positive cone

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.

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.

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 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:

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:

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).

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

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

Revised on October 17, 2013 11:09:50
by Toby Bartels
(64.89.53.234)