The Cone of Positive Operators

Introduction

The positive operators of an ordered space form a convex cone, and the cone carries more information than its definition suggests: its rays distinguish the indecomposable positive operators, its order intervals carry the operator norms, and its order units generate the order ideals on which the operators are bounded. This article studies the cone itself. It determines the extreme rays in the cases in which they are known — the dual cone of an order-unit space, where the extreme rays are the pure states; the operators between function spaces, where they are the weighted composition operators; and the matrix algebras, where they are described by the Størmer theorem — and it identifies the order units of the cone, which are the strictly positive rank-one operators generated by a strictly positive functional and an order unit of the target.

The cone of positive operators is the object the remaining articles of this category act on. The order projections of The Order Projection are the positive idempotents of the cone; the operator of The Order Unit as an Operator is a distinguished element of it; the affine maps of a convex set form a cone of positive operators after the identification of Operators on a Convex Set; and the multiplication operators of an ordered algebra are a subcone of it in The Left and Right Multiplication Operators on an Ordered Algebra. The positivity used is that of Positive Operators on an Ordered Space, the order and the order-unit norm are those of Ordered Vector Spaces and the Order Unit, the cones and their extreme rays are Cones, Extremal Rays and the Choquet Theory, and the completely positive maps, which are a subcone in the matrix case, are Operator Algebras. The duality of the cone with its dual is Duality Theory.

The Cone of Positive Operators

Basic Structure

Let $E$ and $F$ be ordered vector spaces, with positive cones $E_+$ and $F_+$, and let $L_+(E,F)$ be the cone of positive operators of Positive Operators on an Ordered Space.

Proposition. $L_+(E,F)$ is a convex cone in $L(E,F)$. It is pointed when $E_+$ generates $E$; it is closed for the topology of pointwise convergence, hence for every topology finer than that and in particular for the topology of bounded convergence, when $F_+$ is closed in $F$; and it is generating in $L(E,F)$ exactly when every operator is regular, that is, a difference of two positive operators.

Proof. The cone property was established in Positive Operators on an Ordered Space; pointedness is the same proposition. For closedness, if $T_\alpha\to T$ pointwise with every $T_\alpha\geq0$ and $x\in E_+$, then $T_\alpha x\to Tx$ with every $T_\alpha x\in F_+$, so $Tx\in F_+$ by the closedness of $F_+$. The generating statement is the definition of a regular operator.

Proposition (directedness). $L_+(E,F)$ is directed, that is, any two positive operators have an upper bound, exactly when for every pair $S,T\geq0$ the operator $S\vee T$ exists; this is automatic when $F$ admits finite suprema of positive elements, which is the Riesz-space hypothesis of The Order Projection.

Proof. An upper bound of $S$ and $T$ is a positive $U$ with $U - S\geq0$ and $U - T\geq0$; the least such is $S\vee T$. When $F$ has finite suprema of positives, the pointwise formula $(S\vee T)x = Sx\vee Tx$ defines the least upper bound, because both operators are linear and the supremum of linear maps with a lattice target is linear.

The Dual Cone and its Extreme Rays

Theorem (the extreme rays of the cone of functionals are the pure states). Let $E$ be an ordered vector space with order unit $u$, and let $E^*_+ = L_+(E,\mathbb{R})$ be the dual cone, with state space $S = \{f\in E^*_+ : f(u) = 1\}$. Then the extreme rays of $E^*_+$ are the rays through the pure states, the extreme points of $S$.

Proof. The base trace $B = \{f\in E^*_+ : f(u) = 1\}$ is a base of the cone, and the extreme points of a base correspond to the extremal rays of the cone by Cones, Extremal Rays and the Choquet Theory. The extreme points of $S$ are the pure states. For the continuous-function case the extreme points are the point evaluations, and for the finite-dimensional order-unit space they are the vector states, by the same article.

Theorem (the extreme rays between function spaces are the weighted compositions). Let $E = C(X)$ and $F = C(Y)$ for compact Hausdorff $X$ and $Y$. A positive operator $T : C(X)\to C(Y)$ generates an extreme ray of $L_+(C(X),C(Y))$ if and only if

$$ (Tf)(y) = h(y)\,f\bigl(\phi(y)\bigr) $$

for a continuous $\phi : Y\to X$ and a positive $h\in C(Y)$, with the caveat that the representation is unique only up to the points at which $h$ vanishes.

Proof sketch. If $T$ is of the stated form and $T$ is the sum of two positive operators, then on the dense set where $h>0$ each summand is forced to be a multiple of $T$ at each point, by the lattice characterisation of the extreme positive maps of $C(X)$ into a Riesz space, giving the extremality. Conversely an extreme positive operator preserves norm-one extreme points of $C(X)$, i.e. assigns to each point evaluation of $C(X)$ a scalar multiple of a point evaluation of $C(Y)$; that correspondence defines $\phi$ and $h$. The argument rests on the fact that a positive operator carries the extreme rays of the unit ball of the domain into extreme rays of the unit ball of the target unless a strict loss occurs, and a strict loss splits the operator.

Remark (the matrix case). For $E = F = H_n(\mathbb{C})$ with the Loewner order the positive maps form a cone whose extreme rays are the maps $X\mapsto f(X)\,vv^{*}$ for a pure state $f$ of $H_n(\mathbb{C})$ and a rank-one positive $vv^{*}$; the description is the Størmer theorem and it is quoted, since the proof requires the representation theory of Operator Algebras.

The Order Unit Defined by the Cone

Definition. An operator $T\in L_+(E,F)$ is an order unit of the cone of positive operators when every operator lies between two multiples of it: for every $S$ there is $\lambda>0$ with $-\lambda T\leq S\leq\lambda T$.

Theorem (a strictly positive rank-one operator is an order unit). Let $E$ have an order unit $u$, let $f$ be a positive functional that is strictly positive, $f(x)>0$ for every nonzero $x\in E_+$, and let $v$ be an order unit of $F$. Then the rank-one operator

$$ T = f\otimes v, \qquad Tx = f(x)\,v , $$

is an order unit of $L_+(E,F)$, and the order-unit norm it defines dominates the operator norm.

Proof. The base $\{x\in E_+ : f(x) = 1\}$ is compact in the weak topology of the order interval when $E$ is a Banach space in its order-unit norm, and the order unit $u$ is an interior point of $E_+$, so strict positivity of $f$ gives a constant $c>0$ with $f(x)\geq c\|x\|_u$ for $x\in E_+$: otherwise a sequence $x_n$ with $f(x_n)\to0$ and $\|x_n\|_u = 1$ would have a cluster point in the compact base on which $f = 0$. For a bounded $S$ one has $Sx\leq\|S\|\|x\|_u\,v$ and $-Sx\leq\|S\|\|x\|_u\,v$ for $x\in E_+$, by the order-unit estimate of Positive Operators on an Ordered Space; substituting $\|x\|_u\leq f(x)/c$ gives $-\lambda T\leq S\leq\lambda T$ with $\lambda = \|S\|/c$. Hence every bounded operator is dominated by a multiple of $T$, and $T$ is an order unit of the cone of bounded positive operators; an unbounded operator is dominated by no multiple of $T$.

Corollary (the identity is an order unit of the two-sided ideal it generates). In an ordered vector space $E$ with order unit $u$, the identity $I$ of $E$ is an order unit of the order ideal it generates inside $L(E,E)$, namely the set of operators $S$ with $-\lambda I\leq S\leq\lambda I$ for some $\lambda$, and this ideal consists of the operators bounded by a multiple of the identity and is generally a proper subspace of $L(E,E)$.

Proof. The condition $-\lambda I\leq S\leq\lambda I$ is the defining condition of the ideal generated by $I$ in the ordered vector space $L(E,E)$; it is a subspace because it is closed under addition and scaling, and it is contained in the bounded operators with the order-unit norm as the norm.

Worked Cases

The Cone of Probability Densities

Let $E = \mathbb{R}$ and $F = C(X)$, so that $L_+(\mathbb{R},C(X))$ is the cone of nonnegative continuous functions identified with $F_+$ by $T\mapsto T(1)$. The extreme rays of the cone of positive operators are then the extreme rays of the cone of nonnegative functions, which are the rays through the point masses formally, and in the continuous case the rays through the functions vanishing off a point do not exist; the correct statement is for $F = \ell^\infty(S)$, where the extreme rays are the point masses $\delta_s$ and the weighted compositions of the theorem reduce to the evaluation maps.

Order Units of the Cone of Matrices

Let $E = F = \mathbb{R}^n$ with the standard cone, so that $L_+(E,F)$ is the cone of matrices with nonnegative entries. An order unit of this cone is a matrix of strictly positive entries: for every matrix $S$ there is $\lambda$ with $-\lambda T\leq S\leq\lambda T$ entrywise, exactly when every entry of $T$ is positive. The extreme rays of the cone are the matrix units $E_{ij}$, and the order-unit norm defined by a strictly positive $T$ is $\|S\|_T = \max_{ij}\lvert S_{ij}\rvert/T_{ij}$, a weighted supremum norm.

Positive Maps of a Matrix Algebra

For $E = F = H_n(\mathbb{C})$ the positive maps form a cone whose extreme rays are described by the Størmer theorem and whose order units are the strictly positive maps, those carrying every nonzero positive semidefinite matrix to a positive definite one. The completely positive maps form a subcone which is closed and not an extreme-ray face: the transpose map is an extreme ray of the positive maps that is not completely positive, so the two cones have different extreme structure.

Summary

The positive operators $L_+(E,F)$ form a convex cone, pointed when $E$ is directed, closed for pointwise convergence when $F_+$ is closed, and generating exactly when every operator is regular; it is directed when the target admits finite suprema of positive elements. Its extreme rays are known in three cases: for the dual cone of an order-unit space they are the pure states, through the correspondence between extreme points of a base and extreme rays of a cone; for $C(X)\to C(Y)$ they are the weighted composition operators $f\mapsto h\cdot(f\circ\phi)$; and for the matrix algebra they are described by the Størmer theorem, all of them quoted from their standard treatments. The cone defines the operator order on $L(E,F)$, and an order unit of the cone is a strictly positive rank-one operator $f\otimes v$ formed from a strictly positive functional and an order unit of the target: every bounded operator is then between multiples of it, and the identity is an order unit of the two-sided ideal it generates. The positivity is Positive Operators on an Ordered Space, the extreme-ray theory is Cones, Extremal Rays and the Choquet Theory, the order-unit norm is Ordered Vector Spaces and the Order Unit, and the completely positive subcone is Operator Algebras.

Summary of Notation

Symbol	Meaning
$L_+(E,F)$	Cone of positive operators
$E^*_+ = L_+(E,\mathbb{R})$	Dual cone, cone of positive functionals
Pure state	Extreme point of the state space, generating an extreme ray of $E^*_+$
$f\otimes v$	Rank-one operator $x\mapsto f(x)v$
$f\otimes v$ strictly positive	Order unit of the cone of bounded positive operators
$h\cdot(f\circ\phi)$	Weighted composition operator, an extreme ray for $C(X)\to C(Y)$
$-\lambda T\leq S\leq\lambda T$	Domination by the order unit

The Cone of Positive Operators

Introduction