Positive Functionals and Self-Adjointness

Introduction

The involution of an ordered involutive algebra induces an involution on the dual space, $\varphi\mapsto\varphi^{*}$ with

$$ \varphi^{*}(a) = \overline{\varphi(a^{*})} , $$

and the functionals fixed by it are the self-adjoint (Hermitian) functionals, those with $\varphi(a^{*}) = \overline{\varphi(a)}$ for every $a$. The positive functionals are self-adjoint,

$$ \varphi\geq0 \implies \varphi = \varphi^{*} , $$

in a unital algebra, and this is the self-adjointness of the positivity: the positivity of a functional is a property of its real part, the imaginary part of a positive functional is invisible on the self-adjoint elements and vanishes, and the cone of the functionals is contained in the real vector space of the self-adjoint functionals. The article proves this, shows that the self-adjoint functionals form an ordered real vector space whose positive cone is the cone of the positive functionals, and describes the interaction of the self-adjointness with the positivity, the Cauchy–Schwarz inequality and the states.

The reason to isolate the self-adjointness is that it is the dual counterpart of the Hermitian elements: the involution acts on the algebra and on its dual, the self-adjoint elements are the fixed points of the first, the self-adjoint functionals the fixed points of the second, and the order lives on both. The positive functionals are the self-adjoint functionals that are positive on the cone, so the order of the algebra is the order of the self-adjoint functionals read through the duality $a\mapsto\varphi(a)$; this is the functional-level statement of Self-Adjoint Elements and the Order, and it is the reason the states of a physical theory are the positive normalised self-adjoint functionals.

The self-adjoint elements and their order are Self-Adjoint Elements and the Order; the positive functionals, the states and the extreme points are The Cone of Positive Functionals; the forms and the order are Positive Definite Forms and the Order; the Hermitian elements and the order unit are Hermitian Elements and the Order Unit; the ordered involution is Ordered Involutive Algebras; the Jordan order is The Jordan Algebra of Self-Adjoint Elements; the adjoint of a positive operator is The Adjoint of a Positive Operator; and the adjoint of the left multiplication is The Adjoint of the Left Multiplication on an Ordered Algebra. The operator-algebraic duality is Operator Algebras and The Theory of von Neumann Algebras of Part II.

The Involution on the Dual

Definition. The adjoint of a functional is $\varphi^{*}(a) = \overline{\varphi(a^{*})}$; it is again linear, the map $\varphi\mapsto\varphi^{*}$ is conjugate-linear and an involution of the dual space, $(\varphi^{*})^{*} = \varphi$, and a functional is self-adjoint (or Hermitian) when $\varphi^{*} = \varphi$.

Proposition (the decomposition of a functional). Every functional decomposes uniquely as

$$ \varphi = \operatorname{Re}\varphi + i\operatorname{Im}\varphi , \qquad \operatorname{Re}\varphi = \tfrac12(\varphi + \varphi^{*}), \quad \operatorname{Im}\varphi = \tfrac1{2i}(\varphi - \varphi^{*}) , $$

into a sum of self-adjoint functionals; the self-adjoint functionals form a real vector space $A^{*}_{\mathrm{sa}}$, and the map $\varphi\mapsto\varphi^{*}$ is the identity on it and the reflection $\varphi\mapsto -\varphi$ on the imaginary direction.

Proof. The formulas are the standard decomposition of a complex-valued function into its real and imaginary parts with respect to the involution; the two summands are self-adjoint because $(\varphi^{*})^{*} = \varphi$ and $(\operatorname{Re}\varphi)^{*} = \operatorname{Re}\varphi$, $(\operatorname{Im}\varphi)^{*} = \operatorname{Im}\varphi$; the uniqueness is immediate from the definitions.

Proposition (self-adjointness on the Hermitian elements). A functional is self-adjoint if and only if it is real on the self-adjoint elements, $\varphi(h)\in\mathbb{R}$ for every $h = h^{*}$; consequently the self-adjoint functionals are the real-linear functionals on the Hermitian part extended conjugate-linearly to the algebra, and the duality between the self-adjoint elements and the self-adjoint functionals is the ordinary duality of real vector spaces.

Proof. If $\varphi = \varphi^{*}$ and $h = h^{*}$ then $\varphi(h) = \varphi^{*}(h) = \overline{\varphi(h^{*})} = \overline{\varphi(h)}$, so $\varphi(h)$ is real; conversely, if $\varphi$ is real on the self-adjoint elements then for arbitrary $a$ the decomposition $a = h + ik$ gives $\varphi(a^{*}) = \varphi(h - ik) = \varphi(h) - i\varphi(k)$ and $\overline{\varphi(a)} = \overline{\varphi(h) + i\varphi(k)} = \varphi(h) - i\varphi(k)$, so $\varphi(a^{*}) = \overline{\varphi(a)}$.

The Self-Adjointness of the Positive Functionals

Theorem (positivity implies self-adjointness). In a unital involutive algebra with a positive functional $\varphi$ of $\varphi(1) > 0$, the functional is self-adjoint: $\varphi(a^{*}) = \overline{\varphi(a)}$ for every $a$.

Proof. For $\lambda\in\mathbb{C}$ the positivity gives $\varphi((\lambda + a)^{*}(\lambda + a))\geq0$, that is, $\lvert\lambda\rvert^{2}\varphi(1) + \lambda\varphi(a^{*}) + \bar\lambda\varphi(a) + \varphi(a^{*}a)\geq0$. For $\lambda = t$ real the left side is a real quadratic in $t$ that is nonnegative, so its linear coefficient $\varphi(a) + \varphi(a^{*})$ is real; for $\lambda = is$ real the coefficient is $i(\varphi(a^{*}) - \varphi(a))$, which is likewise real, so $\varphi(a^{*}) - \varphi(a)$ is purely imaginary. Hence $\operatorname{Im}\varphi(a^{*}) = -\operatorname{Im}\varphi(a)$ and $\operatorname{Re}\varphi(a^{*}) = \operatorname{Re}\varphi(a)$, which is $\varphi(a^{*}) = \overline{\varphi(a)}$.

Corollary (the positive cone of the functionals). The positive functionals form a convex cone in the real space of the self-adjoint functionals, and they are the self-adjoint functionals that are positive on the positive cone,

$$ A^{*}_{+} = A^{*}_{\mathrm{sa}}\cap\{\varphi : \varphi(A_+)\geq0\} , $$

so the order of the algebra is the order of the self-adjoint functionals restricted to the Hermitian elements.

Proof. The inclusion $A^{*}_+\subseteq A^{*}_{\mathrm{sa}}$ is the theorem; conversely a self-adjoint functional positive on $A_+$ is positive because $a^{*}a\in A_+$; the conic and convexity statements are immediate, and the order statement is the theorem that the order is the intersection of the forms.

Proposition (the Cauchy–Schwarz inequality and the norm). For a positive functional $\varphi$ the Cauchy–Schwarz inequality

$$ \lvert\varphi(b^{*}a)\rvert^{2}\leq\varphi(a^{*}a)\,\varphi(b^{*}b) $$

holds, the functional is continuous for the order-unit norm, and in the unital case $\lVert\varphi\rVert = \varphi(1)$; the states are the positive functionals with $\varphi(1) = 1$, which are self-adjoint and are the normalised points of the cone.

Proof. The Cauchy–Schwarz inequality is that of The Cone of Positive Functionals; the continuity and the norm statement are the standard estimates $\lvert\varphi(a)\rvert\leq\varphi(1)\lVert a\rVert$ and the reverse bound; the identification of the states is the definition, together with the self-adjointness just proved.

The Self-Adjointness and the Order

Proposition (the order of the self-adjoint functionals). The self-adjoint functionals form a real ordered vector space with the cone of the positive functionals as its positive cone; the order is

$$ \varphi\leq\psi \iff \psi - \varphi \ \text{ is positive} \iff \varphi(a)\leq\psi(a) \ \text{ for every } a\in A_+ , $$

and the order is Archimedean when the algebra has an order unit and the functionals are continuous for the order-unit norm; the order unit of the space of the functionals is the trace $\varphi\mapsto\operatorname{tr}(a)$, when it exists.

Proof. The definition of the cone order is the standard one; the equivalence is the definition of the positivity of the difference; the Archimedean property is the order-unit statement of Ordered Vector Spaces and the Order Unit; the order unit is the trace functional, positive and dominating by the definition of the trace.

Theorem (the duality theorem for the order). The order of the self-adjoint elements and the order of the self-adjoint functionals are the two sides of the same duality:

$$ a\geq0 \iff \varphi(a)\geq0 \ \text{ for every } \varphi\in A^{*}_{+} , \qquad \varphi\geq0 \iff \varphi(a)\geq0 \ \text{ for every } a\in A_+ , $$

and the self-adjointness is what makes the two readings consistent: on a non-self-adjoint functional the duality would fail on the imaginary directions of the algebra.

Proof. The first equivalence is the theorem that the order is the intersection of the forms of Positive Definite Forms and the Order; the second is the definition of the positivity of a functional; the consistency statement is the proposition above that a self-adjoint functional is real on the Hermitian elements, so the duality is a duality of real spaces.

Corollary (the positive functionals and the adjoint). The positive functionals are exactly the self-adjoint functionals that are fixed by the adjoint-involution and positive on the cone of the squares; the map $\varphi\mapsto\varphi^{*}$ preserves the positivity of the difference of two functionals, so the order on the self-adjoint functionals is compatible with the involution, exactly as the order on the self-adjoint elements is compatible with the involution of the algebra. This is the functional form of the self-adjointness of Self-Adjoint Elements and the Order.

Proof. The fixed-point statement is the definition of the self-adjointness; the compatibility of the order with the involution is that $\varphi\leq\psi\implies\varphi^{*}\leq\psi^{*}$, which holds because the cone of the positive functionals is contained in the fixed-point set; the identification with the element statement is the duality theorem.

Worked Cases

The Matrix Algebra

Let $A = M_n(\mathbb{C})$ with the trace form. The functionals are $a\mapsto\operatorname{tr}(\rho a)$ for a matrix $\rho$; the adjoint of the functional corresponds to the adjoint of $\rho$, and the self-adjoint functionals are those with $\rho$ self-adjoint. The positive functionals are the ones with $\rho\geq0$, they are automatically self-adjoint, and the states are the density matrices; the duality theorem is the ordinary duality of the Hermitian matrices. This is the finite-dimensional model.

The Continuous Functions

Let $A = C(X,\mathbb{C})$ with the pointwise order. The functionals are the complex measures, the self-adjoint functionals are the real measures, and the positive functionals are the positive measures; the positive functionals are self-adjoint, and the duality theorem is the Riesz representation theorem: a continuous function is pointwise nonnegative if and only if every positive measure integrates it to a nonnegative value. The states are the probability measures.

The Non-Unital Case

For an algebra without a unit the positivity of $\varphi(a^{*}a)$ alone does not force self-adjointness; the article's theorem uses the unit through the test elements $\lambda + a$. When the algebra has an approximate identity and the functional is bounded for the associated seminorm the self-adjointness is recovered, and in a $C^{*}$-algebra every positive functional is self-adjoint even without a unit. The commutative non-unital case $C_0(X)$ shows the mechanism: the positivity is tested at the points of $X$ and the self-adjointness follows pointwise.

Summary

The involution of an ordered involutive algebra induces the adjoint $\varphi^{*}(a) = \overline{\varphi(a^{*})}$ on the dual, and the self-adjoint functionals are its fixed points; they are the functionals real on the self-adjoint elements and form a real vector space into which every functional decomposes as $\varphi = \operatorname{Re}\varphi + i\operatorname{Im}\varphi$. A positive functional is self-adjoint in a unital algebra (proved by the test elements $\lambda + a$), so the positive functionals form a convex cone in the space of the self-adjoint functionals, $A^{*}_+ = A^{*}_{\mathrm{sa}}\cap\{\varphi : \varphi(A_+)\geq0\}$; the Cauchy–Schwarz inequality, the continuity, the norm $\lVert\varphi\rVert = \varphi(1)$ and the identification of the states as the normalised positive functionals follow. The self-adjoint functionals are ordered by the cone of the positive functionals, the order is Archimedean for the continuous functionals, and the duality theorem $a\geq0\iff\varphi(a)\geq0$ for all positive $\varphi$ and $\varphi\geq0\iff\varphi(a)\geq0$ for all $a\geq0$ exhibits the order of the elements and the order of the functionals as two sides of one duality, made consistent by the self-adjointness. The self-adjoint elements are Self-Adjoint Elements and the Order; the positive functionals and the states are The Cone of Positive Functionals; the forms and the order are Positive Definite Forms and the Order; the Hermitian elements are Hermitian Elements and the Order Unit; the ordered involution is Ordered Involutive Algebras; the Jordan order is The Jordan Algebra of Self-Adjoint Elements; and the adjoints are The Adjoint of a Positive Operator and The Adjoint of the Left Multiplication on an Ordered Algebra.

Summary of Notation

Symbol	Meaning
$\varphi^{}(a) = \overline{\varphi(a^{})}$	Adjoint of a functional
$\varphi = \varphi^{*}$	Self-adjoint functional
$\varphi = \operatorname{Re}\varphi + i\operatorname{Im}\varphi$	Decomposition into self-adjoint parts
$\varphi(h)\in\mathbb{R}$ for $h = h^{*}$	Characterisation of self-adjointness
$\varphi\geq0\implies\varphi = \varphi^{*}$	Positivity implies self-adjointness (unital)
$A^{}_+ = A^{}_{\mathrm{sa}}\cap\{\varphi(A_+)\geq0\}$	Cone of the positive functionals
$\lvert\varphi(b^{}a)\rvert^{2}\leq\varphi(a^{}a)\varphi(b^{*}b)$	Cauchy–Schwarz inequality
$\lVert\varphi\rVert = \varphi(1)$	Norm of a positive functional