The Hilbert Cone of an Involutive Algebra

Introduction

A Hilbert algebra is an involutive algebra with an inner product in which the involution is adjoint to the multiplication,

$$ \langle ab,c\rangle = \langle b,\, a^{*}c\rangle , $$

and the Hilbert cone is the closed cone generated by the elements of the form $a^{*}a$,

$$ \Pi = \overline{\Bigl\{\sum_{i} a_i^{*}a_i\Bigr\}} , $$

the closure taken in the Hilbert-space norm. The cone is a proper convex cone exactly when the Hilbert algebra is positive definite, that is, when $\sum a_i^{*}a_i = 0$ forces every $a_i = 0$; then it makes the algebra an ordered involutive algebra, and its order is the Hilbert order. The distinctive feature of the Hilbert cone is that it is self-dual: an element is positive if and only if its inner product with every positive element is nonnegative,

$$ \Pi = \Pi^{\prime} = \{a : \operatorname{Re}\langle a,b\rangle\geq0 \ \text{ for every } b\in\Pi\} , $$

so the order and the inner product determine each other, and the positivity is a purely geometric condition in the Hilbert space. The article proves this autopolarity, describes the positivity it gives, and identifies the resulting structure: the self-adjoint part of the Hilbert algebra is a JBW-algebra, a Jordan algebra that is the dual of a Banach space, and the Hilbert cone is the positive cone of that Jordan algebra.

The Hilbert cone is the origin of the order of an involutive algebra: the abstract cone of The Positive Cone of an Involutive Algebra is the one of a Hilbert algebra, and the Hilbert cone is the concrete model of the self-dual cones of the corpus. Its role is the same as that of the standard cone of a Hilbert space among the cones: it is the cone compatible with the geometry, and it is the reason the positivity of a Hilbert algebra is a theorem about projections and about the JBW-structure rather than a definition.

The Hilbert algebra and the Hermitian sandwich are Hilbert Algebras and The Hermitian Sandwich on a Hilbert Algebra with Hermitian Adjoint of Part II; the positive cone and the ordered involutive algebra are The Positive Cone of an Involutive Algebra and Ordered Involutive Algebras; the Jordan order and the JB-algebras are The Jordan Algebra of Self-Adjoint Elements and JB*-Algebras and the Gelfand–Naimark Theorem; the forms are Positive Definite Forms and the Order; the order and the order unit are Ordered Vector Spaces and the Order Unit; the forms on an ordered space are Positive Definite Forms on an Ordered Space; and the von Neumann algebras and the projections are Operator Algebras and The Theory of von Neumann Algebras of Part II.

The Hilbert Algebra and the Hilbert Cone

Definition. A Hilbert algebra is a complex involutive algebra $A$ with an inner product $\langle\cdot,\cdot\rangle$ and a trace element (the identity in the unital case) such that the involution is adjoint to the multiplication, $\langle ab,c\rangle = \langle b,a^{*}c\rangle$, and the left multiplications are bounded. It is positive definite when $\sum_i a_i^{*}a_i = 0$ implies every $a_i = 0$; the Hilbert cone is

$$ \Pi = \overline{\Bigl\{\sum_{i} a_i^{*}a_i\Bigr\}} . $$

Proposition (the Hilbert cone is a cone). The set $\Pi$ is a convex cone closed under the involution, under the congruences $x\mapsto c^{*}xc$ and under the symmetrised product; it contains the trace and is proper exactly when the algebra is positive definite; the order $a\leq b\iff b - a\in\Pi$ makes the algebra an ordered involutive algebra with the trace as order unit.

Proof. The closure properties are those of the sums $a^{*}a$ together with the closure of the set under the norm, which is a limit of the corresponding properties; the properness is the positive definiteness, by the argument of The Positive Cone of an Involutive Algebra; the order is the order of the cone, and the trace is positive and dominating by the boundedness of the left multiplications.

Proposition (the trace form and the order unit). The trace form $\langle a,b\rangle$ is a positive definite form of Positive Definite Forms and the Order; the trace is the order unit, $\langle a,a\rangle\geq0$ with equality only for $a = 0$, and the order-unit norm of the order is the Hilbert norm.

Proof. The trace form is Hermitian and positive by the definition of the inner product, and it is invariant because the involution is adjoint to the multiplication; the order unit is the trace, which dominates every element by the boundedness; the order-unit-norm identity is the standard one for a self-dual cone.

The Self-Duality of the Hilbert Cone

Definition. The dual cone of $\Pi$ in the Hilbert space is $\Pi^{\prime} = \{a : \operatorname{Re}\langle a,b\rangle\geq0 \ \text{for every } b\in\Pi\}$, and the cone is self-dual when $\Pi^{\prime} = \Pi$.

Theorem (the Hilbert cone is self-dual). In a positive definite Hilbert algebra the Hilbert cone is self-dual, $\Pi = \Pi^{\prime}$; consequently

$$ a\geq0 \iff \langle a,b\rangle\geq0 \ \text{ for every } b\geq0 , $$

and the order of the Hilbert algebra is the order of the inner product.

Proof. One inclusion, $\Pi\subseteq\Pi^{\prime}$, is the standard fact that the inner product of two positive elements of a Hilbert algebra is nonnegative; on the generators it is $\langle c^{*}c, d^{*}d\rangle = \langle cd^{*}, cd^{*}\rangle\geq0$, which follows from the adjointness of the involution to the multiplication by two applications of $\langle ab,e\rangle = \langle b,a^{*}e\rangle$, and the general case follows by continuity in the two variables. The reverse inclusion uses the completion and the enveloping von Neumann algebra: if $a\notin\Pi$ then the Hahn–Banach theorem separates $a$ from the closed convex cone, and the Riesz representation of the separating functional is an element $b$ with $\operatorname{Re}\langle c,b\rangle\geq0$ on $\Pi$ but $\operatorname{Re}\langle a,b\rangle<0$; the polar decomposition of $b$ in the enveloping algebra shows that $b$ may be taken in $\Pi$, so $a\notin\Pi^{\prime}$. Hence $\Pi^{\prime}\subseteq\Pi$, and the two cones are equal.

Corollary (the order is determined by the positivity of the inner products). The positive elements of the Hilbert algebra are exactly those whose inner product with every positive element is real and nonnegative; the trace is positive, the projections (the idempotent positive elements) are positive, and the order interval $[0,1]$ at the trace is the set of the positive elements of norm at most one.

Proof. The self-duality is the theorem; the positivity of the trace is the definition; the projections are the idempotents of the form $p = p^{*} = p^{2}$, which are positive because $p = p^{*}p$; the interval statement is the order-unit-norm identity.

The Positivity and the JBW Structure

Proposition (the positive functionals of the Hilbert algebra). The positive functionals of the Hilbert algebra are the maps $a\mapsto\langle a,b\rangle$ for $b\in\Pi$, so the cone of the functionals is linearly isomorphic to the Hilbert cone itself; consequently the Hilbert algebra is self-dual as an ordered vector space and its order is reflexive.

Proof. The Riesz representation theorem identifies the bounded functionals on the Hilbert space with its elements, and the positive functionals with the elements of the dual cone, which is $\Pi$ by the self-duality; the linearity of the identification is the Hilbert-space duality.

Theorem (the self-adjoint part is a JBW-algebra). The self-adjoint part of a positive definite Hilbert algebra, with the symmetrised product and the Hilbert cone, is a JBW-algebra, that is, a JB-algebra that is the dual of a Banach space; its predual is the space of the positive functionals, its positive cone is the Hilbert cone, and its order is the Hilbert order. The enveloping von Neumann algebra of the Hilbert algebra is the von Neumann algebra generated by its left multiplications, and the self-adjoint part of that algebra is the JBW-completion.

Proof. The self-adjoint part of a Hilbert algebra is a JB-algebra by The Jordan Algebra of Self-Adjoint Elements, and the identification of the functionals with the elements of the cone exhibits it as the dual of the space of the positive functionals with the weak-$\ast$ topology; the completeness in that topology is the von Neumann-algebraic completeness of the left multiplications, which is the definition of the enveloping von Neumann algebra. The claim about the enveloping algebra is the standard double-commutant theorem of the theory.

Corollary (the projections form a complete lattice). The idempotent positive elements of the self-adjoint part, the projections, form a complete lattice in the Hilbert order, and the cones and the faces of the JBW-algebra are determined by them; in the von Neumann case the projections are the orthogonal projections of the enveloping algebra and the lattice is the lattice of the projections of the algebra.

Proof. The projections of a JBW-algebra form a complete lattice, and the order-theoretic statements are the standard ones of the JBW-theory; in the von Neumann case they are the orthogonal projections, whose lattice is complete by the double-commutant theorem.

Worked Cases

The Hilbert–Schmidt Algebra

Let $A = B(H)$ with the Hilbert–Schmidt inner product $\langle a,b\rangle = \operatorname{tr}(b^{*}a)$ on the Hilbert–Schmidt class. The involution is the operator adjoint, the Hilbert cone is the cone of the positive operators of the Hilbert–Schmidt class, and the order is the Loewner order; the cone is self-dual for the Hilbert–Schmidt form, the trace is the order unit, and the self-adjoint part is the JBW-algebra of the self-adjoint Hilbert–Schmidt operators. This is the model instance.

The Finite von Neumann Algebra

Let $A$ be a finite von Neumann algebra with a faithful normal trace $\tau$. The Hilbert algebra is the algebra with the inner product $\langle a,b\rangle = \tau(b^{*}a)$; the Hilbert cone is the cone of the positive elements of the trace class, and it is self-dual; the self-adjoint part is the JBW-algebra of the self-adjoint elements, and its projections form the complete lattice of the projections of $A$. This is the noncommutative measure-theoretic instance.

The Group Algebra

Let $A = L^{1}\cap L^{2}(G)$ for a unimodular group $G$, with the convolution product, the inner product $\langle a,b\rangle = \int b^{*}a$, and the involution $a^{*}(x) = \overline{a(x^{-1})}$. It is a Hilbert algebra whose Hilbert cone is the cone of the functions with a positive-definite Fourier transform; the self-adjoint part is the JBW-algebra of the self-adjoint elements of the group algebra, and the positivity is the classical positive-definiteness of the functions on $G$.

Summary

A Hilbert algebra is an involutive algebra with an inner product in which the involution is adjoint to the multiplication; its Hilbert cone $\Pi$ is the closed cone generated by the elements $a^{*}a$, and it is proper exactly when the algebra is positive definite, in which case it makes the algebra an ordered involutive algebra with the trace as order unit and the Hilbert norm as the order-unit norm. The cone is self-dual, $\Pi = \Pi^{\prime}$, so $a\geq0$ if and only if $\langle a,b\rangle\geq0$ for every $b\geq0$; the positive functionals are the inner products with the elements of the cone, so the algebra is reflexive as an ordered space; and the self-adjoint part with the Hilbert cone is a JBW-algebra, the dual of the space of the positive functionals, with the projections forming a complete lattice and the enveloping von Neumann algebra providing the completion. The models are the Hilbert–Schmidt algebra of $B(H)$, the finite von Neumann algebra with a faithful trace, and the $L^{1}\cap L^{2}$ group algebra. The Hilbert algebra and the Hermitian sandwich are Hilbert Algebras and The Hermitian Sandwich on a Hilbert Algebra with Hermitian Adjoint; the cone and the ordered involution are The Positive Cone of an Involutive Algebra and Ordered Involutive Algebras; the Jordan order and the JB-theory are The Jordan Algebra of Self-Adjoint Elements and JB*-Algebras and the Gelfand–Naimark Theorem; the forms are Positive Definite Forms and the Order; the order is Ordered Vector Spaces and the Order Unit; the forms on an ordered space are Positive Definite Forms on an Ordered Space; and the von Neumann theory is Operator Algebras and The Theory of von Neumann Algebras.

Summary of Notation

Symbol	Meaning
$\langle ab,c\rangle = \langle b,a^{*}c\rangle$	The involution is adjoint to the multiplication
$\Pi = \overline{\{\sum a_i^{*}a_i\}}$	Hilbert cone
$\Pi^{\prime} = \{a : \operatorname{Re}\langle a,b\rangle\geq0\ \forall b\in\Pi\}$	Dual cone
$\Pi = \Pi^{\prime}$	Self-duality of the Hilbert cone
Positive definite	$\sum a_i^{*}a_i = 0$ implies every $a_i = 0$
$a\mapsto\langle a,b\rangle$, $b\in\Pi$	Positive functionals of the Hilbert algebra
JBW-algebra	JB-algebra that is a dual Banach space
$p = p^{*} = p^{2}$	Projections, forming a complete lattice