Order-unit space - Imported Knowlpedia Content

Definition

An Archimedean order-unit space is a real vector space $A$ with a proper convex cone $A_+$ and an element $e\in A_+$ such that:

for every $a\in A$ , some $r>0$ satisfies $-re\leq a\leq re$ ; and
if $a+\varepsilon e\in A_+$ for every $\varepsilon>0$ , then $a\in A_+$ .

The cone defines $a\leq b$ by $b-a\in A_+$ . The element $e$ is the order unit, and the second condition is Archimedeanness. The associated order-unit norm is

\|a\|_e=\inf\{r>0:-re\leq a\leq re\}.

States and the order

A state on $(A,e)$ is a positive real linear functional $\mu:A\to\mathbb R$ satisfying $\mu(e)=1$ . States have norm one for the order-unit norm and detect its order:

a\in A_+\quad\Longleftrightarrow\quad \mu(a)\geq0\ \text{for every state }\mu.

The state space is convex and carries the weak-star topology inherited from the dual of the normed space $A$ .

C*-algebra example

If $B$ is a unital $C^*$ -algebra, its self-adjoint part $B_{\mathrm{sa}}$ , with the positive cone and order unit $1_B$ , is an Archimedean order-unit space. Its order-unit norm is the restricted $C^*$ -norm. Complex $C^*$ -algebra states correspond exactly to the real order-unit states on $B_{\mathrm{sa}}$ .

Conventions and scope

Some authors say “order-unit space” without requiring Archimedeanness and then pass to an Archimedeanization. This knowl includes that axiom. Completeness in $\|\cdot\|_e$ is not part of the definition. An operator system has additional compatible matrix-level cones, so it is more than an order-unit space.

References

Marc A. Rieffel, Metrics on State Spaces, Memoirs of the American Mathematical Society 168, no. 796, 2004. AMS DOI record. Relevant: §2 on order-unit spaces, states, and order-unit norms.
Erik M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer, 1971. Publisher DOI record. Relevant: Chapters I–II on ordered vector spaces, order units, and state spaces.