The Indefinite Modular Operator

Introduction

For a von Neumann algebra on a Hilbert space with a cyclic and separating vector $\xi$, the Tomita operator $S$ is defined by $S(x\xi) = x^{\dagger}\xi$; it is antilinear, densely defined and closable, and its polar decomposition $S = \jmath\Delta^{1/2}$ produces the two objects of the whole theory: the modular operator $\Delta = S^{*}S$, self-adjoint and positive, and the modular conjugation $\jmath$, antilinear and involutive. The entire construction rests on the definiteness of the inner product, through the closure of $S$ and through the positivity of $\Delta$.

On a Krein space the same formula defines the same antilinear operator, but its properties change. The adjoint in the polar decomposition is now the indefinite one, so what is obtained is a $J$-modular operator $\Delta$ that is positive for the definite form of the fundamental symmetry $J$ and self-adjoint for the indefinite one, and a $J$-modular conjugation $\jmath$ that is $J$-unitary and commutes with $J$. The two objects therefore carry $J$ with them: the modular data of an indefinite algebra is $J$-twisted, and the classical theory is the case $J = \mathrm{id}$.

What must be added is a condition: on a Krein space the Tomita operator need not be closable, so the polar decomposition need not exist. The hypotheses that make it exist are that $\xi$ be $J$-cyclic for the algebra and $J$-separating for it, together with the availability of a self-dual cone carried by $\xi$; then $S$ has a closure and the closure factorises. This article fixes the operator $S$, the $J$-cyclic and $J$-separating hypotheses, the polar decomposition into the $J$-modular operator and the $J$-modular conjugation, and the properties that the two inherit.

The Krein space, the form and the operator $J$ are Krein Spaces and The Fundamental Symmetry; the algebra is Krein–von Neumann Algebras; the indefinite representation that produces the algebra is The Indefinite GNS Construction; the theorem that uses the modular data is Krein–Tomita–Takesaki Theory; the definite construction is The Modular Operator and Tomita-Takesaki Theory. Those are cited. The Krein space is $K$ with form $[\cdot,\cdot]$ and fundamental symmetry $J$, the algebra is $\mathcal{M}$, and the vector is $\xi$.

The Tomita Operator for a Krein Space

Definition. Let $\mathcal{M}$ be a Krein–von Neumann algebra on $K$ and $\xi\in K$. The Tomita operator is the antilinear operator

$$ S : \mathcal{M}\xi\to K, \qquad S(x\xi) = x^{\dagger}\xi , $$

with domain the linear span of the orbit $\mathcal{M}\xi$.

Proposition (well-definedness and involution). $S$ is well defined: if $x\xi = 0$ then $x^{\dagger}\xi = 0$ exactly when $\xi$ is separating for $\mathcal{M}$. On its domain $S$ is an involution, $S^{2} = \mathrm{id}$, and hence bijective onto its image.

Proof. If $S$ were ambiguous there would be $x$ with $x\xi = 0$ and $x^{\dagger}\xi\neq0$; the separating hypothesis excludes this, and then $S^{2}(x\xi) = S(x^{\dagger}\xi) = (x^{\dagger})^{\dagger}\xi = x\xi$.

Definition. The vector $\xi$ is $J$-cyclic for $\mathcal{M}$ when the orbit $\mathcal{M}\xi$ is dense in $K$, and $J$-separating when $x\xi = 0$ with $x\in\mathcal{M}$ forces $x = 0$. Cyclicity is density of the domain of $S$ and separatingness is well-definedness of $S$; note that separatingness cannot be read off the orbit, since $\mathcal{M}$ is closed under $\dagger$ and so $\mathcal{M}^{\dagger}\xi = \mathcal{M}\xi$.

Proposition (what changes in the indefinite case). For a Hilbert space the operator $S$ is closed (indeed $S^{**} = S$), and this is the Cauchy–Schwarz inequality in disguise. For a Krein space $S$ need not be closed, and it need not be the case that $S^{**} = S$; the graph of $S$ may fail to be a graph of a closed operator.

Proof. In the definite case $S$ is an isometry of the norm, $\|x\xi\| = \|x^{\dagger}\xi\|$, and that identity is exactly the Cauchy–Schwarz inequality applied to the positive definite form, so the closed graph argument goes through. In the indefinite case the identity becomes $[x\xi,x\xi] = [x^{\dagger}\xi,x^{\dagger}\xi]$, which holds but bounds neither side, since the form is indefinite; the closed graph argument has nothing to stand on and $S$ need not be closed.

Remark (the missing inequality). The whole difference is a single missing inequality: an indefinite form does not bound the norm of a vector by the norm of its image under $S$. All the extra hypotheses of the next section are devices that supply a substitute bound.

The Polar Decomposition and the $J$-Modular Pair

Definition. The vector $\xi$ is modular for the pair $(\mathcal{M},J)$ when it is $J$-cyclic and $J$-separating and the Tomita operator $S$ has a closure $\bar S$ that factorises as

$$ \bar S = \jmath\,\Delta^{1/2}, $$

with $\jmath$ an antilinear isometry of the definite form of $J$ and $\Delta$ a positive definite self-adjoint operator of that form. The operator $\Delta$ is the $J$-modular operator and $\jmath$ the $J$-modular conjugation attached to $\xi$.

Proposition (the $J$-modular operator). When the polar decomposition exists the $J$-modular operator is

$$ \Delta = \bar S^{\dagger}\bar S $$

with $\dagger$ the indefinite adjoint, it is positive for the definite form $\langle\cdot,\cdot\rangle = [J\cdot,\cdot]$, self-adjoint for the indefinite form, and it commutes with $J$; its domain is the set of $u$ with $\bar Su\in\mathrm{dom}\,\bar S^{\dagger}$.

Proof. The polar decomposition of a closed operator gives $\bar S^{\dagger}\bar S = \Delta^{1/2}\jmath^{\dagger}\jmath\Delta^{1/2} = \Delta^{1/2}\Delta^{1/2} = \Delta$, using $\jmath^{\dagger}\jmath = \mathrm{id}$ for a $J$-unitary antilinear isometry; positivity for the definite form is the assignment of $\Delta^{1/2}$ as the positive factor; the commutation with $J$ is the commutation of the factorisation with the fundamental symmetry, which holds because $S$ is defined by the involution and the involution is $J$-compatible in a Krein algebra.

Proposition (the $J$-modular conjugation). With the notation of the polar decomposition, $\jmath$ is antilinear, $J$-unitary, involutive, $\jmath^{2} = \mathrm{id}$, and it commutes with the fundamental symmetry,

$$ \jmath J = J\jmath , \qquad [\jmath u,\jmath v] = \overline{[u,v]} . $$

Proof. The involution $S^{2} = \mathrm{id}$ passes to the closure as $\jmath\Delta^{1/2}\jmath\Delta^{1/2} = \mathrm{id}$, and the uniqueness of the polar decomposition, together with the positivity of $\Delta$, forces $\jmath^{2} = \mathrm{id}$; the $J$-unitarity is the isometry statement of the definition, and the commutation with $J$ is the $J$-compatibility of the involution underlying $S$.

Theorem (the modular flow is the flow of the pair). The one-parameter family

$$ \sigma_t(x) = \Delta^{it}\,x\,\Delta^{-it}, \qquad x\in\mathcal{M}, $$

is a one-parameter group of $J$-automorphisms of $\mathcal{M}$: each $\sigma_t$ is an algebra automorphism of $\mathcal{M}$, each $\sigma_t$ is compatible with the indefinite adjoint, $\sigma_t(x^{\dagger}) = \sigma_t(x)^{\dagger}$, and $\sigma_{s+t} = \sigma_s\sigma_t$.

Proof. $\Delta^{it}$ is a group of definite-unitary operators commuting with $J$, and a $J$-commuting definite-unitary $U$ induces the automorphism $x\mapsto UxU^{-1} = UxU^{\dagger}$ of $\mathcal{M}$, which is compatible with the indefinite adjoint because $U^{\dagger} = JU^{*}J = U^{-1}$; the group law is that of $t\mapsto\Delta^{it}$.

The Role of the $J$-Cyclic and $J$-Separating Conditions

Proposition (cyclic gives density, separating gives well-definedness). $J$-cyclicity makes $\mathcal{M}\xi$ dense and hence makes $S$ densely defined; $J$-separating makes $S$ well defined and injective. Both are needed, and neither can be dropped.

Proof. Density of the domain is cyclicity, well-definedness is separatingness by the first proposition of the article; a cyclic but not separating vector makes $S$ ambiguous, and a separating but not cyclic one leaves $S$ defined only on a non-dense domain, so that it has no closure to speak of.

Proposition (the self-dual cone supplies closability). If in addition $\xi$ carries a self-dual cone $C$ — a closed convex cone with $C = C^{\natural} = \{u : [u,v]\geq0 \text{ for all } v\in C\}$ — then the Tomita operator $S$ is closable, and the polar decomposition of its closure exists.

Proof. The self-dual cone is exactly the structure that replaces the Cauchy–Schwarz-type inequality missing in the indefinite case; it provides a positive definite form with respect to which $S$ is represented by a bounded operator and hence closable. This is the theory of Krein–Tomita–Takesaki Theory.

Remark (three data, not two). In the definite case the modular data is attached to the pair (algebra, cyclic and separating vector). In the indefinite case it is attached to the triple (algebra, $J$-cyclic and $J$-separating vector, self-dual cone), and the fundamental symmetry $J$ is carried along by each of the two objects produced. This is the precise sense in which the modular operator is indefinite: not that its spectrum is complex, but that it is positive for $J$ and self-adjoint for the form whose sign $J$ reverses.

Worked Cases

The Trivial Algebra on $\mathbb{C}^{1,1}$

Let $K = \mathbb{C}^{1,1}$, $\mathcal{M}$ the diagonal $2\times2$ matrices, $J = \mathrm{diag}(1,-1)$ and $\xi = e_1+e_2$. Then $\xi$ is $J$-cyclic, the orbit of $x = \mathrm{diag}(a,b)$ being the point $(a,b)$ and the orbit of $\mathcal{M}$ all of $K$, and it is $J$-separating, since $x\xi = 0$ means $a = b = 0$. Here $x^{\dagger} = \mathrm{diag}(\bar a,\bar b)$, so the Tomita operator is termwise complex conjugation, $$ S(a e_1 + b e_2) = \bar a\,e_1 + \bar b\,e_2 , $$ an antilinear involution with $S^{2} = \mathrm{id}$; it satisfies $[Su,Sv] = \overline{[u,v]}$ and $\jmath J = J\jmath$ with $\jmath = S$, so it is the $J$-modular conjugation, and the polar decomposition $S = \jmath\Delta^{1/2}$ has $\Delta = \mathrm{id}$.

A Pontryagin Space

On $\ell^{2}\oplus-\mathbb{C}$ with the algebra of diagonal operators and $\xi$ with all coordinates nonzero, the vector is $J$-cyclic and $J$-separating as soon as the self-dual cone is chosen; the $J$-modular operator is then diagonal, positive definite for $\langle\cdot,\cdot\rangle$ and self-adjoint for the form, and its $t$-th power generates the modular flow by rotating the phases of the coordinates. The choice of the cone, not the choice of $\xi$, is the extra datum.

The Definite Case

For $J = \mathrm{id}$ the theory collapses to The Modular Operator and Tomita-Takesaki Theory: a cyclic and separating vector suffices, the self-dual cone is automatic (the positive cone of the Hilbert space), the $J$-modular operator is the modular operator and the $J$-modular conjugation is the modular conjugation.

Summary

The Tomita operator of a Krein–von Neumann algebra $\mathcal{M}$ and a vector $\xi$ is the antilinear map $S(x\xi) = x^{\dagger}\xi$; it is well defined exactly when $\xi$ is $J$-separating and densely defined exactly when $\xi$ is $J$-cyclic, and it is involutive. In the definite case $S$ is closed and its polar decomposition gives the modular operator and the modular conjugation; in the indefinite case $S$ need not be closed, and the decomposition exists only under the additional hypothesis of a self-dual cone, and then reads $\bar S = \jmath\Delta^{1/2}$. The factor $\Delta$ is the $J$-modular operator: positive definite for the form of $J$, self-adjoint for the indefinite form, commuting with $J$, equal to $\bar S^{\dagger}\bar S$. The factor $\jmath$ is the $J$-modular conjugation: antilinear, $J$-unitary, involutive, commuting with $J$. The family $\sigma_t(x) = \Delta^{it}x\Delta^{-it}$ is a one-parameter group of $J$-automorphisms of $\mathcal{M}$, and it is the modular flow whose properties — the invariance of the algebra, the identification of the commutant and the $J$-KMS condition — are the Krein–Tomita–Takesaki theory. The algebra is Krein–von Neumann Algebras, the form and $J$ are Krein Spaces and The Fundamental Symmetry, and the definite case is The Modular Operator and Tomita-Takesaki Theory.

Summary of Notation

Symbol	Meaning
$S$, $S(x\xi) = x^{\dagger}\xi$	Tomita operator, antilinear, involutive
$\xi$ $J$-cyclic / $J$-separating	Density of $\mathcal{M}\xi$ / well-definedness of $S$
$C = C^{\natural}$	Self-dual cone, the closability hypothesis
$\bar S = \jmath\Delta^{1/2}$	Polar decomposition
$\Delta = \bar S^{\dagger}\bar S$	$J$-modular operator
$\jmath$	$J$-modular conjugation, $\jmath^{2} = \mathrm{id}$, $\jmath J = J\jmath$
$\sigma_t(x) = \Delta^{it}x\Delta^{-it}$	Modular flow, a group of $J$-automorphisms
$J = \mathrm{id}$	The definite case