},

where $M_+$ is the positive cone of $M$ . Its elements are precisely the positive normal functionals on $M$ . The cone is norm closed, convex, proper, and generating in the self-adjoint part of $M_*$ . It defines the order $\varphi\leq\psi$ exactly when $\psi-\varphi\in(M_*)_+$ .

Norm and Jordan decomposition

For $\varphi\in(M_*)_+$ , positivity gives

\lVert\varphi\rVert=\varphi(1).

Every self-adjoint normal functional has a unique Jordan decomposition

\omega=\omega_+-\omega_-,

where $\omega_\pm\in(M_*)_+$ have orthogonal support projections and $\lVert\omega\rVert=\lVert\omega_+\rVert+\lVert\omega_-\rVert$ . Taking real and imaginary parts then shows that every element of $M_*$ is a linear combination of four positive normal functionals.

Concrete realization

For $M=B(H)$ , the predual is the trace-class operators, with pairing

\varphi_T(x)=\operatorname{Tr}(Tx).

Under this identification, $(M_*)_+$ corresponds exactly to the positive trace-class operators. Normal states correspond to those positive operators whose trace is one. For a general concrete von Neumann algebra $M\subseteq B(H)$ , the predual is a quotient of the trace class, so a normal positive functional can have more than one positive trace-class representative.

Order-theoretic role

The cone $(M_*)_+$ is not the positive cone of $M$ itself: its elements are functionals, not operators in $M$ . It records the order that is compatible with the canonical duality $M=(M_*)^*$ . Normal states form the base

\{\varphi\in(M_*)_+:\varphi(1)=1\}

of the nonzero cone, while unbounded normal weights lie outside $M_*$ .

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III, §2 on the predual, positive normal functionals, and Jordan decomposition.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. AMS record. Relevant: §5.2 on normal functionals and the predual order.