Hermitian Elements and the Order Unit

Introduction

The Hermitian elements of an involutive algebra are the fixed points of the involution, $H(A) = \{a : a^{*} = a\}$; they form a real vector subspace, and the positive cone of the algebra restricts to a cone in it, so that $H(A)$ is an ordered vector space with the order of The Positive Cone of an Involutive Algebra. When the algebra is unital and reduced the identity is Hermitian and positive, $1 = 1^{*}1 > 0$, and it is an order unit: for every Hermitian $h$ there is $\lambda>0$ with $-\lambda1\leq h\leq\lambda1$, the least such $\lambda$ being the order-unit seminorm of $h$, which in a $\ast$-normed algebra coincides with the norm. The order unit is therefore the bridge between the algebraic decomposition of an element and the order: it is the calibration against which the Hermitian elements are measured, and the state space of the algebra is its dual base, the set of the positive functionals that take the value $1$ at the order unit.

The article develops the order and the decomposition. Every element decomposes as $a = \operatorname{Re}a + i\operatorname{Im}a$ into two Hermitian elements, and every Hermitian element decomposes with respect to the order unit into its positive and negative parts, $h = h_+ - h_-$ with $h_+\geq0$, $h_-\geq0$ and $h_+h_- = 0$; the two decompositions are orthogonal in a precise sense, and the order interval $[-1,1]$ is the unit ball of the order-unit norm. The consequential facts are that the Hermitian part is an ordered real vector space whose order interval at the unit is the unit ball, that the involution is an order isomorphism, and that every Hermitian element is the difference of two commuting positive elements — the Jordan decomposition — which is what makes the order of an involutive algebra a lattice-like order on the commuting parts.

The ordered vector space and the order unit are Ordered Vector Spaces and the Order Unit and The Order Unit as an Operator; the positive cone and its closure properties are The Positive Cone of an Involutive Algebra; the states and the extreme points are The Cone of Positive Functionals; the ordered involutive algebra as a structure is Ordered Involutive Algebras; the Hilbert cone and its positivity are The Hilbert Cone of an Involutive Algebra; the self-adjoint elements of a C*-algebra, the model of this article, are The Jordan Algebra of Self-Adjoint Elements; and the $JB$-structures are JB*-Algebras and the Gelfand–Naimark Theorem.

The Hermitian Part as an Ordered Space

Definition. The Hermitian part of the involutive algebra $A$ is $H(A) = \{a : a^{*} = a\}$; the skew part is $\{a : a^{*} = -a\} = iH(A)$. The order of $H(A)$ is that of the cone, $h\leq k\iff k - h\in A_+$.

Proposition (the Hermitian part is an ordered real vector space). $H(A)$ is a real vector subspace of $A$ of real dimension $\dim_{\mathbb{C}}A$, closed under the symmetrised product $\{h,k\} = \frac12(hk + kh)$, and the cone $H(A)\cap A_+$ is a proper cone when $A$ is reduced; the identity is Hermitian and positive, $1 = 1^{*}1\in A_+$.

Proof. If $h^{*} = h$ and $k^{*} = k$ then $(\alpha h + \beta k)^{*} = \alpha h + \beta k$ for real $\alpha,\beta$, so $H(A)$ is a real subspace, and its real dimension is that of $A$ over $\mathbb{C}$ because the map $a\mapsto\operatorname{Re}a$ is a real isomorphism onto $H(A)$; the symmetrised product of two Hermitian elements is Hermitian because $hk + kh$ is, using the anti-multiplicativity; the cone is proper by the reducedness; the identity is Hermitian and $1 = 1^{*}1$.

Proposition (every element is a complex combination of two Hermitian elements). Every $a\in A$ has a unique decomposition

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

with $h,k\in H(A)$; and $a$ is normal, Hermitian, or positive according as $hk = kh$, $k = 0$, or $k = 0$ and $h\geq0$.

Proof. The verification that $h$ and $k$ are Hermitian and that $a = h + ik$ is direct; conversely if $a = h + ik$ with $h,k$ Hermitian then $\tfrac12(a + a^{*}) = h$ and $\tfrac1{2i}(a - a^{*}) = k$, so the decomposition is unique. The normality criterion is $a^{*}a = aa^{*}$, which reads $hk = kh$ for the decomposition; the Hermitian and the positive cases are specialisations.

The Order Unit

Definition. An order unit of $H(A)$ is an element $u > 0$ such that for every $h$ there is $\lambda>0$ with $-\lambda u\leq h\leq\lambda u$; an order unit is strong when in addition every nonempty majorised set has a least upper bound, and the order-unit seminorm is

$$ \lVert h\rVert_u = \inf\{\lambda>0 : -\lambda u\leq h\leq\lambda u\} . $$

Proposition (the identity is an order unit). When $A$ is a unital reduced involutive algebra the identity is a positive order unit of $H(A)$; in a $\ast$-normed algebra with $\lVert a^{*}a\rVert = \lVert a\rVert^{2}$ the order-unit norm at the identity is the given norm, $\lVert h\rVert_1 = \lVert h\rVert$, and the order interval $[-1,1]$ is the unit ball of $H(A)$ in that norm.

Proof. $1>0$; for a Hermitian $h$ the bound $-\lVert h\rVert1\leq h\leq\lVert h\rVert1$ is the spectral bound in a $\ast$-normed algebra, so $1$ is an order unit and $\lVert h\rVert_1\leq\lVert h\rVert$; conversely the order interval estimate $-\lambda1\leq h\leq\lambda1$ gives $\lVert h\rVert\leq\lambda$ by the norm estimate on the positive cone, so $\lVert h\rVert = \lVert h\rVert_1$. The interval $[-1,1]$ is the set of Hermitian elements of norm at most one, which is the unit ball.

Proposition (the states are the unit's dual base). The states, the positive functionals with $\varphi(1) = 1$, are the base of the dual cone $A_+^{*}$ at the order unit, and they separate the points of $H(A)$; the order of $H(A)$ is the order of the evaluation functionals, $h\leq k\iff\varphi(h)\leq\varphi(k)$ for every state $\varphi$.

Proof. The normalisation $\varphi(1) = 1$ cuts the dual cone in a base because $\varphi(1)>0$ for every nonzero positive functional in the unital case; separation and the order representation are the Hahn–Banach theorem together with the fact that a Hermitian element that is nonpositive has a state seeing it, which is a consequence of the Hahn–Banach theorem applied in the ordered space $H(A)$ with the order unit.

The Decomposition with Respect to the Order Unit

Theorem (the Jordan decomposition). Let $A$ be a unital reduced involutive algebra that is $\ast$-normed with $\lVert a^{*}a\rVert = \lVert a\rVert^{2}$, and let $h\in H(A)$. Then there are unique commuting positive elements $h_+,h_-$ with

$$ h = h_+ - h_-, \qquad h_+h_- = 0, \qquad \lVert h\rVert = \max(\lVert h_+\rVert,\lVert h_-\rVert) , $$

the positive part $h_+$ and the negative part $h_-$; the decomposition is produced by the functional calculus, $h_\pm = \tfrac12(\lvert h\rvert\pm h)$ with $\lvert h\rvert = (h^{2})^{1/2}$, and it is the decomposition of $h$ with respect to the order unit.

Proof. The element $\lvert h\rvert$ is defined by the functional calculus and commutes with $h$; then $h_\pm = \tfrac12(\lvert h\rvert\pm h)$ are positive, commute, satisfy $h_+ - h_- = h$ and $h_+h_- = \tfrac14(\lvert h\rvert^{2} - h^{2}) = 0$, so the decomposition exists; the norm identity is the spectral identity $\lVert h\rVert = \max(\lVert h_+\rVert,\lVert h_-\rVert)$, because the spectra are separated by the sign. Uniqueness: if $h = p - q$ with $p,q\geq0$ commuting and $pq = 0$, then $p - q = h$ and $p + q = \lvert h\rvert$ by the square $\lvert h\rvert^{2} = h^{2} = (p+q)^{2}$ and the positivity of $p+q$, so $p = \tfrac12(\lvert h\rvert + h) = h_+$ and $q = h_-$.

Corollary (the order interval and the unit ball; the absolute value). The order interval $[-1,1]$ consists of the Hermitian elements with $\lVert h\rVert\leq1$; every Hermitian element has an absolute value $\lvert h\rvert = (h^{2})^{1/2}$ with $\lvert h\rvert\geq\pm h$, and $a^{*}a\geq0$ for every $a$, so the absolute value of an element is $(\operatorname{Re}(a^{*}a))^{1/2}$.

Proof. The interval identification is the proposition above; the absolute value is the positive square root of $h^{2}$, which satisfies $\lvert h\rvert\pm h\geq0$ by the decomposition; and $a^{*}a\in A_+$ by the definition of the cone, with $a^{*}a$ Hermitian.

Proposition (the involution preserves the order unit and the decomposition). The involution fixes $1$, is an order isomorphism of $H(A)$, and carries the decomposition $a = h + ik$ to $a^{*} = h - ik$; on a Hermitian element it preserves the positive and the negative parts, $(h_+)^{*} = h_+$ and $(h_-)^{*} = h_-$.

Proof. $1^{*} = 1$ by the definition of the involution of a unital algebra; the order isomorphy is the positivity of the involution of The Positive Cone of an Involutive Algebra; the decomposition statement is the conjugate-linearity on the imaginary part and the Hermitian property of the parts.

Worked Cases

The Self-Adjoint Operators

Let $A = B(H)$ with the operator adjoint. The Hermitian elements are the self-adjoint operators, the order is the Loewner order, the order unit is the identity, and the Jordan decomposition is the spectral decomposition into the positive and the negative spectral parts: $h_+ = \int_{0}^{\infty}\lambda\, \mathrm{d}E_\lambda$ and $h_- = -\int_{-\infty}^{0}\lambda\, \mathrm{d}E_\lambda$ for the spectral measure $E$ of $h$. The order interval $[-1,1]$ is the set of the self-adjoint contractions, and the states are the density matrices. This is the model instance.

The Continuous Functions

Let $A = C(X,\mathbb{C})$ with pointwise conjugation. The Hermitian elements are the real-valued continuous functions, the order is pointwise, the order unit is the constant function $1$, and the Jordan decomposition is the pointwise decomposition $h_+ = \max(h,0)$, $h_- = \max(-h,0)$. The Hermitian part is the lattice-ordered $C(X,\mathbb{R})$, in which every pair has a least upper bound; this is the commutative extreme, and in it the order unit is strong.

The Group Algebra

Let $A = \mathbb{C}[G]$ with $g^{*} = g^{-1}$. The Hermitian elements are the elements with $a^{*} = a$, the order unit is the identity of the unit group, and the Jordan decomposition is the decomposition of a self-adjoint element of the group algebra into its positive and negative parts in the Hilbert cone of The Hilbert Cone of an Involutive Algebra; the states are the positive-definite functions on $G$ normalised to $1$ at the identity, which is the classical correspondence between the states and the positive-definite functions.

Summary

The Hermitian part $H(A)$ of an involutive algebra is a real ordered vector space of real dimension $\dim_{\mathbb{C}}A$, closed under the symmetrised product, with the cone $H(A)\cap A_+$; every element decomposes uniquely as $a = \operatorname{Re}a + i\operatorname{Im}a$ into Hermitian parts. The identity is a positive order unit in the unital reduced algebra, $\lVert h\rVert_1 = \lVert h\rVert$ in a $\ast$-normed algebra, and the order interval $[-1,1]$ is the unit ball; the states are the base of the dual cone at the order unit and separate the points. Every Hermitian element has the Jordan decomposition $h = h_+ - h_-$ with $h_\pm\geq0$, $h_+h_- = 0$ and $\lVert h\rVert = \max(\lVert h_+\rVert,\lVert h_-\rVert)$, produced by the functional calculus through the absolute value $\lvert h\rvert = (h^{2})^{1/2}$, and the involution fixes the order unit and preserves the decomposition. The order and the order unit are Ordered Vector Spaces and the Order Unit and The Order Unit as an Operator; the cone is The Positive Cone of an Involutive Algebra; the states are The Cone of Positive Functionals; the ordered involutive algebra is Ordered Involutive Algebras; the Hilbert cone is The Hilbert Cone of an Involutive Algebra; the self-adjoint model is The Jordan Algebra of Self-Adjoint Elements; and the JB-theory is JB*-Algebras and the Gelfand–Naimark Theorem.

Summary of Notation

Further Reading

• Richard Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. 1 (Academic Press, 1983), for the Hermitian part, the order unit and the states of an operator algebra.
• Gert K. Pedersen, C*-Algebras and their Automorphism Groups (Academic Press, 1979), for the functional calculus, the absolute value and the polar decomposition.
• Erik M. Alfsen and Frederik W. Shultz, State Spaces of Operator Algebras (Birkhäuser, 2001), for the order unit, the state space and its facial structure.
• Shoichiro Sakai, C*-Algebras and W*-Algebras (Springer, 1971), for the states, the positive functionals and the extreme points.
• Charalambos D. Aliprantis and Owen Burkinshaw, Positive Operators (Academic Press, 1985), for the order-unit seminorm and the ordered vector spaces of an involutive algebra.