The Order Unit as an Operator

Introduction

An order unit is an element, but it acts. Reading it as an operator produces the two maps that carry the whole metric theory of an ordered vector space. The first is the canonical embedding

$$ \iota_u : \mathbb{R}\to E, \qquad \iota_u(1) = u, $$

which is positive because $u\in E_+$ and which puts the line through $u$ into $E$ as an ordered subspace. The second is the evaluation at the order unit,

$$ \mathrm{ev}_u : L(E,F)\to F, \qquad \mathrm{ev}_u(T) = Tu, $$

an operator between the space of operators and the target; on the dual it is the functional $f\mapsto f(u)$ that defines the state space, and on the operators it is the map whose norm is the order-unit norm of a positive operator. The two maps are transposes of each other, and together they are what the order unit is as an operator: a vector that probes the space and its operators.

The article collects these facts and draws their consequences. The order-unit norm is the gauge of the interval $[-u,u]$, hence the operator norm induced by the evaluation at $u$; the order topology is the initial topology of the evaluation maps, so it is determined by the order unit alone; and the state space is the intersection of the positive cone of the dual with the level set $\mathrm{ev}_u = 1$, a weak- compact base. The order unit is therefore not one more datum beside the order and the topology but the single element from which the other two are read, and this is the sense in which an order-unit space* is a complete description of its ordered normed structure.

The order, the order-unit norm, the order topology and the state space are those of Ordered Vector Spaces and the Order Unit; the positivity of the maps is that of Positive Operators on an Ordered Space; the extreme points of the state space are The Cone of Positive Operators and Cones, Extremal Rays and the Choquet Theory; the weak- topology and the polar are Duality Theory, and the bounded operators are Bounded Operators and the Operator Norm. The order-unit norm is the notation of Jordan Algebras and the Positive Cone. The multiplication operators of an ordered algebra — where the order unit is the algebra identity and acts by multiplication — are deferred to The Left and Right Multiplication Operators on an Ordered Algebra* later in this category, and the Saks space and the order-unit semigroup are not entered.

The Two Canonical Maps

The Embedding of the Scalars

Proposition. Let $E$ be an ordered vector space over $\mathbb{R}$ with order unit $u$. The map $\iota_u : \mathbb{R}\to E$, $\iota_u(\lambda) = \lambda u$, is an injective positive linear map; it is an order isomorphism onto its image, and it identifies $\mathbb{R}$ with the one-dimensional ordered subspace spanned by $u$ when $u>0$ and the order of $\mathbb{R}$ is the usual one.

Proof. Linearity is immediate; positivity is $\lambda\geq0\Rightarrow\lambda u\in E_+$ by the closure of the cone under nonnegative scaling. Injectivity is $u\neq0$, which holds because an order unit is positive and the cone is pointed. An order isomorphism onto its image is the same statement two-sided: $\lambda u\leq\mu u$ if and only if $(\mu-\lambda)u\in E_+$, which for $\mu-\lambda$ of either sign is equivalent to $\mu\geq\lambda$, using pointedness for the reverse direction.

The Evaluation at the Order Unit

Definition. The evaluation at the order unit is

$$ \mathrm{ev}_u : L(E,F)\to F, \qquad \mathrm{ev}_u(T) = T(u), $$

for an ordered vector space $F$ carrying an order unit $v$.

Proposition (the evaluation is positive and linear, with the right kernel). The map $\mathrm{ev}_u$ is linear and positive: $T\geq0$ implies $T(u)\in F_+$. Its kernel is the subspace of the operators vanishing at $u$, and on the positive operators it is surjective onto the positive elements of $F$ that the order-unit structure of $E$ allows, namely onto $F_+$ when $E$ is the free order-unit space generated by $u$ with the interval $[-u,u]$ as its unit ball.

Proof. Linearity is the linearity of $T$; positivity is $Tu\in F_+$ for $T\geq0$ and $u\in E_+$. The kernel is the preimage of zero. For the surjectivity, a positive $w\in F_+$ is the image of the operator $x\mapsto \lambda(x)\,w$ when every $x$ admits a scalar $\lambda(x)$ with $x = \lambda(x)u$, which is the free case; in general the functional that reads off the coefficient may fail to be linear, and the image is the set of $w$ for which such a linear reading exists.

Proposition (the transposes). The transpose of $\iota_u$ is $\mathrm{ev}_u$ restricted to the dual, in the sense that for $f\in E^*$,

$$ \iota_u'(f) = f(u) = \mathrm{ev}_u(f); $$

consequently the state space is the intersection of the positive cone with the level set of the transpose of the embedding,

$$ S = E^*_+\cap\bigl\{f : \iota_u'(f) = 1\bigr\}, $$

and it is a base of the cone $E^*_+$.

Proof. The transpose is defined by $(\iota_u'f)(\lambda) = f(\lambda u) = \lambda f(u)$, so $\iota_u'f = f(u)$ after the identification of $\mathbb{R}^*$ with $\mathbb{R}$. The state space is the stated intersection by Ordered Vector Spaces and the Order Unit, and the base property is that every positive $g$ with $g(u)>0$ is $g = g(u)\,(g/g(u))$.

The Order-Unit Norm as an Operator Norm

The Gauge of the Order Interval

Proposition (the norm is the gauge). The order-unit norm of Ordered Vector Spaces and the Order Unit is the gauge of the order interval $[-u,u]$,

$$ \|x\|_u = \inf\{\lambda>0 : x\in\lambda[-u,u]\}, $$

and the interval is the closed unit ball, convex, symmetric about the origin and absorbing.

Proof. The set $\{\lambda>0 : x\in\lambda[-u,u]\}$ is exactly $\{\lambda>0 : -\lambda u\leq x\leq\lambda u\}$, which is the set in the definition of the norm; the gauge of a convex absorbing symmetric set is the norm of the set, and the interval has these properties because the cone is convex, pointed and has $u$ as an order unit.

Proposition (the order unit norm of a positive operator is the norm of its value at the unit). For positive $T\in L_+(E,F)$,

$$ \|T\| = \|\mathrm{ev}_u(T)\|_v = \|Tu\|_v, $$

the operator norm being taken for the order-unit norms.

Proof. This is the order-unit estimate of Positive Operators on an Ordered Space, stated there and recalled here in the language of the evaluation: the supremum defining the operator norm is attained at the order unit, so the norm is the norm of the evaluation.

Determination of the Topology and the State Space

Proposition (the order topology is given by the state space). The order topology of $E$ is the topology of the norm $\|x\|_u$, which is the supremum of the evaluations over the state space,

$$ \|x\|_u = \sup\bigl\{\lvert f(x)\rvert : f\in S\bigr\}. $$

Proof. Positivity is detected by the states: an element $y$ lies in $E_+$ if and only if $f(y)\geq0$ for every $f\in S$, by the order-unit form of the Hahn–Banach separation theorem of Duality Theory applied to a point outside the cone. Hence for $x\in E$ and $\lambda>0$ the two inequalities $x\leq\lambda u$ and $-x\leq\lambda u$ are equivalent to $f(x)\leq\lambda$ and $-f(x)\leq\lambda$ for every state, that is, to $\lvert f(x)\rvert\leq\lambda$ for every state. Taking the infimum over $\lambda$ and using the gauge formula for the norm gives the displayed identity.

Proposition (the state space as an operator level set). With $S = \mathrm{ev}_u^{-1}(1)\cap E^*_+$ as above, the evaluation $f\mapsto f(x)$ embeds $E$ into the space of affine continuous functions on $S$, and it is an isometry onto its image when the norm is the order-unit norm. When $E$ is a Banach space in that norm the state space is weak- compact and convex, so it has extreme points, the pure states*, and the image of $E$ in the affine continuous functions on $S$ is norm closed and generally a proper subspace; it is all of the continuous affine functions exactly in the case in which $E$ is recovered from its state space, which is the Kadison representation theorem and is quoted here.

Proof. The embedding is linear and its norm is the order-unit norm by the previous proposition. Compactness of $S$ is the Banach–Alaoglu theorem applied to the polar of the unit ball; convexity is the linearity of the defining conditions; the extreme points exist by the Krein–Milman theorem of Convex Sets and the Convex Hull. The identification of the image is the Kadison representation theorem; the integral representation behind it is Cones, Extremal Rays and the Choquet Theory.

Remark (the order unit is the whole structure). The three data — the cone $E_+$, the order unit $u$, and the order-unit norm — determine one another: the cone is recovered as $\{x : \|x - \lambda u\|_u\leq\lambda \text{ for all large } \lambda\}$, the unit is recovered as the element of norm one that is interior to the cone, and the norm is the gauge of the interval. An order-unit space is thus a Banach space together with either of the three, and the passage between them is the content of this article.

Worked Cases

The Continuous Functions Again

For $E = C(X)$ with $u = 1$ the embedding $\iota_u$ is the inclusion of the constants, the evaluation $\mathrm{ev}_u$ is integration against the probability measures when restricted to the dual, and the order-unit norm is the supremum norm, which is the supremum of the evaluations at the point states. The state space is the simplex of probability measures, and the recovery of $C(X)$ from its state space is the Riesz representation theorem and the Kadison theorem in the commutative case.

The Symmetric Matrices

For $E = H_n(\mathbb{R})$ with $u = 1$ the embedding is the inclusion of the scalar matrices, the evaluation sends a positive map $T$ to the matrix $T(1)$, and the order-unit norm of a positive map is the operator norm of that matrix. The state space is the set of positive linear functionals normalised at the identity, that is, the density matrices under the trace pairing; the pure states are the vector states, and the evaluation of a matrix at a pure state is its quadratic form.

Summary

An order unit acts by two maps. The embedding $\iota_u:\mathbb{R}\to E$, $1\mapsto u$, is injective, positive and an order isomorphism onto its image; its transpose is the evaluation $\mathrm{ev}_u$, which on the dual is $f\mapsto f(u)$ and on the operators is $T\mapsto Tu$, positive in each case and with kernel the maps vanishing at $u$. The order-unit norm is the gauge of the interval $[-u,u]$ and the operator norm induced by the evaluation: $\|T\| = \|Tu\|_v$ for a positive operator, and $\|x\|_u = \sup\{\lvert f(x)\rvert : f\in S\}$ for every element. The order topology is the initial topology of the evaluations, so it is determined by the order unit alone; the state space is the level set $\mathrm{ev}_u = 1$ in the positive cone of the dual, a weak- compact convex base whose extreme points are the pure states. The cone, the order unit and the norm determine one another, and an order-unit space is the resulting Banach space together with any one of them. The structure is Ordered Vector Spaces and the Order Unit, the positivity is Positive Operators on an Ordered Space, the extreme points are The Cone of Positive Operators and Cones, Extremal Rays and the Choquet Theory, the duality is Duality Theory, and the algebra case, where the order unit multiplies, is The Left and Right Multiplication Operators on an Ordered Algebra* later in this category.

Summary of Notation

Symbol	Meaning
$\iota_u:\mathbb{R}\to E$	Embedding of the scalars at the order unit, $\iota_u(1) = u$
$\mathrm{ev}_u(T) = Tu$	Evaluation at the order unit
$\iota_u'(f) = f(u)$	Transpose of the embedding, the evaluation on the dual
$\\|x\\|_u = \inf\{\lambda>0 : x\in\lambda[-u,u]\}$	Order-unit norm as the gauge of the interval
$\\|T\\| = \\|Tu\\|_v$	Norm of a positive operator is the norm of its value at the unit
$\\|x\\|_u = \sup_{f\in S}\lvert f(x)\rvert$	Norm as the supremum over the state space
$S = \mathrm{ev}_u^{-1}(1)\cap E^*_+$	State space, a weak-* compact base
Order-unit space	The cone, the unit and the norm, mutually determined