Self-Adjoint Elements and the Positive Cone

Introduction

The involution singles out the self-adjoint elements, those fixed by $\dagger$, and they form a real vector space inside the algebra; among them the positive elements are those that can be written as sums of elements $y^{\dagger}y$. The positive elements generate an order by $a\leq b$ whenever $b - a$ is positive, and the order is the algebraic substitute for the comparison of magnitudes: it is reflexive and transitive, it is invariant under the inner conjugations $x\mapsto y^{\dagger}xy$, and its positive part is exactly the positive cone. The order structures the algebra of a Hilbert algebra, and it is the object on which the positivity of states and of weights is decided.

Two facts give the order its power. The positive cone is proper in a concrete algebra of operators — an element and its negative are both positive only if the element is zero — so the order is a partial order and not merely a preorder; and every positive element has a unique positive square root, which produces the absolute value $|x| = (x^{\dagger}x)^{1/2}$ and the polar decomposition $x = u|x|$. Positivity therefore carries a metric structure: lengths, moduli and decompositions.

This article fixes the self-adjoint elements, the positive cone, the order and its invariance, and the square roots with the polar decomposition, in the setting of a $\ast$-algebra of bounded operators on a Hilbert space, which is the setting of the Hilbert algebras of the corpus.

The involution and the adjoint axiom are Hilbert Algebras; the states built on the positive elements are The GNS Construction; the indefinite version of the cone, where positivity depends on the fundamental symmetry, is Krein Algebras; the operator algebras are Von Neumann Algebras and the Hilbert Algebra Completeness; the spectral theorem that produces the square roots in the general case is deferred to Analysis on Linear Spaces (Part III). Those are cited. The algebra is a unital $\ast$-algebra $A$ of bounded operators on a Hilbert space with adjoint $*$, and the involution is denoted $*$ when the Hilbert adjoint is meant.

Self-Adjoint Elements

Definition. An element $x$ is self-adjoint, or Hermitian, when $x^{*} = x$. The set $A_{\mathrm{sa}}$ of self-adjoint elements is the fixed set of the involution.

Proposition (the real vector space). $A_{\mathrm{sa}}$ is a real vector space; every $x$ decomposes uniquely as $x = a + ib$ with $a, b$ self-adjoint, namely $a = \frac{1}{2}(x + x^{*})$ and $b = \frac{1}{2i}(x - x^{*})$; and $A$ is the complexification of $A_{\mathrm{sa}}$.

Proof. The fixed set of an antilinear involution is a real vector space; the decomposition is the standard one, and uniqueness follows by applying the involution.

Proposition (products and commutation). The product of two self-adjoint elements is self-adjoint exactly when they commute; $x^{*}x$ and $xx^{*}$ are always self-adjoint, and they have the same norm when the norm is that of a $\ast$-algebra of operators.

Proof. $(ab)^{*} = b^{*}a^{*} = ba$, which equals $ab$ exactly when $ab = ba$; $(x^{*}x)^{*} = x^{*}x$ always.

Remark (the real form). The self-adjoint part is the real form of the algebra: the involution is the conjugation with respect to it, the norm is determined on it, and every positivity statement in the theory is a statement about $A_{\mathrm{sa}}$.

The Positive Cone

Definition. The positive cone is

$$ A_{+} = \Bigl\{\sum_{i=1}^{n} y_i^{*}y_i : n\geq1,\ y_i\in A\Bigr\}, $$

the set of finite sums of elements $y^{*}y$; an element of $A_{+}$ is positive and one writes $x\geq0$.

Proposition (positivity is positivity of the quadratic form). $x\in A_{+}$ if and only if $x$ is self-adjoint and $\langle xu,u\rangle\geq0$ for every $u$ in the Hilbert space, and this is also the condition that $x = y^{2}$ for a self-adjoint $y$.

Proof. For $x = \sum y_i^{*}y_i$ one has $\langle xu,u\rangle = \sum\|y_iu\|^{2}\geq0$; conversely a self-adjoint operator with nonnegative quadratic form is a limit of polynomials in itself with nonnegative values, hence a sum of squares in the von Neumann algebra generated by $x$, by the functional calculus.

Proposition (properties of the cone). $A_{+}$ is a convex cone, closed under $x\mapsto y^{*}xy$ for every $y$, closed under sums and under multiplication by nonnegative real scalars, closed in the norm; and it is proper: $x\in A_{+}$ and $-x\in A_{+}$ imply $x = 0$.

Proof. Convexity and stability under sums and scalars are immediate from the definition; the inner invariance is $\sum(y_i y)^{*}(y_iy)$; closure is a limit of sums of squares; properness is $\langle xu,u\rangle\geq0$ and $\leq0$, hence $= 0$, for every $u$, which forces $x = 0$.

Proposition (the cone of a Hilbert algebra). In a Hilbert algebra the elements $y^{\dagger}y$ are positive, and the positive cone is the closure for the form of the sums of such elements; the involution axiom makes the cone invariant under the inner conjugations, exactly as in the operator case.

Proof. $\langle y^{\dagger}yu,u\rangle = \langle yu,yu\rangle\geq0$ by the adjoint axiom, and the closure statement is the positivity of the form.

The Order

Definition. The order is $a\leq b$ when $b - a\in A_{+}$, and $a < b$ when moreover $b - a$ is not zero.

Proposition (it is a partial order). The relation $\leq$ is reflexive, transitive and antisymmetric; it is a partial order on $A_{\mathrm{sa}}$ and its positive part is $A_{+}$.

Proof. Reflexivity is $0\in A_{+}$; transitivity is the closure of the cone under addition; antisymmetry is properness: $a\leq b$ and $b\leq a$ give $b - a\in A_{+}$ and $a - b\in A_{+}$, hence $b - a = 0$.

Proposition (compatibility with the algebra). The order is invariant under the inner conjugations and compatible with the involution:

$$ a\leq b \ \Longrightarrow\ y^{*}ay\leq y^{*}by, \qquad a\leq b \ \Longleftrightarrow\ -b\leq -a . $$

Proof. Both are the stability and the linearity of the cone.

Proposition (order and norm). In a unital $\ast$-algebra of operators the unit is an order unit and the order is Archimedean: for every self-adjoint $x$ one has $-\|x\|1\leq x\leq\|x\|1$, and if $x\leq\epsilon1$ for every $\epsilon>0$ then $x\leq0$.

Proof. For a self-adjoint operator $x$ the operator $\|x\|1 - x$ has nonnegative quadratic form, giving the first bound with its negative; the second is the same bound applied to $x + \delta 1$.

Remark (what the order is not). The order is not total: there are self-adjoint elements neither of which dominates the other, because the positive cone is not all of $A_{\mathrm{sa}}$ in dimension greater than one; and it is not compatible with the product: $a\leq b$ does not imply $xa\leq xb$ for a self-adjoint $x$ unless $x$ is positive, and even for positive $x$ the product inequality fails unless the factors commute.

Square Roots and the Polar Decomposition

Theorem (unique positive square root). Every $x\in A_{+}$ has a unique square root in $A_{+}$: there is exactly one $y\in A_{+}$ with $y^{2} = x$.

Proof. Uniqueness: if $y, z\in A_{+}$ and $y^{2} = z^{2}$ then $y$ and $z$ commute with $x$ and with each other, and taking the quadratic form at $u$ gives $\|yu\| = \|zu\|$, whence $y = z$. Existence: for a positive element of a von Neumann algebra the square root is produced by the functional calculus, $y = f(x)$ with $f(t) = t^{1/2}$, and the functional calculus is the spectral theorem of Analysis on Linear Spaces (Part III); in the finite-dimensional case it is the diagonalisation of a positive matrix.

Definition. The absolute value of $x$ is $|x| = (x^{*}x)^{1/2}$, and $x$ has the polar decomposition

$$ x = u\,|x| , \qquad u^{*}u = \text{the projection onto } \overline{\mathrm{im}\,|x|} , $$

with $u$ a partial isometry.

Proof. The identity $\|xu\|^{2} = \langle x^{*}xu,u\rangle = \||x|u\|^{2}$ shows that the map $|x|u\mapsto xu$ is isometric from the image of $|x|$ onto the image of $x$, and it extends by continuity to the partial isometry $u$.

Proposition (the square root is order preserving and the cone is the set of squares). $a\leq b$ implies $a^{1/2}\leq b^{1/2}$ when $a, b$ commute, and $A_{+}$ is exactly the set of squares of self-adjoint elements: $x\in A_{+}$ if and only if $x = y^{2}$ for a self-adjoint $y$.

Proof. The square root of a positive element is self-adjoint and positive, giving the second statement; the first is the monotonicity of $t\mapsto t^{1/2}$ on commuting positive elements, again by the functional calculus.

Remark (the three faces of positivity). An element is positive exactly when it is a sum of squares, when its quadratic form is nonnegative, and when it is the square of a self-adjoint element; the three descriptions are used in different contexts, and the equivalence between them is the content of the two propositions above together with the square root theorem.

Worked Cases

Matrices

For $A = M_n(\mathbb{C})$ the self-adjoint elements are the Hermitian matrices, the positive cone is the set of positive semidefinite matrices, the order is the Loewner order, and every positive matrix has a unique positive semidefinite square root; the polar decomposition is the classical one.

The Abelian Case

For $A = C(X)$ the self-adjoint elements are the real-valued functions, $A_{+}$ is the set of nonnegative functions, the order is pointwise, and the square root is the pointwise one; the cone is proper and the order is Archimedean with the constant function $1$ as order unit.

A Hilbert Algebra

In a Hilbert algebra the elements $y^{\dagger}y$ generate the positive cone and the order is the one determined by the form; the involution carries the cone to itself and the inner conjugations preserve it, so the order is the order of Hilbert Algebras read in the operator representation of The Left and the Right Regular Representation.

Summary

The self-adjoint elements of a $\ast$-algebra of operators form the real form $A_{\mathrm{sa}}$, and the positive cone $A_{+}$ is the set of finite sums of elements $y^{*}y$; by the functional calculus it is also the set of self-adjoint elements with nonnegative quadratic form and the set of squares of self-adjoint elements. The cone is a convex cone, stable under the inner conjugations $x\mapsto y^{*}xy$, closed in the norm, and proper, so the order $a\leq b$ defined by $b - a\in A_{+}$ is a partial order, invariant under the inner conjugations, Archimedean and with the unit as order unit when the algebra is unital. Every positive element has a unique positive square root, which gives the absolute value $|x| = (x^{*}x)^{1/2}$ and the polar decomposition $x = u|x|$ with $u$ a partial isometry; the square roots are produced by the functional calculus, whose general form — the spectral theorem — is deferred to Analysis on Linear Spaces (Part III), while the finite-dimensional case is diagonalisation. The involution and the Hilbert-algebra positivity are Hilbert Algebras, the states built on the cone are The GNS Construction, and the indefinite analogue where positivity depends on $J$ is Krein Algebras.

Summary of Notation

Symbol	Meaning
$A_{\mathrm{sa}}$, $x^{*} = x$	Self-adjoint elements, the real form
$x = a+ib$	Unique decomposition into self-adjoint real and imaginary parts
$A_{+} = \{\sum y_i^{*}y_i\}$	Positive cone
$\langle xu,u\rangle\geq0$	Positivity as nonnegativity of the quadratic form
$a\leq b \iff b-a\in A_{+}$	The order
$y^{}ay\leq y^{}by$	Invariance under inner conjugations
$x^{1/2}$, $\|x\| = (x^{*}x)^{1/2}$	Square root and absolute value
$x = u\|x\|$	Polar decomposition, $u$ a partial isometry