Extended positive cone of a von Neumann algebra

Definition

Let $M$ be a von Neumann algebra and $M_*^+$ its cone of normal positive functionals. The extended positive cone $\widehat M_+$ consists of the maps

m:M_*^+\longrightarrow[0,\infty]

that are additive, positively homogeneous, and lower semicontinuous for the norm topology on $M_*^+$ . Thus $m(\omega_1+\omega_2)=m(\omega_1)+m(\omega_2)$ and $m(t\omega)=t\,m(\omega)$ for $t\geq0$ , with $0\cdot\infty=0$ . It is ordered pointwise. Every $x\in M_+$ defines an extended positive element by $m_x(\omega)=\omega(x)$ , but elements of $\widehat M_+$ may take the value $\infty$ .

Operator realization

Extended positive elements admit a spectral realization by positive self-adjoint operators affiliated with $M$ , together with an allowed infinite part supported on a projection. This realizes $\widehat M_+$ as a completion of $M_+$ that is closed under increasing suprema Haagerup, §1. The functional description is intrinsic and does not require choosing a particular representation of $M$ .

Operations and examples

Addition, multiplication by nonnegative scalars, and increasing suprema are computed pointwise on $M_*^+$ . If $a\in M$ and $m\in\widehat M_+$ , then $a^*ma$ is defined by

(a^*ma)(\omega)=m(a\omega a^*), \qquad (a\omega a^*)(x)=\omega(a^*xa).

An ordinary positive element gives a finite-valued example. At the opposite extreme, the formal value $+\infty$ on a nonzero projection produces an extended element that cannot belong to $M_+$ .

Role in integration

The extended cone is the natural codomain for an operator-valued weight: integrating a positive element over one part of a noncommutative space can produce an unbounded positive object over another part. Ordinary $N_+$ -valued maps cannot express this behavior without an artificial boundedness assumption.

References

Uffe Haagerup, “Operator-Valued Weights in von Neumann Algebras I,” Journal of Functional Analysis 32 (1979), 175–206. DOI record. Relevant: §1 on the extended positive part and its operations.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter IX, §4 on conditional expectations and operator-valued weights.