The Two-Sided Operators and the Modular Conjugation

Introduction

A two-sided operator of a Hilbert algebra is a sandwich $\Theta_x(y) = xyx^{\dagger}$, and the family of two-sided operators is closed under composition, adjunction and the passage to invertible parameters. The modular conjugation $\jmath$ of the standard form is the antiunitary involution that exchanges the algebra with its commutant. The relation between the two is a single invariance: every two-sided operator is fixed by the modular conjugation, $\jmath\Theta_x\jmath = \Theta_x$, because the conjugation exchanges the left factor with the right one and the sandwich is symmetric between them.

The invariance is not a coincidence but the definition of a two-sided operator in operator terms. A left multiplication is carried by the conjugation to a right multiplication and back, so the only operators naturally attached to an element and unchanged by the exchange are the sandwiches. This is why the sandwiches are the operators of the standard form: in the quadruple $(\mathcal{M},H,\jmath,P)$ the sandwiches preserve the self-dual cone $P = \overline{\{\Theta_x\xi\}}$ and the modular conjugation fixes them, so the standard form is precisely the receptacle in which two-sided operators, commutant and self-duality fit together. The self-duality of the cone — $P$ equal to its dual $P^{\natural}$ — is the deepest of these facts and the one that makes the indefinite theory work; in this article the cone is described and its relation to the two-sided operators is fixed.

This article fixes the two-sided operators, their invariance under the modular conjugation, the standard form, and the self-dual cone with its relation to the sandwiches.

The sandwich and its adjoint are The Adjoint of the Sandwich on a Hilbert Algebra; the left and right multiplications are The Adjoint of the Left and the Right Multiplication; the modular conjugation and the cone are The Modular Operator and Tomita-Takesaki Theory; the standard form is Von Neumann Algebras and the Hilbert Algebra Completeness. Those are cited. The algebra is $A$, the completion $H$, the standard form $(\mathcal{M},H,\jmath,P)$.

The Two-Sided Operators

Definition. The two-sided operators of $A$ are the sandwiches $\Theta_x(y) = xyx^{\dagger}$ for $x\in A$, together with their limits on $H$; they form the semigroup $\Theta = \{\bar\Theta_x : x\in A\}$ closed under composition.

Proposition (the structure of the family). $\Theta$ is closed under composition, $\Theta_x\Theta_z = \Theta_{xz}$; closed under adjunction, $\Theta_x^{*} = \Theta_{x^{\dagger}}$; and the invertible two-sided operators are the inner automorphisms of the algebra implemented by invertible elements, with the unitary ones exactly the form-preserving inner automorphisms.

Proof. The composition law is associativity; the adjoint law is The Adjoint of the Sandwich on a Hilbert Algebra; the automorphism statement is the defining property of an inner automorphism and the unitarity condition is the isometry theorem.

Proposition (the two-sided operators are conjugation-invariant). Every sandwich satisfies

$$ \jmath\,\bar\Theta_x\,\jmath = \bar\Theta_x , $$

and the conjugation-invariant bounded operators form a subalgebra of $B(H)$ containing all sandwiches; a one-sided multiplication $\bar L_x$ is conjugation-invariant only when the algebra is commutative, in which case $\bar L_x = \bar R_x = \bar\Theta_x$ for self-adjoint $x$.

Proof. The invariance is the next section; the subalgebra statement is the closedness of the fixed set of an involution under products and limits; the last statement is $\jmath\bar L_x\jmath = \bar R_{x^{\dagger}}$, which equals $\bar L_x$ exactly when $\bar L_{x^{\dagger}} = \bar L_x$ and $R = L$, that is when the algebra is commutative and $x$ is self-adjoint.

The Modular Conjugation

Theorem (the invariance). Let $\jmath$ be the modular conjugation of the standard form. Then for every $x$

$$ \jmath\,\bar L_x\,\jmath = \bar R_{x^{\dagger}} , \qquad \jmath\,\bar R_{x^{\dagger}}\,\jmath = \bar L_x , \qquad \jmath\,\bar\Theta_x\,\jmath = \bar\Theta_x . $$

So the modular conjugation exchanges the two factors of a two-sided operator and leaves the operator itself unchanged.

Proof. The first two identities are the exchange theorem of The Adjoint of the Left and the Right Multiplication; multiplying them gives $\jmath\bar L_x\bar R_{x^{\dagger}}\jmath = \bar R_{x^{\dagger}}\bar L_x = \bar\Theta_x$.

Corollary (the fixed algebra of the conjugation). The operators fixed by $T\mapsto\jmath T\jmath$ form the commutant of the pair, and the two-sided operators lie in it: so $\Theta_x$ belongs to the commutant of the algebra generated by the modular conjugation and the conjugation-invariant elements.

Proof. The fixed set of an involution is a subalgebra; the sandwiches lie in it by the theorem.

Remark (why the invariance is structural). The invariance is the operator statement that a two-sided object depends on an element and its involution only through the pair, and it is the reason the modular conjugation is the natural symmetry of the theory of two-sided operators: the sandwiches are exactly the operators whose left and right parts are exchanged by the conjugation without changing the operator.

The Standard Form and the Self-Duality

Definition. The standard form of the algebra is the quadruple $(\mathcal{M}, H, \jmath, P)$ where $\mathcal{M}$ is the von Neumann algebra generated by the left multiplications, $\jmath$ the modular conjugation, and

$$ P = \overline{\{\Theta_x\xi : x\in\mathcal{M}\}} = \overline{\mathcal{M}_{+}\xi} $$

the closure of the image of the positive cone under the cyclic vector.

Theorem (self-duality). The cone $P$ is self-dual:

$$ P = P^{\natural} := \{u\in H : [u,v]\geq0 \text{ for every } v\in P\} , $$

the modular conjugation fixes $P$ pointwise in the sense $\jmath P = P$, and $\Theta_xP\subseteq P$ for every $x\in\mathcal{M}$.

Proof. The self-duality is the cone theorem of the modular theory: $P^{\natural}\subseteq P$ because a vector in the dual cone is a limit of positive elements applied to $\xi$, and $P\subseteq P^{\natural}$ by the positivity of the sandwiches; the invariance $\jmath P = P$ follows from the matrix-valued version of $\jmath\Theta_x\jmath = \Theta_x$, which is $\jmath x\jmath = x^{*}$ acting on the cone; and the invariance of the cone under the sandwiches is the stabilisation of the positive cone under inner conjugation.

Proposition (two-sided operators preserve the cone). Every two-sided operator maps $P$ into $P$, and every operator preserving $P$ and commuting with the modular conjugation is a limit of two-sided operators.

Proof. The first statement is the last identity of the theorem; the second is the identification of the cone-preserving operators with the positive sandwiches.

Remark (three equivalent structures). In the standard form the algebra, the commutant and the self-dual cone are three aspects of one structure: the left multiplications generate the algebra, the modular conjugation exchanges it with the commutant, and the cone encodes the positivity that the modular theory needs to replace the missing Cauchy–Schwarz inequality. The two-sided operators are the maps that respect all three, which is why they are the natural morphisms of the standard form.

Worked Cases

The Group Algebra

For $A = \mathbb{C}[G]$ the two-sided operators are $\Theta_g(y) = gyg^{-1}$, the modular conjugation exchanges the left and the right regular representations, and the cone is the closure of the image of the positive cone of the group algebra under the cyclic vector; the form is tracial and the conjugation is the involution.

Matrices

For $A = M_n(\mathbb{C})$ with the Hilbert–Schmidt form, $\Theta_a(b) = aba^{*}$ and $\jmath(u) = u^{*}$ satisfies $\jmath\Theta_a\jmath = \Theta_a$; the cone $P$ is the cone of positive matrices applied to the unit, and its self-duality is the statement that the positive matrices are exactly those pairing nonnegatively with the positive matrices.

The Trivial Algebra

For $A = \mathbb{C}$ the only two-sided operator is the identity, the modular conjugation is $z\mapsto\bar z$ and the cone is the ray of the positive reals; the four data of the standard form are the algebra, its commutant, the conjugation and the cone, and all are one-dimensional.

Summary

The two-sided operators $\Theta_x(y) = xyx^{\dagger}$ are closed under composition and adjunction, and their invertible members are the inner automorphisms, with the unitary members exactly the form-preserving ones. Every two-sided operator is invariant under the modular conjugation, $\jmath\Theta_x\jmath = \Theta_x$, because the conjugation exchanges its two factors; this is the operator statement that a two-sided operator depends on an element and its involution. The standard form $(\mathcal{M},H,\jmath,P)$ consists of the generated algebra, the Hilbert space, the modular conjugation $J\mathcal{M}J = \mathcal{M}'$ and the cone $P = \overline{\{\Theta_x\xi : x\in\mathcal{M}\}}$, and the theorem of the standard form is that $P$ is self-dual, $P = P^{\natural}$, that $\jmath P = P$ and that every two-sided operator preserves $P$. So the algebra, its commutant and the cone are three aspects of one structure, and the two-sided operators are the maps respecting all three. The sandwich and its adjoint are The Adjoint of the Sandwich on a Hilbert Algebra, the exchange is The Adjoint of the Left and the Right Multiplication, and the modular objects and the cone are The Modular Operator and Tomita-Takesaki Theory and Von Neumann Algebras and the Hilbert Algebra Completeness.

Summary of Notation

Symbol	Meaning
$\Theta_x$, $\Theta_x\Theta_z = \Theta_{xz}$	Two-sided operators and their composition
$\jmath\bar\Theta_x\jmath = \bar\Theta_x$	Invariance under the modular conjugation
$\jmath\bar L_x\jmath = \bar R_{x^{\dagger}}$	The conjugation exchanges the factors
$(\mathcal{M},H,\jmath,P)$	The standard form
$P = \overline{\{\Theta_x\xi\}}$	The cone of the standard form
$P = P^{\natural}$	Self-duality of the cone
$\Theta_xP\subseteq P$	Two-sided operators preserve the cone
$\jmath P = P$	The conjugation preserves the cone