nLab positive cone

Idea

Geometrically, the positive elements of a vector space with a partial order form a cone, called the positive cone. The concept makes sense even more generally but is particularly important in operator algebras.

Definition

Let $V$ be a partially ordered set and let $0$ be an element of $V$. Then the positive cone $V^+$ of the pointed poset $(V,0)$ is the up set of $0$:

$$V^+ \coloneqq \{x\colon V \;|\; 0 \leq x\} .$$

If $V$ is an operator algebra, or more generally an ordered group, then we use the usual identity element $0$ here.

Properties

Let $V$ be an ordered group; that is, $V$ is a poset (as above) and also a group (written additively but not necessarily commutative) with this compatibility property:

• If $a \leq b$, then $a + c \leq b + c$ and $c + a \leq c + b$.

Then the positive cone $V^+$ satisfies these properties:

• If $a, -a \in V^+$ (where $-a$ is the inverse of $a$), then $a = 0$;
• Conversely, $0 \in V^+$;
• If $a, b \in V^+$, then $a + b \in V^+$;
• If $a \in V^+$, then $b + a - b \in V^+$ (which is trivial if $V$ is commutative).

Conversely, if $V^+$ is any subset of the group $V$ with these properties, then $V$ becomes an ordered group with either of these equivalent definitions:

• $a \leq b$ iff $b - a \in V^+$,
• $a \leq b$ iff $-a + b \in V^+$.

In this way, we have a bijection between ordered group structures and positive cones in a group.

The extended positive cone

Motivation

The finite measures on a given measurable space form an ordered vector space $V$, and the positive cone $V^+$ consists precisely of the finite positive measures. But we often want to allow positive measures to take infinite values. The space of (possibly infinite) positive measures is the extended positive cone of $V$.

I only know the general definition in some rather limited cases:

Definition (positive cone of a von Neumann algebra)

Let $V$ be a $W^*$-algebra, and let $V_*$ be its predual. Recall that $V$ is the space of continuous linear maps from $V_*$ to the base field. The extended positive cone $\bar{V}^+$ of $V$ is the space of lower semicontinuous linear maps from the positive cone $V_*^+$ of $V_*$ to the space $\bar{\mathbb{R}}^+ = [0,\infty]$ of extended positive real numbers.

The extended positive real numbers are really showing up in their guise as the nonnegative lower real numbers (the appropriate targets for a lower semicontinuous map), and the extended positive cone is really a generalisation of the nonnegative upper reals. In particular, the extended positive cone of $\mathbb{R}$ itself is $[0,\infty]$.

This doesn't include the motivating example, but the following generalisation does:

Definition (positive cone of an ordered module over a von Neumann algebra)

Let $V$ be a module over the $W^*$-algebra $A$, and let $V$ have the structure of an ordered group such that the action of $A$ preserves order (in that a positive element acting on a positive element gives a positive element). Then the extended positive cone $\bar{V}^+$ of $V$ consists of formal infinitary $\bar{A}^+$-linear combinations of positive elements of $V$ modulo the (hopefully) obvious equivalence relation.

When applied to the space of finite measures on a localisable measurable space $X$ (acted on by the $W^*$-algebra $L^\infty(X)$), this should give the positive measures on $X$ (but I need to check the details).

References

The extended positive cone of a $W^*$-algebra is Definition 4.4 of:

• Masamichi Takesaki, Theory of Operator Algebras II, Springer; Google books.