# Modular Completely Dirichlet forms as Squares of Derivations

Melchior Wirth  Institute of Science and Technology Austria (ISTA), Am Campus
1, 3400 Klosterneuburg, Austria<EMAIL_ADDRESS>
###### Abstract.

We prove that certain closable derivations on the GNS Hilbert space associated
with a non-tracial weight on a von Neumann algebra give rise to GNS-symmetric
semigroups of contractive completely positive maps on the von Neumann algebra.

## Introduction

The interplay between derivations and symmetric semigroups of unital (or
contractive) completely positive maps has proven fruitful for applications in
quantum information theory [Bar17, CM17, GR22], operator algebras [Pet09,
Pet09a, DI16, Cas21] and beyond. Using the framework of completely Dirichlet
forms, this connection is particularly well-understood in the case of
tracially symmetric semigroups after the seminal work of Cipriani and
Sauvageot [Sau89, Sau90, CS03].

In many situations however one encounters non-tracial reference states or
weights: In quantum statistical mechanics, the reference state is typically a
Gibbs state, which is not a trace at finite temperature; in quantum
probability in the study of Lévy processes on compact quantum groups, the
natural reference state is the Haar state, which is only a trace for the class
of compact quantum groups of Kac type; and in the structure theory of von
Neumann algebras, one is faced with non-tracial states when the von Neumann
algebra has a non-trivial type III summand.

In the non-tracial setting, the connection between derivations and symmetric
semigroups of completely positive maps is much less understood. Recently, it
was shown by the author that every GNS-symmetric semigroup of unital
completely positive maps gives rise to a canonical derivation via its
associated Dirichlet form [Wir22]. This result was (partially) extended to
KMS-symmetric semigroups by Vernooij and the author [VW23].

There has also been work in the opposite direction – starting with a
derivation to construct a completely Dirichlet form [Cip97, Par00, BK02, CZ21,
LM22, Wir22]. However, these results all rely on additional structural
assumptions on the derivation, usually some form of (approximate) innerness.
This means that natural examples like derivations arising from cocycles on
non-unimodular groups or Voiculescu’s derivation in non-tracial free
probability could not be treated in this framework.

In this article, we prove in a general context that closable derivations give
rise to GNS-symmetric semigroups of completely bounded maps. More precisely,
our main result is the following.

######  Theorem.

Let $\mathfrak{A}$ be a Tomita algebra, $\mathscr{H}$ a normal Tomita bimodule
over $\mathfrak{A}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ a closable
symmetric derivation. Let $\mathscr{E}$ be the closure of the quadratic form
$\mathscr{E}_{0}$ given by
$\operatorname{dom}(\mathscr{E}_{0})=\operatorname{dom}(\delta)$ and
$\mathscr{E}_{0}(a)=\lVert\delta(a)\rVert_{\mathscr{H}}^{2}$. Then the
strongly continuous semigroup associated with $\mathscr{E}$ is the GNS
implementation of a GNS-symmetric semigroup of contractive completely positive
maps on the left von Neumann algebra generated by $\mathfrak{A}$.

Here a normal Tomita bimodule is a bimodule over a Tomita algebra that
additionally carries a complex one-parameter group $(\mathscr{U}_{z})$ and an
involution $\mathscr{J}$ satisfying some compatibility conditions, and a
symmetric derivation $\delta\colon\mathfrak{A}\to\mathscr{H}$ is a map that
intertwines the complex one-parameter groups and involutions on $\mathfrak{A}$
and $\mathscr{H}$ and satisfies the product rule

$\delta(ab)=a\delta(b)+\delta(a)b.$

These objects were introduced in [Wir22] and appear to be the natural non-
tracial analogs of the Hilbert bimodule and derivation occurring in the
context of completely Dirichlet forms on tracial von Neumann algebras.

Combined with the results from [Wir22], we thus obtain a comprehensive picture
of GNS-symmetric quantum Markov semigroups analogous to the result of Cipriani
and Sauvageot for tracially symmetric semigroups. Among other potential
applications, we hope that this result opens the gate for applications to non-
tracial free probability and deformation/rigidity theory of type III von
Neumann algebras similar to recent work in this direction in the tracial case.

One main difficulty when trying to prove that closable derivations generate
completely Dirichlet forms (or semigroups of completely positive maps) is that
the property defining derivations, the product rule, is an algebraic property,
while Dirichlet forms are defined in terms of order properties, and the domain
of a derivation is not necessarily closed under order operations. As such, the
problem of properly dealing with domains is crucial. Note that it is
unavoidable to allow for unbounded derivations as everywhere defined
derivations yield norm continuous semigroups of completely positive maps,
which is too restrictive for many applications.

In the tracial case, this difficulty can be overcome since order operations
such as taking the positive part can be expressed in terms of functional
calculus and as such can be approximated by polynomials. In the non-tracial
case, the order operations can still be expressed in terms of functional
analysis in the setup of Haagerup $L^{p}$ spaces, but the product rule is
formulated in terms of Hilbert algebra multiplication, which is different from
the product of two operators in Haagerup $L^{2}$ (which is only in $L^{2}$ if
it is zero). Therefore it is not clear how to connect the two.

Instead of trying to follow the proof in the tracial setting, our proof
strategy instead relies on Haagerup reduction method, which allows to embed a
von Neumann algebra as an expected subalgebra of a bigger von Neumann algebra
that can be approximated by finite von Neumann algebras. As it turns out, this
reduction method is well-suited to reduce the problem at hand to the known
case of tracial von Neumann algebras. One key challenge are again domain
issues: For the Haagerup construction one has to extend the derivation to a
domain on a crossed product that is sufficiently big, but such that the
extension still satisfies the product rule. The essential new technical
ingredient to overcome this kind of domain problems lies in the introduction
of a new locally convex topology on the domain of a derivation that allows to
extend derivations to derivations on a completion.

As a final note, considering the results from [VW23], it is a natural question
whether the results from the present article can be extended to cover KMS-
symmetric semigroups. For one, our methods crucially use commutation with the
modular group, which fails for KMS-symmetric maps if they are not GNS-
symmetric. But more severely, it seems like there are additional algebraic
obstructions, already in finite dimensions: It is shown in [VW23] that if
$\mathscr{E}$ is a completely Dirichlet form on
$L^{2}(M_{n}(\mathbb{C}),\varphi)$, then there exist self-adjoint matrices
$v_{j}\in M_{n}(\mathbb{C})$ such that

$\mathscr{E}(\rho^{1/4}x\rho^{1/4})=\sum_{j}\mathrm{tr}(\lvert\rho^{1/4}[v_{j},x]\rho^{1/4}\rvert^{2}),$

where $\rho$ is the density matrix inducing the state $\varphi$ on
$M_{n}(\mathbb{C})$. However, without further assumptions on the operators
$v_{j}$, the quadratic form on the right side of the previous equation is not
necessarily a completely Dirichlet form.

### Outline of the article

In Section 1 we recall some basics regarding modular theory, completely
Dirichlet forms on standard forms of von Neumann algebras and Tomita bimodules
and derivations. In Section 2 we introduce a topology on the domain of a
derivation, the $\delta$-topology, and show that derivations can be extended
to derivations on the completion in the $\delta$-topology. In Section 3 we
give a closability criterion for derivations in our setting. In Section 4 we
discuss how derivations can be extended to crossed products and discuss how
completely Dirichlet forms behave with respect to change of the reference
weight. Then we state and prove the main result of this article, Theorem 4.4,
showing that the quadratic form associated with a closable derivation is a
modular completely Dirichlet form. Finally, in Section 5 we discuss several
classes of examples, including inner derivations, derivations arising in non-
tracial free probability and derivation induced by cocycles on (possibly non-
unimodular) locally compact groups.

### Acknowledgments

The author was funded by the Austrian Science Fund (FWF) under the Esprit
Programme [ESP 156]. For the purpose of Open Access, the authors have applied
a CC BY public copyright licence to any Author Accepted Manuscript (AAM)
version arising from this submission.

##  1\. Basics

In this section we briefly recap some material concerning modular theory and
in particular Hilbert and Tomita algebras, completely Dirichlet forms, Tomita
bimodules and derivations that is used in the later sections.

###  1.1. Modular theory

As our approach is formulated in the language of Hilbert and Tomita algebras,
we summarize the relevant definitions here. Our treatment mostly follows
[Tak03, Chapters VI–VII].

An algebra $\mathfrak{A}$ with involution ♯ (resp. ♭) and inner product
$\langle\,\cdot\,,\cdot\,\rangle$ is called _left (resp. right) Hilbert
algebra_ if

  * •

for every $a\in\mathfrak{A}$ the map
$\pi_{l}(a)\colon\mathfrak{A}\to\mathfrak{A}$, $b\mapsto ab$ (resp. $b\mapsto
ba$) is bounded,

  * •

$\langle ab,c\rangle=\langle b,a^{\sharp}c\rangle$ (resp. $\langle
ab,c\rangle=\langle b,ca^{\flat}\rangle$) for all $a,b,c\in\mathfrak{A}$,

  * •

the involution ♯ (resp. ♭) is closable,

  * •

the linear span of all products $ab$ with $a,b\in\mathfrak{A}$ is dense in
$\mathfrak{A}$.

Let $M$ be a von Neumann algebra and $\varphi$ a normal semi-finite faithful
weight on $M$. We write $\mathfrak{n}_{\varphi}$ for the definition ideal
$\\{x\in M\mid\varphi(x^{\ast}x)<\infty\\}$ and
$(\pi_{\varphi},L^{2}(M,\varphi),\Lambda_{\varphi})$ for the associated semi-
cyclic representation.

The prototypical example of a left Hilbert algebra is
$\mathfrak{A}=\Lambda_{\varphi}(\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast})$
with the product
$\Lambda_{\varphi}(x)\Lambda_{\varphi}(y)=\Lambda_{\varphi}(xy)$, the
involution $\Lambda_{\varphi}(x)^{\sharp}=\Lambda_{\varphi}(x^{\ast})$ and the
inner product inherited from $L_{2}(M,\varphi)$, that is,
$\langle\Lambda_{\varphi}(x),\Lambda_{\varphi}(y)\rangle=\varphi(x^{\ast}y)$.
In this case, $\pi_{l}(\mathfrak{A})^{\prime\prime}=\pi_{\varphi}(M)$. We
write $\mathfrak{A}_{\varphi}$ for this left Hilbert algebra.

Conversely, every left Hilbert algebra $\mathfrak{A}$ gives rise to a von
Neumann algebra $\pi_{l}(\mathfrak{A})^{\prime\prime}$ acting on the
completion of $\mathfrak{A}$ and a weight

$\varphi\colon\pi_{l}(\mathfrak{A})^{\prime\prime}_{+}\to[0,\infty],\,\varphi(x)=\begin{cases}\lVert\xi\rVert^{2}&\text{if
}x^{1/2}=\pi_{l}(\xi),\\\ \infty&\text{otherwise}.\end{cases}$

If $\mathfrak{A}$ is a full left Hilbert algebra [Tak03, Definition VI.1.16],
then $\varphi$ is a normal semi-finite faithful weight on
$\pi_{l}(\mathfrak{A})^{\prime\prime}$, and $\mathfrak{A}$ is canonically
isomorphic to $\mathfrak{A}_{\varphi}$.

Let $\mathfrak{H}$ be the completion of the left Hilbert algebra
$\mathfrak{A}$. Since the involution ♯ on $\mathfrak{A}$ is closable, its
closure $S$ on $\mathfrak{H}$ exists and has a polar decomposition
$S=J\Delta^{1/2}$. The operator $\Delta$ is a non-singular positive self-
adjoint operator, called the _modular operator_ , and $J$ is an anti-unitary
involution, called the _modular conjugation_. If $\mathfrak{A}$ is the left
Hilbert algebra associated with a weight $\varphi$, we write
$\Delta_{\varphi}$ and $J_{\varphi}$ for the associated modular operator and
modular conjugation. We write
$\Lambda_{\varphi}^{\prime}\colon\mathfrak{n}_{\varphi}^{\ast}\to
L_{2}(M,\varphi)$ for the map $x\mapsto
J_{\varphi}\Lambda_{\varphi}(x^{\ast})$.

If $\mathfrak{A}$ is full, the modular conjugation $J$ gives rise to the
positive self-dual cone $P=\overline{\\{\pi_{l}(a)Ja\mid a\in\mathfrak{A}\\}}$
and $\pi_{l}(\mathfrak{A})^{\prime\prime}$ is in standard form [Tak03,
Definition IX.1.13].

The modular operator $\Delta$ gives rise to a point weak∗ continuous group of
automorphisms $x\mapsto\Delta^{it}x\Delta^{-it}$ on
$\pi_{l}(\mathfrak{A})^{\prime\prime}$. If $\varphi$ is a normal semi-finite
faithful weight on $M$, the group $\sigma^{\varphi}$ given by
$\sigma^{\varphi}_{t}(x)=\pi_{\varphi}^{-1}(\Delta_{\varphi}^{it}\pi_{\varphi}(x)\Delta_{\varphi}^{-it})$
is called the _modular group_ associated with $\varphi$.

If $(\alpha_{t})_{t\in\mathbb{R}}$ is a point weak∗ continuous group of
$\ast$-automorphisms on $M$, then an element $x\in M$ is called _entire
analytic_ if the map $t\mapsto\alpha_{t}(x)$ has an extension
$z\mapsto\alpha_{z}(x)$ to the complex plane such that
$z\mapsto\omega(\alpha_{z}(x))$ is analytic for every $\omega\in M_{\ast}$.
The entire analytic elements form a weak∗ dense $\ast$-subalgebra of $M$.

A _Tomita algebra_ is a left Hilbert algebra $\mathfrak{A}$ endowed with a
complex one-parameter group $(U_{z})_{z\in\mathbb{C}}$ of algebra automorphism
such that

  * •

$z\mapsto\langle a,U_{z}b\rangle$ is analytic for all $a,b\in\mathfrak{A}$,

  * •

$(U_{z}a)^{\sharp}=U_{\bar{z}}(a^{\sharp})$ for all $a\in\mathfrak{A}$,
$z\in\mathbb{C}$,

  * •

$\langle U_{z}a,b\rangle=\langle a,U_{-\bar{z}}b\rangle$ for all
$a,b\in\mathfrak{A}$, $z\in\mathbb{C}$,

  * •

$\langle a^{\sharp},b^{\sharp}\rangle=\langle U_{-i}b,a\rangle$ for all
$a,b\in\mathfrak{A}$.

Note that every Tomita algebra becomes a right Hilbert algebra when endowed
with the involution

$\mathfrak{A}\to\mathfrak{A},a\mapsto a^{\flat}=U_{-i}(a^{\sharp}).$

For a full left Hilbert algebra $\mathfrak{A}$ let

$\mathfrak{A}_{0}=\left\\{\xi\in\bigcap_{n\in\mathbb{Z}}D(\Delta^{n})\,\bigg{|}\,\Delta^{n}\xi\in\mathfrak{A}\text{
for all }n\in\mathbb{Z}\right\\}.$

For every $\xi\in\mathfrak{A}_{0}$ the map $t\mapsto\Delta^{it}\xi$ has an
entire analytic extension $z\mapsto U_{z}\xi$ with
$U_{z}\xi\in\mathfrak{A}_{0}$ for all $z\in\mathbb{C}$. This makes
$\mathfrak{A}_{0}$ into a Tomita algebra such that
$\pi_{l}(\mathfrak{A}_{0})^{\prime\prime}=\pi_{l}(\mathfrak{A})^{\prime\prime}$.

In particular,

$(\mathfrak{A}_{\varphi})_{0}=\\{\Lambda_{\varphi}(x)\mid
x\in\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast},\,x\text{ entire
analytic for }\sigma^{\varphi}\\}.$

###  1.2. Completely Dirichlet forms

Completely Dirichlet forms in the non-tracial setting were introduced by
Goldstein and Lindsay [GL95, GL99] in the language of GNS Hilbert spaces of
states (or weights) and by Cipriani [Cip97] in the language of standard forms
with a fixed cyclic vector. Our approach is somewhat different from both of
these formulations in that we use left Hilbert algebras, but in view of the
previous subsection it is equivalent to the formulation by Goldstein–Lindsay
(and to that of Cipriani in case the left Hilbert algebra has a unit).

Let $\mathfrak{A}$ be a full left Hilbert algebra with completion
$\mathfrak{H}$. Let $C$ be the closure of $\\{\Delta^{1/4}a\mid
a\in\mathfrak{A},\,0\leq\pi_{l}(a)\leq 1\\}$ and let $P_{C}$ be the metric
projection onto $C$. We say that a closed densely defined quadratic form
$\mathscr{E}$ on $\mathfrak{H}$ is a _Dirichlet form_ with respect to
$\mathfrak{A}$ if $\mathscr{E}\circ J=\mathscr{E}$ and
$\mathscr{E}(P_{C}(a))\leq\mathscr{E}(a)$ for all $a\in\mathfrak{H}$ with
$Ja=a$.

The Dirichlet form $\mathscr{E}$ is called _completely Dirichlet form_ if for
every $n\in\mathbb{N}$ the quadratic form

$\mathscr{E}^{(n)}\colon\mathfrak{H}\otimes
M_{n}(\mathbb{C})\to[0,\infty],\,\mathscr{E}^{(n)}([\xi_{ij}])=\sum_{i,j=1}^{n}\mathscr{E}(\xi_{ij})$

is a Dirichlet form with respect to $\mathfrak{A}\odot M_{n}(\mathbb{C})$.
Here $M_{n}(\mathbb{C})$ carries the normalized Hilbert–Schmidt inner product
and the multiplication and involution on $\mathfrak{A}\odot M_{n}(\mathbb{C})$
are given by $[a_{ij}][b_{ij}]=[\sum_{k}a_{ik}b_{kj}]$,
$[a_{ij}]^{\sharp}=[a_{ji}^{\sharp}]$.

A (completely) Dirichlet form with respect to $\mathfrak{A}$ is called
_modular_ (or GNS-symmetric) if $\mathscr{E}\circ U_{t}=\mathscr{E}$ for all
$t\in\mathbb{R}$.

Completely Dirichlet forms are of particular interest for their connection to
semigroups of contractive completely positive maps on von Neumann algebras.
Let us briefly sketch this correspondence. Proofs can be found in [GL99,
Theorems 4.9, 5.7] for the wider class of KMS-symmetric semigroups. The result
for GNS-symmetric semigroups follows from the fact that GNS symmetry is
equivalent to KMS symmetry and commutation with the modular group (see [AC82,
Proposition 6.1] for example).

Let $M$ be a von Neumann algebra. A _quantum dynamical semigroup_ is a
semigroup of normal contractive completely positive operators on $M$ that is
continuous in the point weak∗ topology. If $\varphi$ is a normal semi-finite
faithful weight on $M$, a quantum dynamical semigroup $(P_{t})$ is called
_GNS-symmetric_ with respect to $\varphi$ if $\varphi\circ P_{t}\leq\varphi$
for all $t\geq 0$ and

$\varphi(P_{t}(x)^{\ast}y)=\varphi(x^{\ast}P_{t}(y))$

for all $x,y\in\mathfrak{n}_{\varphi}$ and $t\geq 0$.

Every GNS-symmetric quantum dynamical semigroup gives rise to a strongly
continuous semigroup $(T_{t})$ on $L^{2}(M,\varphi)$, its _GNS implementation_
, acting by $T_{t}\Lambda_{\varphi}(x)=\Lambda_{\varphi}(P_{t}(x))$ for
$x\in\mathfrak{n}_{\varphi}$, and the associated quadratic form is a modular
completely Dirichlet form with respect to $\mathfrak{A}_{\varphi}$. Vice
versa, the strongly continuous semigroup associated with a modular completely
Dirichlet form is the GNS implementation of a GNS-symmetric quantum dynamical
semigroup.

We call a completely Dirichlet form a _quantum Dirichlet form_ if the
associated quantum dynamical semigroups consists of unital maps. A criterion
in terms of the form itself is given in [Wir22, Proposition 3.2].

###  1.3. Tomita bimodules and derivations

Tomita bimodules were introduced in [Wir22] as codomains of the derivations
associated with modular completely Dirichlet forms.

Let $\mathfrak{A}$ be a Tomita algebra. A _Tomita bimodule_ over
$\mathfrak{A}$ is an inner product space $\mathscr{H}$ endowed with non-
degenerate commuting left and right actions of $\mathfrak{A}$, an anti-
isometric involution $\mathscr{J}\colon\mathscr{H}\to\mathscr{H}$ and a
complex one-parameter group $(\mathscr{U}_{z})$ of isometries such that

  * •

$\lVert a\xi
b\rVert\leq\lVert\pi_{l}(a)\rVert\lVert\pi_{r}(b)\rVert\lVert\xi\rVert$ for
$a,b\in\mathfrak{A}$, $\xi\in\mathscr{H}$,

  * •

$\langle a\xi b,\eta\rangle=\langle\xi,a^{\sharp}\eta b^{\flat}\rangle$ for
$a,b\in\mathfrak{A}$, $\xi,\eta\in\mathscr{H}$,

  * •

$\mathscr{U}_{z}(a\xi b)=(U_{z}a)(\mathscr{U}_{z}\xi)(U_{z}b)$ for
$a,b\in\mathfrak{A}$, $\xi\in\mathscr{H}$, $z\in\mathbb{C}$,

  * •

$\mathscr{J}(a\xi b)=(Jb)(\mathscr{J}\xi)(Ja)$ for $a,b\in\mathfrak{A}$,
$\xi\in\mathscr{H}$,

  * •

$\mathscr{U}_{z}\mathscr{J}=\mathscr{J}\mathscr{U}_{\bar{z}}$ for
$z\in\mathbb{C}$.

Let $\bar{\mathscr{H}}$ be the completion of $\mathscr{H}$. The first two
bullet points imply that $\pi_{l}(a)\mapsto(\xi\mapsto a\xi)$ extends to a
non-degenerate $\ast$-homorphism from $\pi_{l}(\mathfrak{A})$ to
$B(\bar{\mathscr{H}})$. If this map can be extended to a normal
$\ast$-homomorphism from $\pi_{l}(\mathfrak{A})^{\prime\prime}$ to
$B(\bar{\mathscr{H}})$, then we say that $\mathscr{H}$ is a _normal Tomita
bimodule_. Requiring normality for the right action instead leads to the same
notion of normal Tomita bimodule.

If $\mathfrak{A}$ is a Tomita algebra and $\mathscr{H}$ a bimodule over
$\mathfrak{A}$, we call a linear map $\delta\colon\mathfrak{A}\to\mathscr{H}$
a _derivation_ if it satisfies the product rule

$\delta(ab)=a\delta(b)+\delta(a)b$

for $a,b\in\mathfrak{A}$. If $\mathscr{H}$ is a Tomita bimodule over
$\mathfrak{A}$, we say that a derivation
$\delta\colon\mathfrak{A}\to\mathscr{H}$ is _symmetric_ if $\delta\circ
J=\mathscr{J}\circ\delta$ and $\delta\circ U_{z}=\mathscr{U}_{z}\circ\delta$
for all $z\in\mathbb{C}$.

If $\mathfrak{A}$ is a full left Hilbert algebra and $\mathscr{E}$ a modular
quantum Dirichlet form with respect to $\mathfrak{A}$, it is shown in [Wir22,
Theorem 6.3] that

$\mathfrak{A}_{\mathscr{E}}=\\{a\in\mathfrak{A}_{0}\mid
U_{z}a\in\operatorname{dom}(\mathscr{E})\text{ for all }z\in\mathbb{C}\\}$

is a Tomita subalgebra of $\mathfrak{A}_{0}$ and a core for $\mathscr{E}$.
Moreover, by [Wir22, Theorem 6.8] there exists a Tomita bimodule $\mathscr{H}$
over $\mathfrak{A}$ and a symmetric derivation
$\delta\colon\mathfrak{A}_{\mathscr{E}}\to\mathscr{H}$ such that

$\mathscr{E}(a,b)=\langle\delta(a),\delta(b)\rangle_{\mathscr{H}}$

for $a,b\in\mathfrak{A}_{\mathscr{E}}$.

Under the minimality condition $\mathscr{H}=\mathrm{lin}\\{\delta(a)b\mid
a,b\in\mathfrak{A}_{\mathscr{E}}\\}$, the pair $(\mathscr{H},\delta)$ is
uniquely determined by $\mathscr{E}$ up to isometric isomorphism preserving
the Tomita bimodule structure and intertwining the derivations [Wir22, Theorem
6.9]. By a slight abuse of notation, any such pair $(\mathscr{H},\delta)$ is
called the _first-order differential structure associated with $\mathscr{E}$_.
If $\mathscr{H}$ is a normal Tomita bimodule, the quantum Dirichlet form
$\mathscr{E}$ is called _$\Gamma$ -regular_. A characterization in terms of
the carré du champ is given in [Wir22, Theorem 7.2].

##  2\. $\delta$-topology and completeness

In this section we introduce a locally convex topology on the domain of a
closable symmetric derivation, called the $\delta$-topology. This topology is
strong enough to ensure that the derivation extends to a derivation on the
completion, which is a key technical ingredient in the proof of the main
theorem later.

For the definition of the $\delta$-topology recall that the Mackey topology
$\tau(M,M_{\ast})$ on a von Neumann algebra $M$ is the finest linear topology
$\mathscr{T}$ on $M$ such that the topological dual of $(M,\mathscr{T})$ is
$M_{\ast}$. Equivalently, it is the finest locally convex topology on $M$ that
coincides with the strong∗ topology on norm bounded sets [Coo87, Proposition
2.11, Remark 2.12]. It has the advantage over the other usual locally convex
topologies on $M$ of being complete, which is convenient for several of the
following arguments.

Let $\mathfrak{A}$ be a Tomita algebra with completion $\mathfrak{H}$, let
$\mathscr{H}$ be a normal Tomita bimodule over $\mathfrak{A}$ and
$\delta\colon\mathfrak{A}\to\mathscr{H}$ a symmetric derivation. We define the
$\delta$-topology $\mathscr{T}_{\delta}$ on $\mathfrak{A}$ as the coarsest
locally convex topology that makes the maps

$\mathfrak{A}\to\mathfrak{H}\oplus\bar{\mathscr{H}},\,a\mapsto(\Delta^{n}a,\delta(\Delta^{n}a))$

continuous with respect to the norm topology on
$\mathfrak{H}\oplus\bar{\mathscr{H}}$ for all $n\in\mathbb{Z}$ and the maps

$\mathfrak{A}\to B(\mathfrak{H}),\,a\mapsto\pi_{l}(\Delta^{n}a)$

continuous for the Mackey topology on $B(\mathfrak{H})$ for all
$n\in\mathbb{Z}$. Clearly, the $\delta$-topology is stronger than the topology
induced by the graph norm
$(\lVert\cdot\rVert_{\mathfrak{H}}^{2}+\lVert\delta(\cdot)\rVert_{\mathscr{H}}^{2})^{1/2}$.

######  Lemma 2.1.

If $\mathscr{E}$ is a $\Gamma$-regular modular quantum Dirichlet form and
$(\mathscr{H},\delta)$ the associated first-order differential structure, then
$\mathfrak{A}_{\mathscr{E}}$ is complete in the $\delta$-topology.

###### Proof.

Let $(a_{j})$ be a Cauchy net in $\mathfrak{A}_{\mathscr{E}}$ with respect to
the $\delta$-topology. In particular,
$(\Delta^{n}a_{j},\delta(\Delta^{n}a_{j}))_{j}$ is Cauchy in
$\mathfrak{H}\oplus\bar{\mathscr{H}}$ for all $n\in\mathbb{Z}$. Since
$\Delta^{n}$ and $\delta$ are closable on $\mathfrak{A}_{\mathscr{E}}$, it
follows that there exists $a\in\mathfrak{H}$ such that
$(\Delta^{n}a_{j},\delta(\Delta^{n}a_{j}))\to(\Delta^{n}a,\bar{\delta}(\Delta^{n}a))$
for all $n\in\mathbb{Z}$. In particular,
$a\in\bigcap_{n\in\mathbb{Z}}\operatorname{dom}(\Delta^{n})$ and
$\Delta^{n}a\in\operatorname{dom}(\bar{\delta})=\operatorname{dom}(\mathscr{E})$
for all $n\in\mathbb{Z}$.

Moreover, as the Mackey topology is complete, for $n\in\mathbb{Z}$ there
exists $x_{n}\in B(\mathfrak{H})$ such that $\pi_{l}(\Delta^{n}a_{j})\to
x_{n}$ with respect to $\tau(B(\mathfrak{H}),B(\mathfrak{H})_{\ast})$. For
$b\in\mathfrak{A}_{\mathscr{E}}$ we have

$x_{n}b=\lim_{j}\pi_{l}(\Delta^{n}a_{j})b=\lim_{j}\pi_{r}(b)\Delta^{n}a_{j}=\pi_{r}(b)\Delta^{n}a.$

Since $\mathfrak{A}_{\mathscr{E}}$ is dense in $\mathfrak{H}$, it follows that
$\Delta^{n}a\in\mathfrak{A}_{\mathscr{E}}^{\prime\prime}$ and
$\pi_{l}(\Delta^{n}a_{j})\to\pi_{l}(\Delta^{n}a)$ for all $n\in\mathbb{Z}$.

Altogether we conclude that $a\in\mathfrak{A}_{\mathscr{E}}$ and $a_{j}\to a$
in the $\delta$-topology. ∎

######  Lemma 2.2.

If $\mathfrak{A}$ is a Tomita algebra, $\mathscr{H}$ a Tomita bimodule over
$\mathfrak{A}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ a closable
symmetric derivation, the inclusion of $\mathfrak{A}$ into its completion
$\mathfrak{H}$ extends to an injective map from the completion of
$\mathfrak{A}$ in the $\delta$-topology to $\mathfrak{H}$.

###### Proof.

We have to show that if $(a_{j})$ is a Cauchy net in $\mathfrak{A}$ with
respect to the $\delta$-topology and $a_{j}\to 0$ in $\mathfrak{H}$, then
$a_{j}\to 0$ in the $\delta$-topology. Since $\Delta^{n}$, $n\in\mathbb{Z}$,
and $\delta$ are closable, we have
$(\Delta^{n}a_{j},\delta(\Delta^{n}a_{j}))\to 0$ for all $n\in\mathbb{Z}$.
Furthermore, using the completeness of the Mackey topology and a similar
argument as in the previous lemma, one sees that $\pi_{l}(\Delta^{n}a_{j})\to
0$ in $\tau(B(\mathfrak{H}),B(\mathfrak{H})_{\ast})$ for all $n\in\mathbb{Z}$.
Hence $a_{j}\to 0$ in the $\delta$-topology. ∎

Let $\widehat{\mathfrak{A}}^{\delta}$ denote the set of all elements
$a\in\mathfrak{H}$ for which there exists a net $(a_{j})$ in $\mathfrak{A}$
such that $a_{j}\to a$ in $\mathfrak{H}$ and $(a_{j})$ is Cauchy in the
$\delta$-topology. By the previous lemma, $\widehat{\mathfrak{A}}^{\delta}$ is
a completion of $\mathfrak{A}$ in the $\delta$-topology, and we call it simply
the $\delta$-completion of $\mathfrak{A}$. It is not hard to see that
$\widehat{\mathfrak{A}}^{\delta}$ is a Tomita subalgebra of
$(\mathfrak{A}^{\prime\prime})_{0}$ and contained in
$\operatorname{dom}(\bar{\delta})$.

Recall that if $\mathfrak{H}$ is a normal Tomita bimodule over $\mathfrak{A}$,
we can continuously extend the left and right action of $\mathfrak{A}$ and the
maps $\mathscr{J}$ and $\mathscr{U}_{t}$, $t\in\mathbb{R}$, to the Hilbert
completion $\bar{\mathscr{H}}$. This is usually not possible for
$\mathscr{U}_{z}$, $z\in\mathbb{C}\setminus\mathbb{R}$. We define

$\mathscr{H}^{\mathrm{a}}=\\{\xi\in\bar{\mathscr{H}}\mid
t\mapsto\bar{\mathscr{U}}_{t}\xi\text{ has an entire extension}\\}.$

If it exists, this entire extension is unique and will be denoted by
$z\mapsto\bar{\mathscr{U}}_{z}\xi$. Clearly,
$\mathscr{H}\subset\mathscr{H}^{\mathrm{a}}$ and
$\mathscr{U}_{z}\subset\bar{\mathscr{U}}_{z}$ for all $z\in\mathbb{C}$.

######  Lemma 2.3.

If we endow $\mathscr{H}^{\mathrm{a}}$ with the coarsest locally convex
topology that makes

$\mathscr{H}^{\mathrm{a}}\to\bar{\mathscr{H}},\,\xi\mapsto\bar{\mathscr{U}}_{in}\xi$

continuous for all $n\in\mathbb{Z}$, then $\mathscr{H}^{\mathrm{a}}$ is
complete.

###### Proof.

If $A$ is the unique non-singular positive self-adjoint operator in
$\bar{\mathscr{H}}$ such that $\bar{\mathscr{U}}_{t}=A^{it}$ for
$t\in\mathbb{R}$, then
$\mathscr{H}^{\mathrm{a}}=\bigcap_{n\in\mathbb{Z}}\operatorname{dom}(A^{n})$
and $\mathscr{H}^{\mathrm{a}}$ is the projective limit of the Banach spaces
$(\operatorname{dom}(A^{n})\cap\operatorname{dom}(A^{-n}),\lVert\cdot\rVert_{\bar{\mathscr{H}}}+\lVert
A^{n}\,\cdot\,\rVert_{\bar{\mathscr{H}}}+\lVert
A^{-n}\,\cdot\,\rVert_{\bar{\mathscr{H}}})$ in the topology described in the
lemma. In particular, $\mathscr{H}^{\mathrm{a}}$ is complete. ∎

Since $\mathscr{H}$ is a _normal_ Tomita bimodule over $\mathfrak{A}$, the
Hilbert completion $\bar{\mathscr{H}}$ has a canonical structure of a
$\pi_{l}(\mathfrak{A})^{\prime\prime}$-$\pi_{l}(\mathfrak{A})^{\prime\prime}$
correspondence determined by

$\pi_{l}(a)\cdot\xi\cdot J\pi_{r}(b)^{\ast}J=a\xi b$

for $a,b\in\mathfrak{A}$ and $\xi\in\mathscr{H}$.

If $a\in\widehat{\mathfrak{A}}^{\delta}$, $(a_{j})$ is a net in $\mathfrak{A}$
such that $a_{j}\to a$ in the $\delta$-topology and $\xi\in\bar{\mathscr{H}}$,
then

$\bar{\mathscr{U}}_{t}(\pi_{l}(a)\cdot\xi)=\lim_{j}\bar{\mathscr{U}}_{t}(\pi_{l}(a_{j})\xi)=\lim_{j}\pi_{l}(U_{t}a_{j})\bar{\mathscr{U}}_{t}\xi=\pi_{l}(U_{t}a)\cdot\bar{\mathscr{U}}_{t}\xi.$

Thus, if $\xi\in\mathscr{H}^{\mathrm{a}}$, then
$z\mapsto\pi_{l}(U_{z}a)\cdot\bar{\mathscr{U}}_{z}\xi$ is an entire
continuation of $t\mapsto\bar{\mathscr{U}}_{t}(\pi_{l}(a)\cdot\xi)$, which
implies $\pi_{l}(a)\xi\in\mathscr{H}^{\mathrm{a}}$. Likewise, if
$b\in\widehat{\mathfrak{A}}^{\delta}$, then $\xi\cdot
J\pi_{r}(b)^{\ast}J\in\mathscr{H}^{\mathrm{a}}$. It is then routine to check
that the bimodule structure given by $a\xi b=\pi_{l}(a)\cdot\xi\cdot
J\pi_{r}(b)^{\ast}J$, then group $(\bar{\mathscr{U}}_{z})_{z\in\mathbb{C}}$
and the restriction of $\bar{\mathscr{J}}$ make $\mathscr{H}^{\mathrm{a}}$
into a Tomita bimodule over $\widehat{\mathfrak{A}}^{\delta}$.

######  Lemma 2.4.

If $\mathfrak{A}$ is a Tomita algebra, $\mathscr{H}$ is a Tomita bimodule over
$\mathfrak{A}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ is a closable
symmetric derivation with closure $\bar{\delta}$, then
$\bar{\delta}(\widehat{\mathfrak{A}}^{\delta})\subset\mathscr{H}^{\mathrm{a}}$
and
$\bar{\delta}\colon\widehat{\mathfrak{A}}^{\delta}\to\mathscr{H}^{\mathrm{a}}$
is a symmetric derivation.

###### Proof.

If $a\in\widehat{\mathfrak{A}}^{\delta}$ and $(a_{j})$ is a net in
$\mathfrak{A}$ such that $a_{j}\to a$ in the $\delta$-topology, then

$\bar{\mathscr{U}}_{t}\bar{\delta}(a)=\lim_{j}\mathscr{U}_{t}\delta(a_{j})=\delta(U_{t}a_{j})=\bar{\delta}(U_{t}a).$

It follows that $t\mapsto\bar{\mathscr{U}}_{t}\bar{\delta}(a)$ has the entire
continuation $z\mapsto\bar{\delta}(U_{z}a)$, which implies
$\bar{\delta}(a)\in\bar{\mathscr{H}}^{\mathrm{a}}$. Again, routine
computations show that the restriction of $\bar{\delta}$ to
$\widehat{\mathfrak{A}}^{\delta}$ is a symmetric derivation from
$\widehat{\mathfrak{A}}^{\delta}$ to $\mathscr{H}^{\mathrm{a}}$. ∎

##  3\. Closability of derivations

In this section we give a simple criterion for the closability of derivations
inspired by a well-known result (see [Voi98, Section 4] and [Nel17, Lemma 3.9]
for the non-tracial case) on the closability of the derivation used in free
probability.

If $\mathfrak{A}$ is a Tomita algebra and $\mathscr{H}$ is a Tomita bimodule
over $\mathfrak{A}$, we say that $\xi\in\mathscr{H}$ is a _bounded vector_ if
there exists $C>0$ such that $\lVert a\xi b\rVert\leq C\lVert a\rVert\lVert
b\rVert$ for all $a,b\in\mathfrak{A}$. In this case, the maps $a\mapsto a\xi$
and $b\mapsto\xi b$ extend to bounded linear operators from the completion
$\mathfrak{H}$ of $\mathfrak{A}$ to $\mathscr{H}$, which we denote by $R(\xi)$
and $L(\xi)$, respectively.

######  Lemma 3.1.

Let $\mathfrak{A}$ be a Tomita algebra, $\mathscr{H}$ a normal Tomita bimodule
over $\mathfrak{A}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ a derivation.
If $\delta(\mathfrak{A})$ is contained in the space of bounded vectors, then
$\operatorname{dom}(\delta^{\ast})$ is a subbimodule of $\mathscr{H}$ and

$\delta^{\ast}(a\xi
b)=a^{\ast}\delta^{\ast}(\xi)b-L(\delta(a^{\ast}))^{\ast}(\xi
b)-R(\delta(b^{\ast}))^{\ast}(a\xi)$

for $a,b\in\mathfrak{A}$ and $\xi\in\operatorname{dom}(\delta^{\ast})$.

###### Proof.

Let $a,b,c\in\mathfrak{A}$ and $\xi\in\operatorname{dom}(\delta^{\ast})$. By
the product rule,

$\displaystyle\langle a\xi b,\delta(c)\rangle$
$\displaystyle=\langle\xi,a^{\ast}\delta(c)b^{\ast}\rangle$
$\displaystyle=\langle\xi,\delta(a^{\ast}cb^{\ast})-\delta(a^{\ast})cb^{\ast}-a^{\ast}c\delta(b^{\ast})\rangle$
$\displaystyle=\langle a\delta^{\ast}(\xi)b-L(\delta(a^{\ast}))^{\ast}(\xi
b)-R(\delta(b^{\ast}))^{\ast}(a\xi),c\rangle.$

Thus $a\xi b\in\operatorname{dom}(\delta^{\ast})$ and the claimed identity for
$\delta^{\ast}(a\xi b)$ holds. ∎

######  Lemma 3.2.

Let $\mathfrak{A}$ be a Tomita algebra, $\mathscr{H}$ a normal Tomita bimodule
over $\mathscr{H}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ a derivation.
If $\delta(\mathfrak{A})$ is contained in the space of bounded vectors and
$\operatorname{dom}(\delta^{\ast})$ is a cyclic subset, then $\delta$ is
closable.

###### Proof.

By the previous lemma, $\operatorname{dom}(\delta^{\ast})$ is a subbimodule of
$\mathscr{H}$. Hence, if $\operatorname{dom}(\delta^{\ast})$ is cyclic, then
it is dense in $\mathscr{H}$. Therefore, $\delta$ is closable. ∎

##  4\. Completely Dirichlet forms associated with closable derivations

In this section we prove the main theorem of this article, Theorem 4.4,
showing that the closure of the quadratic form associated with a closable
symmetric derivation is a modular completely Dirichlet form.

As mentioned in the introduction, we rely on Haagerup’s reduction method. To
set up the stage for its use, we first discuss crossed products of Tomita
algebras and Tomita bimodules. To extend closable symmetric derivations to a
sufficiently large domains on the crossed product, we use the
$\delta$-completion technique developed in Section 2. Further, to reduce the
problem to the tracial case, we need a “change of reference weight” argument
and an analysis of approximation properties of completely Dirichlet forms.
This will be dealt with in the following lemmas. Finally, in Proposition 4.6
we discuss the relation between the derivation we started with and the first-
order differential structure of the associated completely Dirichlet form.

Let $\mathfrak{A}$ be a Tomita algebra, $\mathscr{H}$ a normal Tomita bimodule
over $\mathfrak{A}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ a closable
symmetric derivation. Throughout this section we endow $\mathfrak{A}$ with the
$\delta$-topology and $\mathscr{H}$ with the projective topology induced by
the maps
$\mathscr{H}\to\bar{\mathscr{H}},\,\xi\mapsto\bar{\mathscr{U}}_{in}\xi$ for
$n\in\mathbb{Z}$, and we assume that $\mathfrak{A}$ and $\mathscr{H}$ are
complete in these topologies. As discussed in Section 2, this can always be
achieved by passing to the completions.

Let $G$ be a countable subgroup of $\mathbb{R}$, viewed as discrete group. The
vector space $C_{c}(G;\mathfrak{A})\cong C_{c}(G)\odot\mathfrak{A}$ can be
made into a Tomita algebra by the operations

$\displaystyle(a\ast b)(g)$ $\displaystyle=\sum_{h\in G}U_{-h}a(g-h)b(h),$
$\displaystyle a^{\sharp}(g)$ $\displaystyle=U_{-g}(a(-g)^{\sharp}),$
$\displaystyle(U_{z}a)(g)$ $\displaystyle=U_{z}a(g).$

Moreover, the vector space $C_{c}(G;\mathscr{H})$ becomes a normal Tomita
bimodule over $C_{c}(G;\mathfrak{A})$ with the operations

$\displaystyle(a\xi)(g)$ $\displaystyle=\sum_{h\in G}U_{-h}a(g-h)\xi(h)$
$\displaystyle(\xi b)(g)$ $\displaystyle=\sum_{h\in
G}\mathscr{U}_{-h}\xi(g-h)b(h)$ $\displaystyle(\tilde{\mathscr{J}}\xi)(g)$
$\displaystyle=\mathscr{U}_{-g}\mathscr{J}\xi(-g)$
$\displaystyle(\tilde{\mathscr{U}}_{z}\xi)(g)$
$\displaystyle=\mathscr{U}_{z}\xi(g).$

Furthermore, $1_{C_{c}(G)}\odot\delta\colon C_{c}(G;\mathfrak{A})\to
C_{c}(G;\mathscr{H})$ is a closable symmetric derivation, whose closure we
denote by $1\otimes\bar{\delta}$.

We write $\tilde{\mathfrak{A}}$ for the $(1\odot\delta)$-completion of
$C_{c}(G;\mathfrak{A})$, $\tilde{\mathscr{H}}$ for
$C_{c}(G;\mathscr{H})^{\mathrm{a}}$ and $\tilde{\delta}$ for the restriction
of $1\otimes\bar{\delta}$ to $\tilde{\mathfrak{A}}$. By Lemma 2.4 the map
$\tilde{\delta}$ is a (closable) symmetric derivation from
$\tilde{\mathfrak{A}}$ to $\tilde{\mathscr{H}}$.

######  Lemma 4.1.

If $x\in L(G)\otimes\mathbb{C}1_{\bar{\mathscr{H}}}$ and
$a\in\tilde{\mathfrak{A}}$, then $xa,ax\in\tilde{\mathfrak{A}}$ and
$\tilde{\delta}(xa)=x\tilde{\delta}(a)$,
$\tilde{\delta}(ax)=\tilde{\delta}(a)x$.

###### Proof.

Let $x=y\otimes 1$ with $y\in L(G)$, let $(y_{i})$ be a bounded net in
$\mathbb{C}[G]$ such that $y_{i}\to y$ in the strong∗ topology and let
$x_{i}=y_{i}\otimes 1$. Clearly, $x_{i}\to x$ in the Mackey topology.

If $a\in C_{c}(G;\mathfrak{A})$, then $x_{i}a\in C_{c}(G;\mathfrak{A})$ and
$\Delta^{n}(x_{i}a)=x_{i}\Delta^{n}a$,
$(1\odot\delta)(x_{i}a)=x_{i}(1\odot\delta)(a)$,
$\pi_{l}(\Delta^{n}(x_{i}a))=x_{i}\pi_{l}(\Delta^{n}a)$. It follows that
$x_{i}a\to xa$ in $\ell^{2}(G;\mathfrak{H})$, the net $(x_{i}a)$ is Cauchy in
the $\tilde{\delta}$-topology and $(1\odot\delta)(x_{i}a)\to
x(1\odot\delta)(a)$. Thus $xa\in\tilde{\mathfrak{A}}$ and
$\tilde{\delta}(xa)=x\tilde{\delta}(a)$.

A similar argument shows that if $(a_{j})$ is a Cauchy net in
$C_{c}(G;\mathfrak{A})$ with respect to $\mathscr{T}_{\tilde{\delta}}$ and
$a_{j}\to a$ in $\ell^{2}(G;\mathscr{H})$, then $(xa_{j})$ is Cauchy with
respect to $\mathscr{T}_{\tilde{\delta}}$ and $xa_{j}\to xa$ in
$\ell^{2}(G;\mathscr{H})$. Hence if $a\in\tilde{\mathfrak{A}}$, then
$xa\in\tilde{\mathfrak{A}}$ and $\tilde{\delta}(xa)=x\tilde{\delta}(a)$. The
statement for $ax$ can be proven analogously. ∎

For the next lemma recall that
$\mathfrak{A}_{\varphi}=\Lambda_{\varphi}(\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast})$
is the full left Hilbert algebra induced by the weight $\varphi$, the cone
$C_{\varphi}$ is the closure of $\\{\Delta_{\varphi}^{1/4}a\mid
a\in\mathfrak{A}_{\varphi},0\leq\pi_{l}(a)\leq 1\\}$, and $M_{\varphi}$
denotes the centralizer of $\varphi$.

######  Lemma 4.2.

Let $M$ be a von Neumann algebra, $\varphi$ a normal semi-finite faithful
weight on $M$ and $x\in M_{\varphi}$ be positive and invertible. Let
$\psi=\varphi(x^{1/2}\cdot x^{1/2})$.

If $\mathscr{E}$ is a modular (completely) Dirichlet form on $L^{2}(M)$ with
respect to $\mathfrak{A}_{\varphi}$,
$x\operatorname{dom}(\mathscr{E})\subset\operatorname{dom}(\mathscr{E})$ and
$\mathscr{E}(xa,b)=\mathscr{E}(a,xb)$ for all
$a,b\in\operatorname{dom}(\mathscr{E})$, then $\mathscr{E}$ is also a modular
(completely) Dirichlet from with respect to $\mathfrak{A}_{\psi}$.

###### Proof.

Since $x$ is invertible, the weight $\psi$ is faithful and
$J_{\psi}=J_{\varphi}$, and since $x$ commutes with $(\Delta_{\varphi}^{it})$,
we have
$\Delta_{\psi}^{it}=x^{it}\Delta_{\varphi}^{it}(\cdot)x^{-it}=\Delta_{\varphi}^{it}(x^{it}\cdot
x^{-it})$.

Let $A$ be the positive self-adjoint operator associated with $\mathscr{E}$
and $T_{t}=e^{-tA}$. The commutation relation
$x\operatorname{dom}(\mathscr{E})\subset\operatorname{dom}(\mathscr{E})$ and
$\mathscr{E}(xa,b)=\mathscr{E}(a,xb)$ for
$a,b\in\operatorname{dom}(\mathscr{E})$ implies that $x$ commutes strongly
with $A^{1/2}$. Hence $T_{t}(xa)=xT_{t}(a)$ for all $a\in H$ and $t\geq 0$.
Since $T_{t}$ commutes with $J_{\varphi}$, we also have $T_{t}(ax)=T_{t}(a)x$
for $a\in H$, $t\geq 0$. In particular, $(T_{t})$ and $(\Delta_{\psi}^{is})$
commute.

Moreover, $C_{\psi}=x^{1/4}C_{\varphi}x^{1/4}$. Indeed, a direct computation
shows that $\mathfrak{n}_{\psi}=\mathfrak{n}_{\varphi}$ and
$\Lambda_{\psi}(y)=\Lambda_{\varphi}(y)x^{1/2}$ for
$y\in\mathfrak{n}_{\varphi}$. Hence, if $y\in\mathfrak{n}_{\varphi}$ with
$0\leq y\leq 1$, then

$\Delta_{\psi}^{1/4}\Lambda_{\psi}(y)=x^{1/4}\Delta_{\varphi}^{1/4}\Delta_{\varphi}(y)x^{1/4}\in
x^{1/4}C_{\varphi}x^{1/4}.$

The converse inclusion follows by swapping the roles of $\varphi$ and $\psi$.

Therefore, if $a\in C_{\psi}$, then

$T_{t}(a)=x^{1/4}T_{t}(x^{-1/4}ax^{-1/4})x^{1/4}\in C_{\psi}.$

Thus $\mathscr{E}$ is a Dirichlet form with respect to $\mathfrak{A}_{\psi}$
by [GL99, Theorem 5.7]. The result for completely Dirichlet forms follows
easily by applying the same argument to the forms $\mathscr{E}^{(n)}$ on
$L^{2}(M\otimes M_{n}(\mathbb{C}))$. ∎

######  Lemma 4.3.

Let $M$ be a von Neumann algebra and $\varphi$ a normal semi-finite faithful
weight on $M$. Let $(M_{n})$ be an increasing sequence of von Neumann
subalgebras with weak∗ dense union and assume that $M_{n}$ is the range of a
$\varphi$-preserving conditional expectation $E_{n}$ on $M$. Let $H_{n}$
denote the closure of $\Lambda_{\varphi}(\mathfrak{n}_{\varphi}\cap M_{n})$
and let $P_{n}$ denote the orthogonal projection from $H$ to $H_{n}$.

If $\mathscr{E}$ is a closed densely defined quadratic form on $H$ such that
for every $n\in\mathbb{N}$ the quadratic form $\mathscr{E}|_{H_{n}}$ is a
Dirichlet form with respect to
$\Lambda_{\varphi}(\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast}\cap
M_{n})$ and $\mathscr{E}\circ P_{n}\leq\mathscr{E}$, then $\mathscr{E}$ is a
Dirichlet form with respect to $\mathfrak{A}_{\varphi}$.

###### Proof.

Let $(T_{t})$ be the strongly continuous semigroup associated with
$\mathscr{E}$. Since $\mathscr{E}\circ P_{n}\leq\mathscr{E}$, we have
$T_{t}(H_{n})\subset H_{n}$ by Ouhabaz’ theorem [Ouh96, Corollary 2.4]. Thus
$T_{t}$ commutes with $P_{n}$, and it is easy to see that $(T_{t}P_{n})$ is
the semigroup associated with $\mathscr{E}|_{H_{n}}$, viewed as semigroup on
$H$. In particular, $T_{t}P_{n}J_{\varphi}=J_{\varphi}T_{t}P_{n}$. In the
limit we obtain $T_{t}J_{\varphi}=J_{\varphi}T_{t}$.

It remains to show that $T_{t}(C_{\varphi})\subset C_{\varphi}$ for all $t\geq
0$. A direct computation shows that $E_{n}$ and $P_{n}$ are related by
$P_{n}\Lambda_{\varphi}(x)=\Lambda_{\varphi}(E_{n}(x))$. Moreover, $E_{n}$ is
GNS-symmetric with respect to $\varphi$, which implies that $P_{n}$ commutes
with $(\Delta_{\varphi}^{it})$. Thus $P_{n}(C_{\varphi})$ is the closure of
$\\{\Delta_{\varphi}^{1/4}\Lambda_{\varphi}(x)\mid
x\in\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast}\cap M_{n},\,0\leq
x\leq 1\\}$. In particular, $P_{n}(C_{\varphi})\subset C_{\varphi}$.

Since $\mathscr{E}_{n}$ is a Dirichlet form with respect to
$\Lambda_{\varphi}(\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast}\cap
M_{n})$ and $(T_{t}P_{n})$ is the associated semigroup, we have
$T_{t}P_{n}(C_{\varphi})\subset P_{n}(C_{\varphi})$. Moreover, since
$\bigcup_{n}M_{n}$ is weak∗ dense in $M$, we have $P_{n}\to 1$ strongly by
Kaplansky’s density theorem. Therefore, $T_{t}(C_{\varphi})\subset
C_{\varphi}$. ∎

To prove that the quadratic form associated with a closable symmetric
derivation is a completely Dirichlet form, we will reduce the problem to the
tracially symmetric case by means of Haagerup’s reduction method. We only
recall the necessary definitions here and refer to [HJX10] for proofs in the
case of states and to [Cas+15] for the extension to weights.

Let $M=\pi_{l}(\mathfrak{A})^{\prime\prime}$, let $\varphi$ be the weight
induced by the full left Hilbert algebra $\mathfrak{A}^{\prime\prime}$ on $M$,
let $G=\bigcup_{n\in\mathbb{N}}2^{-n}\mathbb{Z}$, let
$\tilde{M}=M\rtimes_{\sigma^{\varphi}}G=\pi_{l}(\tilde{\mathfrak{A}})^{\prime\prime}$
and let $\tilde{\varphi}$ be the dual weight of $\varphi$ on $\tilde{M}$. Let
$(a_{n})$ be a sequence of self-adjoint elements of
$L(G)\otimes\mathbb{C}1\subset\mathscr{Z}(\tilde{M}_{\tilde{\varphi}})$,
$\varphi_{n}=\varphi e^{-a_{n}}$, $M_{n}=\tilde{M}_{\varphi_{n}}$ and
$\tau_{n}=\varphi_{n}|_{M_{n}}$. Here $N_{\psi}$ denotes the centralizer of
the weight $\psi$ on $N$ and $\mathscr{Z}(M)$ is the center of the von Neumann
algebra $N$.

By [Cas+15, Theorem 8.1] the sequence $(a_{n})$ can be chosen such that

  * •

$M_{n}$ is semi-finite with normal semi-finite faithful trace $\tau_{n}$,

  * •

for each $n\in\mathbb{N}$ there exists a conditional expectation $E_{n}$ from
$M$ onto $M_{n}$ such that $\tilde{\varphi}\circ E_{n}=\tilde{\varphi}$ and
$\sigma^{\tilde{\varphi}}_{t}\circ
E_{n}=E_{n}\circ\sigma^{\tilde{\varphi}}_{t}$ for all $t\in\mathbb{R}$,

  * •

$E_{n}(x)\to x$ strongly∗ for every $x\in M$.

In the following we fix a sequence $(a_{n})$ with these properties. The
concrete construction is irrelevant for our purposes.

######  Theorem 4.4.

Let $\mathfrak{A}$ be a Tomita algebra with completion $\mathfrak{H}$, let
$\mathscr{H}$ be a normal Tomita bimodule over $\mathfrak{A}$ and
$\delta\colon\mathfrak{A}\to\mathscr{H}$ a closable symmetric derivation. The
closure $\mathscr{E}$ of the quadratic form

$\mathfrak{H}\to[0,\infty],\,a\mapsto\begin{cases}\lVert\delta(a)\rVert_{\mathscr{H}}^{2}&\text{if
}a\in\mathfrak{A},\\\ \infty&\text{otherwise}\end{cases}$

is a modular completely Dirichlet form with respect to
$\mathfrak{A}^{\prime\prime}$. If moreover $\mathfrak{A}$ is unital, then
$\mathscr{E}$ is a modular quantum Dirichlet form.

###### Proof.

We continue to use the notation from the previous discussion. The derivation
$\tilde{\delta}\colon\tilde{\mathfrak{A}}\to\tilde{\mathscr{H}}$ is a
restriction of $1\otimes\bar{\delta}$. Let $\tilde{\mathscr{E}}$ denote the
closure of the quadratic form

$\ell^{2}(G;\mathfrak{H})\to[0,\infty],\,a\mapsto\begin{cases}\lVert\tilde{\delta}(a)\rVert_{\tilde{\mathscr{H}}}^{2}&\text{if
}a\in\tilde{\mathfrak{A}},\\\ \infty&\text{otherwise}.\end{cases}$

It is clear that
$\tilde{\mathscr{E}}(a)=\lVert(1\otimes\bar{\delta})(a)\rVert_{\ell^{2}(G;\bar{\mathscr{H}})}^{2}$
for $a\in\operatorname{dom}(\tilde{\mathscr{E}})$. Furthermore,
$\operatorname{dom}(\tilde{\mathscr{E}})=\operatorname{dom}(1\otimes\bar{\delta})$
and the strongly continuous semigroups $(T_{t})$ and $(\tilde{T}_{t})$
associated with $\mathscr{E}$ and $\tilde{\mathscr{E}}$, respectively, are
related by $\tilde{T}_{t}=\mathrm{id}_{\ell^{2}(G)}\otimes T_{t}$.

The map
$\iota\colon\mathfrak{H}\to\ell^{2}(G;\mathfrak{H}),\,a\mapsto\mathds{1}_{0}\otimes
a$ is an isometric embedding such that
$\iota(C_{\mathfrak{A}^{\prime\prime}})=C_{\tilde{\mathfrak{A}}^{\prime\prime}}\cap\iota(\mathfrak{H})$.
Thus, if $\tilde{\mathscr{E}}$ is a (completely) Dirichlet form with respect
to $\tilde{\mathfrak{A}}^{\prime\prime}$, then $\mathscr{E}$ is a (completely)
Dirichlet form with respect to $\mathfrak{A}^{\prime\prime}$.

Since $M$ is in standard form on $\mathfrak{H}$ and $\tilde{M}$ is in standard
form on $\ell^{2}(G;\mathfrak{H})$, these spaces can be canonically identified
with $L^{2}(M,\varphi)$ and $L^{2}(\tilde{M},\tilde{\varphi})$, respectively,
and we will tacitly do so in the following. Under these identifications,
$\mathfrak{A}^{\prime\prime}=\mathfrak{A}_{\varphi}$,
$\tilde{\mathfrak{A}}^{\prime\prime}=\mathfrak{A}_{\tilde{\varphi}}$ and
$\Delta_{\tilde{\varphi}}^{it}=\mathrm{id}_{\ell^{2}(G)}\otimes\Delta_{\varphi}^{it}$.

Let

$\mathscr{A}_{n}=\\{x\in\mathfrak{n}_{\tilde{\varphi}}\cap\mathfrak{n}_{\tilde{\varphi}}^{\ast}\cap
M_{n}\mid\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2})\in\tilde{\mathfrak{A}}\\}.$

Since $e^{a_{n}/2}\in\tilde{M}_{\tilde{\varphi}}$, if $x\in\mathscr{A}_{n}$,
then

$\Lambda_{\tilde{\varphi}}(x)=\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2})e^{a_{n}/2}\in\tilde{\mathfrak{A}}$

by Lemma 4.1. Reversing the roles of $e^{-a_{n}/2}$ and $e^{a_{n}/2}$ we get

$\mathscr{A}_{n}=\\{x\in\mathfrak{n}_{\tilde{\varphi}}\cap\mathfrak{n}_{\tilde{\varphi}}^{\ast}\cap
M_{n}\mid\Lambda_{\tilde{\varphi}}(x)\in\tilde{\mathfrak{A}}\\}.$

Since $\tilde{\mathfrak{A}}$ is a Tomita algebra, it follows easily that
$\mathscr{A}_{n}$ is a $\ast$-algebra. Define an
$\mathscr{A}_{n}$-$\mathscr{A}_{n}$-bimodule structure on
$\ell^{2}(G;\bar{\mathscr{H}})$ by

$x\xi
y=\Lambda_{\tilde{\varphi}}(x)\cdot\xi\cdot\Lambda_{\tilde{\varphi}}^{\prime}(y).$

Using that $\tilde{\mathscr{H}}$ is a Tomita bimodule over
$\tilde{\mathfrak{A}}$, it is not hard to see that this left and right action
are contractive (anti-) $\ast$-homomorphisms. Moreover, $\tilde{\mathscr{J}}$
extends to an anti-unitary involution on $\ell^{2}(G;\bar{\mathscr{H}})$
intertwining the left and right action. We still denote this extension by
$\tilde{\mathscr{J}}$.

Let

$\partial_{n}\colon\mathscr{A}_{n}\to
L^{2}(\tilde{M},\tilde{\varphi}),\,\partial_{n}(x)=\tilde{\delta}(\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2})).$

Since $e^{-a_{n}/2}\in\tilde{M}_{\tilde{\varphi}}$ and
$x\in\tilde{M}_{\varphi_{n}}$, we have

$\displaystyle\partial_{n}(x^{\ast})$
$\displaystyle=\tilde{\delta}(\Lambda_{\tilde{\varphi}}(x^{\ast}e^{-a_{n}/2}))$
$\displaystyle=\tilde{\delta}(\Lambda_{\tilde{\varphi}}^{\prime}(e^{-a_{n}/2}x^{\ast}))$
$\displaystyle=\tilde{\delta}(\tilde{J}\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2}))$
$\displaystyle=\tilde{\mathscr{J}}\tilde{\delta}(\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2}))$
$\displaystyle=\tilde{\mathscr{J}}\partial_{n}(x).$

Moreover, it follows from Lemma 4.1 combined with
$e^{-a_{n}/2}\in\tilde{M}_{\tilde{\varphi}}$ and
$x,y\in\tilde{M}_{\varphi_{n}}$ that

$\displaystyle\partial_{n}(xy)$
$\displaystyle=\tilde{\delta}(\Lambda_{\tilde{\varphi}}(xye^{-a_{n}/2}))$
$\displaystyle=\Lambda_{\tilde{\varphi}}(x)\cdot\tilde{\delta}(\Lambda_{\tilde{\varphi}}(ye^{-a_{n}/2}))+\tilde{\delta}(\Lambda_{\tilde{\varphi}}(x))\cdot\Lambda_{\tilde{\varphi}}(ye^{-a_{n}/2})$
$\displaystyle=\Lambda_{\tilde{\varphi}}(x)\cdot\tilde{\delta}(\Lambda_{\tilde{\varphi}}(ye^{-a_{n}/2}))+\tilde{\delta}(\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2}))\cdot\Lambda_{\tilde{\varphi}}(e^{a_{n}/2}ye^{-a_{n}/2})$
$\displaystyle=\Lambda_{\tilde{\varphi}}(x)\cdot\tilde{\delta}(\Lambda_{\tilde{\varphi}}(ye^{-a_{n}/2}))+\tilde{\delta}(\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2}))\cdot\Lambda_{\tilde{\varphi}}^{\prime}(\sigma^{\varphi_{n}}_{-i/2}(y))$
$\displaystyle=\Lambda_{\tilde{\varphi}}(x)\cdot\tilde{\delta}(\Lambda_{\tilde{\varphi}}(ye^{-a_{n}/2}))+\tilde{\delta}(\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2}))\cdot\Lambda_{\tilde{\varphi}}^{\prime}(y)$
$\displaystyle=x\partial_{n}(y)+\partial_{n}(x)y.$

The operator $\partial_{n}$ is closable when viewed as operator in
$L^{2}(M_{n},\tau_{n})$ since $\tilde{\delta}$ is closable and the map
$\Lambda_{\tau_{n}}(x)\mapsto\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2})$ extends
to an isometry $\iota_{n}$ from $L^{2}(M_{n},\tau_{n})$ to
$L^{2}(\tilde{M},\tilde{\varphi})$.

Since $\tau_{n}$ is a trace, [Cip08, Theorem 4.5] implies that the closure
$Q_{n}$ of the quadratic form

$L^{2}(M_{n},\tau_{n})\to[0,\infty],\,a\mapsto\begin{cases}\lVert\partial_{n}(x)\rVert^{2}&\text{if
}a=\Lambda_{\tau_{n}}(x),\,x\in\mathscr{A}_{n},\\\
\infty&\text{otherwise}\end{cases}$

is a completely Dirichlet form.

Let
$H_{n}=\overline{\Lambda_{\tilde{\varphi}}(\mathfrak{n}_{\tilde{\varphi}}\cap\mathfrak{n}_{\tilde{\varphi}}^{\ast}\cap
M_{n})}$ and let $\mathscr{E}_{n}$ be the closure of the quadratic form

$H_{n}\to[0,\infty],\,a\mapsto\begin{cases}\lVert\tilde{\delta}(a)\rVert^{2}&\text{if
}a\in\tilde{\mathfrak{A}},\\\ \infty&\text{otherwise}.\end{cases}$

In other words, $\mathscr{E}_{n}=Q_{n}\circ\iota_{n}^{-1}$.

Note that $\iota_{n}$ maps $\\{\Lambda_{\tau_{n}}(x)\mid
x\in\mathfrak{n}_{\tilde{\varphi}}\cap\mathfrak{n}_{\tilde{\varphi}}^{\ast}\cap
M_{n},0\leq x\leq 1\\}$ onto $\\{\Lambda_{\tilde{\varphi}}(xe^{-a_{n}/2})\mid
x\in\mathfrak{n}_{\tilde{\varphi}}\cap\mathfrak{n}_{\tilde{\varphi}}^{\ast}\cap
M_{n},0\leq x\leq 1\\}$. Since $\varphi_{n}$ is a trace on $M_{n}$, the latter
set coincides with $\\{\Lambda_{\varphi_{n}}(x)\mid
x\in\mathfrak{A}_{\varphi_{n}}\cap H_{n},\,0\leq x\leq 1\\}$. It follows that
$\mathscr{E}_{n}$ is a completely Dirichlet form with respect to
$\mathfrak{A}_{\varphi_{n}}\cap H_{n}$.

Moreover,

$\displaystyle\mathscr{E}_{n}(\Delta_{\varphi_{n}}^{it}a)$
$\displaystyle=\mathscr{E}_{n}(e^{-ia_{n}/2t}(\Delta_{\tilde{\varphi}}^{it}a)e^{ia_{n}/2t})$
$\displaystyle=\lVert
e^{-ia_{n}/2t}(\tilde{\mathscr{U}}_{t}\tilde{\delta}(a))e^{ia_{n}/2t}\rVert^{2}$
$\displaystyle=\lVert\tilde{\delta}(a)\rVert^{2}$
$\displaystyle=\mathscr{E}_{n}(a)$

for $a\in\tilde{\mathfrak{A}}$. This can easily be extended to the closure so
that $\mathscr{E}_{n}$ is a modular completely Dirichlet form.

By Lemma 4.1 we have
$e^{-a_{n}/2}\operatorname{dom}(\mathscr{E}_{n})\subset\operatorname{dom}(\mathscr{E}_{n})$
and $\mathscr{E}_{n}(e^{a_{n}/2}a,b)=\mathscr{E}_{n}(a,e^{-a_{n}/2}b)$ for
$a,b\in\operatorname{dom}(\mathscr{E}_{n})$. Furthermore,
$e^{-a_{n}/2}\in\tilde{M}_{\tilde{\varphi}}$. Hence $\mathscr{E}_{n}$ is also
a modular completely Dirichlet form with respect to
$\mathfrak{A}_{\tilde{\varphi}}\cap
H_{n}=\Lambda_{\tilde{\varphi}}(\mathfrak{n}_{\tilde{\varphi}}\cap\mathfrak{n}_{\tilde{\varphi}}^{\ast}\cap
M_{n})$ by Lemma 4.2.

Let $P_{n}$ denote the orthogonal projection from $\ell^{2}(G;\mathfrak{H})$
onto $H_{n}$. By definition, $\tilde{\mathscr{E}}|_{H_{n}}=\mathscr{E}_{n}$.
To apply Lemma 4.3, we have to check that $\mathscr{E}\circ
P_{n}\leq\mathscr{E}$.

Let $(T_{t})$ be the strongly continuous semigroup associated with
$\mathscr{E}$. As discussed above, $(\mathrm{id}_{\ell^{2}(G)}\otimes T_{t})$
is the strongly continuous semigroup associated with $\tilde{\mathscr{E}}$.
The modular group of $\varphi_{n}$ is given by
$\Delta_{\varphi_{n}}^{it}=e^{-ita_{n}}(\mathrm{id}_{\ell^{2}(G)}\otimes\Delta_{\varphi}^{it})(\cdot)e^{ita_{n}}$.
Since $(T_{t})$ commutes with $(\Delta_{\varphi}^{it})$ and $e^{a_{n}}\in
L(G)\otimes\mathbb{C}1_{\mathfrak{H}}$, the semigroup
$(\mathrm{id}_{\ell^{2}(G)}\otimes T_{t})$ commutes with
$(\Delta_{\varphi_{n}}^{it})$.

Since $M_{n}$ is the centralizer of $\varphi_{n}$, the subspace $H_{n}$ is the
fixed-point set of $(\Delta_{\tilde{\varphi}_{n}}^{it})$. In particular,
$(\mathrm{id}_{\ell^{2}(G)}\otimes T_{t})(H_{n})\subset H_{n}$. From Ouhabaz’s
theorem [Ouh96, Corollary 2.4] we deduce $\mathscr{E}\circ
P_{n}\leq\mathscr{E}$.

Now Lemma 4.3 shows that $\mathscr{E}$ is a modular completely Dirichlet form
with respect to $\mathfrak{A}_{\tilde{\varphi}}$.

If $\mathfrak{A}$ is unital with unit $1_{\mathfrak{A}}$, then the left and
right action of $\mathfrak{A}$ on $\mathscr{H}$ are unital since they are non-
degenerate by definition. Thus

$\delta(1_{\mathfrak{A}})=1_{\mathfrak{A}}\cdot\delta(1_{\mathfrak{A}})+\delta(1_{\mathfrak{A}})\cdot
1_{\mathfrak{A}}-\delta(1_{\mathfrak{A}})=0$

and hence $\mathscr{E}(1_{\mathfrak{A}})=0$. Thus $T_{t}(1_{\mathfrak{A}})=0$,
which implies that $\mathscr{E}$ is a quantum Dirichlet form. ∎

In the situation of the previous theorem, we call $\mathscr{E}$ the
_completely Dirichlet form associated with $\delta$_.

######  Remark 4.5.

If $\mathfrak{A}$ is not unital, the completely Dirichlet form associated with
a derivation is not necessarily a quantum Dirichlet form, even in the
commutative case. For example, this is the case for the standard Dirichlet
energy $\mathscr{E}(f)=\int_{\Omega}\lvert\nabla f\rvert^{2}$ with domain
$H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$ if $\Omega$ is a bounded Lipschitz
domain.

######  Proposition 4.6.

If $\mathfrak{A}$ is a unital Tomita algebra, $\mathscr{H}$ a normal Tomita
bimodule over $\mathfrak{A}$ and $\delta\colon\mathfrak{A}\to\mathscr{H}$ a
closable symmetric derivation with associated completely Dirichlet form
$\mathscr{E}$, then the first-order differential calculus associated with
$\mathscr{E}$ is a corestriction of
$(\mathscr{H}^{\mathrm{a}},\bar{\delta}|_{\mathfrak{A}_{\mathscr{E}}})$. In
particular,

$\bar{\delta}(ab)=a\bar{\delta}(b)+\bar{\delta}(a)b$

for $a,b\in\mathfrak{A}_{\mathscr{E}}$.

###### Proof.

Since $\mathfrak{A}$ is unital, $\mathscr{E}$ is a modular quantum Dirichlet
form by 4.4. Let $(\mathscr{H}_{\mathscr{E}},\delta_{\mathscr{E}})$ be a
first-order differential calculus associated with $\mathscr{E}$. By
definition,
$\mathfrak{A}\subset\mathfrak{A}_{\mathscr{E}}\subset\mathfrak{A}^{\prime\prime}$
and the graph norm of $\bar{\delta}$ coincides with the graph norm of
$\delta_{\mathscr{E}}$ on $\mathfrak{A}_{\mathscr{E}}$. Thus
$\pi_{l}(\mathfrak{A})^{\prime\prime}$ is strong∗ dense in
$\pi_{l}(\mathfrak{A}_{\mathscr{E}})^{\prime\prime}$ and $\mathfrak{A}$ is a
core for $\delta_{\mathscr{E}}$. It follows that the linear hull of
$\\{\delta_{\mathscr{E}}(a)b\mid a,b\in\mathfrak{A}\\}$ is dense in
$\mathscr{H}_{\mathscr{E}}$.

Let

$U\colon\mathrm{lin}\\{\delta_{\mathscr{E}}(a)b\mid
a,b\in\mathfrak{A}\\}\to\mathscr{H},\,U(\delta_{\mathscr{E}}(a)b)=\delta(a)b.$

By [Wir22, Lemma 5.9] the map $U$ is well-defined and extends to an isometric
$\mathfrak{A}_{\mathscr{E}}$-bimodule map from
$\bar{\mathscr{H}}_{\mathscr{E}}$ to $\bar{\mathscr{H}}$ such that
$U(\delta_{\mathscr{E}}(a))=\delta(a)$ for $a\in\mathfrak{A}$.

If $a\in\mathfrak{A}_{\mathscr{E}}$, let $(a_{n})$ be a sequence in
$\mathfrak{A}$ such that $\lVert a_{n}-a\rVert_{\bar{\delta}}\to 0$. As
discussed above, this implies
$\delta_{\mathscr{E}}(a_{n})\to\delta_{\mathscr{E}}(a)$. Hence
$U(\delta_{\mathscr{E}}(a))=\bar{\delta}(a)$. If
$a,b\in\mathfrak{A}_{\mathscr{E}}$, then

$\displaystyle\bar{\delta}(ab)$ $\displaystyle=U(\delta_{\mathscr{E}}(ab))$
$\displaystyle=U(a\delta_{\mathscr{E}}(b)+\delta_{\mathscr{E}}(a)b)$
$\displaystyle=aU(\delta_{\mathscr{E}}(b))+U(\delta_{\mathscr{E}}(a))b$
$\displaystyle=a\bar{\delta}(b)+\bar{\delta}(a)b.$

Moreover, $\delta J=\mathscr{J}\delta$ can be extended by continuity to
$\bar{\delta}J=\mathscr{J}\bar{\delta}$, and

$\mathbb{C}\to\bar{\mathscr{H}},\,z\mapsto\bar{\delta}(U_{z}a)$

is an entire continuation of $t\mapsto\mathscr{U}_{t}\bar{\delta}(a)$ for
$a\in\mathfrak{A}_{\mathscr{E}}$ by [Wir22, Lemma 6.6].

Thus $\bar{\delta}(\mathfrak{A}_{\mathscr{E}})\subset\mathscr{H}^{\mathrm{a}}$
and $\bar{\delta}$ is a symmetric derivation on $\mathfrak{A}_{\mathscr{E}}$.
The statement now follows from the uniqueness of the first-order differential
calculus associated with a modular completely Dirichlet form [Wir22, Theorem
6.9]. ∎

######  Remark 4.7.

The previous result holds more generally with the same proof if $\mathfrak{A}$
is not necessarily unital, but the completely Dirichlet form associated with
$\delta$ is still a quantum Dirichlet form.

######  Remark 4.8.

In the light of Lemma 2.2, one has
$\widehat{\mathfrak{A}}^{\delta}\subset\mathfrak{A}_{\mathscr{E}}$ in the
situation of the previous proposition. It is an interesting question if one
always has equality or if different derivations with $\delta$-complete domains
can have the same associated completely Dirichlet form.

##  5\. Examples

In this section we present several classes of derivations that give rise to
modular completely Dirichlet forms according to Theorem 4.4. The first three
classes of examples concern inner derivations, before we treat derivations
arising in non-tracial free probability in Example 5.4 and derivations induced
by cocycles on locally compact groups in Example 5.5.

######  Example 5.1.

Let $\mathfrak{A}$ be a Tomita algebra, $\mathscr{H}$ a normal Tomita bimodule
over $\mathfrak{A}$ and $\xi\in\mathscr{H}$ be a bounded vector. Assume that
there exists $\omega\in\mathbb{R}$ such that $\mathscr{U}_{t}\xi=e^{i\omega
t}\xi$ for all $t\in\mathbb{R}$. The map

$\delta\colon\mathfrak{A}\to\mathscr{H}\oplus\mathscr{H},\,a\mapsto i(\xi
a-a\xi,(\mathscr{J}\xi)a-a(\mathscr{J}\xi))$

is a bounded derivation, and it is symmetric when
$\mathscr{H}\oplus\mathscr{H}$ is endowed with the involution
$(\eta,\zeta)\mapsto(\mathscr{J}\zeta,\mathscr{J}\eta)$ and the complex one-
parameter group $(e^{-i\omega z}\mathscr{U}_{z},e^{i\omega
z}\mathscr{U}_{z})$.

It follows that the closure of the quadratic form

$\mathfrak{A}\to[0,\infty),\,a\mapsto\lVert\xi
a-a\xi\rVert_{\mathscr{H}}^{2}+\lVert(\mathscr{J}\xi)a-a(\mathscr{J}\xi)\rVert_{\mathscr{H}}^{2}$

is a (bounded) modular completely Dirichlet form with respect to
$\mathfrak{A}^{\prime\prime}$. In the case $\mathscr{H}=\mathfrak{A}$, this
was first proven by Cipriani [Cip97, Proposition 5.3]. See also [Wir22,
Proposition 5.5] for arbitrary Tomita bimodules $\mathscr{H}$ over
$\mathfrak{A}$.

The next example is a (partial) extension of the previous example allowing for
vectors implementing the inner derivation that are not necessarily bounded.

######  Example 5.2.

Let $\mathfrak{A}$ be a Tomita algebra with Hilbert completion $\mathfrak{H}$.
For $\xi\in\operatorname{dom}(\Delta^{1/2})$ the operator

$\pi_{l}^{0}(\xi)\colon\mathfrak{A}\to\mathfrak{H},\,a\mapsto\xi a$

is closable since $\pi_{l}^{0}(\xi^{\sharp})\subset\pi_{l}^{0}(\xi)^{\ast}$.
Likewise, if $\xi\in\operatorname{dom}(\Delta^{-1/2})$, then

$\pi_{r}^{0}(\xi)\colon\mathfrak{A}\to\mathfrak{H},\,a\mapsto a\xi$

is closable with $\pi_{r}^{0}(\xi^{\flat})\subset\pi_{r}^{0}(\xi)^{\ast}$.
Hence if
$\xi\in\operatorname{dom}(\Delta^{-1/2})\cap\operatorname{dom}(\Delta^{1/2})$,
then $\pi_{l}^{0}(\xi)-\pi_{r}^{0}(\xi)$ is closable with
$\pi_{l}^{0}(\xi^{\sharp})-\pi_{r}^{0}(\xi^{\flat})\subset(\pi_{l}^{0}(\xi)-\pi_{r}^{0}(\xi))^{\ast}$.

Now assume that there exists $\omega\in\mathbb{R}$ such that
$\Delta^{it}\xi=e^{i\omega t}\xi$ for all $t\in\mathbb{R}$. This implies in
particular
$\xi\in\operatorname{dom}(\Delta^{-1/2})\cap\operatorname{dom}(\Delta^{1/2})$.

Similar to the last example, one can turn $\mathfrak{A}\oplus\mathfrak{A}$
into a Tomita bimodule over $\mathfrak{A}$ if one equips it with the usual
bimodule structure, the involution $(\eta,\zeta)\mapsto(J\zeta,J\eta)$ and the
complex one-parameter group $(e^{-i\omega z}U_{z},e^{i\omega z}U_{z})$.Then
the map

$\delta\colon\mathfrak{A}\to\mathfrak{A}\oplus\mathfrak{A},\,a\mapsto i(\xi
a-a\xi,(J\xi)a-a(J\xi))$

is a closable symmetric derivation.

Thus the closure of the quadratic form

$\mathfrak{A}\to[0,\infty),\,a\mapsto\lVert\xi
a-a\xi\rVert_{\mathfrak{H}}^{2}+\lVert(J\xi)a-a(J\xi)\rVert_{\mathfrak{H}}^{2}$

is a modular completely Dirichlet form with respect to
$\mathfrak{A}^{\prime\prime}$. This result has first been obtained by Cipriani
and Zegarlinski [CZ21, Theorem 2.5].

The previous examples require eigenvectors of the modular group to construct a
symmetric derivation, which may be hard to find. In the following examples we
show that in certain situations one can start with an arbitrary element if one
“averages” the action of the modular group to ensure modularity.

######  Example 5.3.

Let $M$ be a von Neumann algebra with separable predual. A normal semi-finite
weight faithful weight $\varphi$ on $M$ is called _integrable_ [CT77,
Definition II.2.1] if

$\mathfrak{q}_{\varphi}=\left\\{x\in
M:\int_{\mathbb{R}}\sigma^{\varphi}_{t}(x^{\ast}x)\,dt\text{ exists in the
$\sigma$-strong topology}\right\\}$

is weak∗ dense in $M$.

If $\varphi$ is integrable, the set

$\mathfrak{A}=\\{\Lambda_{\varphi}(x)\mid x\in M\text{ analytic for
}\sigma^{\varphi},\sigma^{\varphi}_{z}(x)\in\mathfrak{q}_{\varphi}\cap\mathfrak{q}_{\varphi}^{\ast}\cap\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast}\text{
for all }z\in\mathbb{C}\\}$

is a Tomita subalgebra of $(\mathfrak{A}_{\varphi})_{0}$ with Hilbert
completion $L^{2}(M)$ and $\pi_{l}(\mathfrak{A})^{\prime\prime}=M$, as can be
seen from [CT77, Lemma II.2.3] together with a standard mollifying argument.

Let $(V_{t})$ be the translation group on $L^{2}(\mathbb{R})$, that is,
$V_{t}f(s)=f(s+t)$, and let $L^{2}(\mathbb{R})^{\mathrm{a}}$ be the set of all
entire analytic elements for $(V_{t})$. Endow
$L^{2}(\mathbb{R})^{\mathrm{a}}\odot\mathfrak{A}$ with the left and right
action of $\mathfrak{A}$ given by $a(f\otimes b)c=f\otimes abc$, the complex
one-parameter group $(V_{z}\odot U_{z})_{z\in\mathbb{C}}$ and the involution
$f\otimes a\mapsto\bar{f}\otimes Ja$. It can be checked that this makes
$L^{2}(\mathbb{R})^{\mathrm{a}}\odot\mathfrak{A}$ into a normal Tomita
bimodule, which we denote by $\mathscr{H}$.

Let
$a\in\operatorname{dom}(\Delta_{\varphi}^{1/2})\cap\operatorname{dom}(\Delta_{\varphi}^{-1/2})$
with $Ja=a$ and define

$\delta\colon\mathfrak{A}\to
L^{2}(\mathbb{R};L^{2}(M)),\,\delta(b)(s)=(U_{-s}a)b-b(U_{-s}a).$

We have

$\displaystyle\delta(U_{t}b)(s)$
$\displaystyle=(U_{-s}a)(U_{t}b)-(U_{t}b)(U_{-s}a)$
$\displaystyle=U_{t}((U_{-(s+t)}a)b-b(U_{-(s+t)}a))$
$\displaystyle=U_{t}\delta(b)(s+t).$

Thus $\delta\circ U_{t}=(U_{t}\otimes V_{t})\circ\delta$. In particular,
$\delta$ maps into $\mathscr{H}^{\mathrm{a}}$.

It is not hard to check that
$\delta\colon\mathfrak{A}\to\mathscr{H}^{\mathrm{a}}$ is a symmetric
derivation. To show closability, first note that for every fixed
$s\in\mathbb{R}$ the map $b\mapsto\delta(b)(s)$ is closable as seen in the
previous example. If $b_{n}\to 0$ and $\delta(b_{n})\to\xi$, then there exists
a subsequence such that $\delta(b_{n_{k}})(s)\to\xi(s)$ for a.e.
$s\in\mathbb{R}$. Closability of the map $b\mapsto\delta(b)(s)$ implies
$\xi(s)=0$ for a.e. $s\in\mathbb{R}$, which proves the closability of
$\delta$.

Thus the closure of the quadratic form

$\mathfrak{A}\to[0,\infty),\,b\mapsto\int_{\mathbb{R}}\lVert(U_{-s}a)b-b(U_{-s}a)\rVert^{2}\,ds$

is a modular completely Dirichlet form.

If we drop the assumption $Ja=a$, a similar argument shows that the closure of
the quadratic form

$\mathfrak{A}\to[0,\infty),\,b\mapsto\int_{\mathbb{R}}(\lVert(U_{-s}a)b-b(U_{-s}a)\rVert^{2}+\lVert(U_{-s}Ja)b+b(U_{-s}Ja)\rVert^{2})\,ds$

is a modular completely Dirichlet form with respect to
$\mathfrak{A}^{\prime\prime}$.

A similar construction is possible if one starts with a weight with periodic
modular group instead of an integrable weight and integrates over a period of
the modular group.

The following class of examples of derivations was introduced by Nelson
[Nel17] in the context of non-tracial free probability.

######  Example 5.4.

Let $M$ be a von Neumann algebra, $\varphi$ a normal faithful state on $M$ and
$B\subset M$ a $\ast$-subalgebra. Let $\partial\colon B\to
M\overline{\otimes}M$ be a linear map such that

$\partial(xy)=(x\otimes 1)\cdot\partial(y)+\partial(x)\cdot(1\otimes y)$

for $x,y\in B$. Note that Nelson works with
$M\overline{\otimes}M^{\mathrm{op}}$ instead, but under the identification
$x\otimes y\mapsto x\otimes y^{\mathrm{op}}$, the $M$-bimodules
$M\overline{\otimes}M$ and $M\overline{\otimes}M^{\mathrm{op}}$ (with the
bimodule structure used in [Nel17]) are isomorphic.

Let $\omega\in\mathbb{R}$ and write $M_{\infty}$ for the set of entire
analytic elements for $\sigma^{\varphi}$. Nelson [Nel17, Definition 3.2] calls
the map $\partial$ an $e^{\omega}$-modular derivation if $B\subset
M_{\infty}$, $B$ is invariant under $\sigma^{\varphi}_{z}$ for all
$z\in\mathbb{C}$, $\partial(B)\subset M_{\infty}\odot M_{\infty}$ and

$\partial(\sigma^{\varphi}_{z}(x))=e^{i\omega
z}(\sigma^{\varphi}_{z}\otimes\sigma^{\varphi}_{z})(\partial(x))$

for all $x\in B$ and $z\in\mathbb{C}$.

One example given by Nelson is the free difference quotient from free
probability (see [Nel17, Definition 3.4] in the non-tracial case). Given a
$\ast$-subalgebra $B$ of $M$ and an element $a\in M$ that is algebraically
free from $B$ (and $a^{\ast}$ is algebraically free from $a$ if $a\neq
a^{\ast})$, let

$\partial_{a}\colon B[a]\to B[a]\odot B[a],\,\partial_{a}(a)=1\otimes
1,\,\delta|_{B}=0$

(and $\delta_{a}(a^{\ast})=0$ if $a^{\ast}\neq a$). If $a$ is an eigenvector
of $\Delta_{\varphi}$ to the eigenvalue $e^{\omega}$, then $\partial_{a}$ is
an $e^{\omega}$-modular derivation.

Let us see how an $e^{\omega}$ derivation gives rise to a symmetric derivation
in our sense. For $x,y\in M$ let $(x\otimes y)^{\dagger}=y^{\ast}\otimes
x^{\ast}$. The conjugate derivation of $\partial$ is the map

$\hat{\partial}\colon B\to M_{\infty}\odot
M_{\infty},\,\hat{\partial}(x)=\partial(x^{\ast})^{\dagger}.$

Let $\mathfrak{A}=\Lambda_{\varphi}(B)$. Since $B$ is consists of the analytic
elements for $\sigma^{\varphi}$ and is invariant under $\sigma^{\varphi}_{z}$
for $z\in\mathbb{C}$, the set $\mathfrak{A}$ is a Tomita subalgebra of
$(\mathfrak{A}_{\varphi})_{0}=\Lambda_{\varphi}(M_{\infty})$.

Let
$\mathscr{H}=(\Lambda_{\varphi}(M_{\infty})\odot\Lambda_{\varphi}(M_{\infty}))^{\oplus
2}$ with left and right action of $\mathfrak{A}$ given by

$a(\xi_{1}\otimes\eta_{1},\xi_{2}\otimes\eta_{2})b=(a\xi_{1}\otimes\eta_{1}b,a\xi_{2}\otimes\eta_{2}b),$

involution $\mathscr{J}$ given by
$(\xi_{1}\otimes\eta_{1},\xi_{2}\otimes\eta_{2})\mapsto(J\eta_{2}\otimes
J\xi_{2},J\eta_{1}\otimes J\xi_{1})$, and complex one-parameter group
$(\mathscr{U}_{z})=(e^{i\omega
z}\Delta_{\varphi\otimes\varphi}^{iz},e^{-i\omega
z}\Delta_{\varphi\otimes\varphi}^{iz})$. One can check that this makes
$\mathscr{H}$ into a normal Tomita bimodule over $\mathfrak{A}$.

Let

$\delta\colon\mathfrak{A}\to\mathscr{H},\,\delta(\Lambda_{\varphi}(x))=(\Lambda_{\varphi\otimes\varphi}(\partial(x)),\Lambda_{\varphi\otimes\varphi}(\partial^{\dagger}(x))).$

The product rule for $\partial$ and $\hat{\partial}$ translate to the product
rule for $\delta$, the $e^{\omega}$ modularity of $\partial$ ensures
$\delta\circ\Delta_{\varphi}^{iz}=\mathscr{U}_{z}\circ\delta$ and the
definition of $\hat{\partial}$ and $\mathscr{J}$ are tailored to guarantee
$\delta\circ J_{\varphi}=\mathscr{J}\circ\delta$.

All of these properties follow by routine calculations, let us just show the
product rule (for the first component of) $\delta$ as illustration. Let
$\delta_{1}(\Lambda_{\varphi}(x))=\Lambda_{\varphi\otimes\varphi}(\partial(x))$.
By the product rule for $\partial$ we have

$\displaystyle\delta_{1}(\Lambda_{\varphi}(xy))$
$\displaystyle=\Lambda_{\varphi\otimes\varphi}((x\otimes
1)\partial(y)+\partial(x)(1\otimes y))$
$\displaystyle=\Lambda_{\varphi\otimes\varphi}((x\otimes
1)\Lambda_{\varphi\otimes\varphi}(\partial(y))+\partial(x))(1\otimes\sigma^{\varphi}_{-i/2}(y))$
$\displaystyle=(\pi_{l}(\Lambda_{\varphi}(x))\otimes
1)\delta_{1}(\Lambda_{\varphi}(y))+(1\otimes\pi_{r}(\Lambda_{\varphi}(y)))\delta_{1}(\Lambda_{\varphi}(x)).$

Thus, if $\delta$ is closable, the closure of the associated quadratic form is
a completely Dirichlet form with respect to $\mathfrak{A}_{\varphi}$ on the
GNS Hilbert space $L^{2}(M,\varphi)$.

To compare that to the result of Nelson, he showed [Nel17, Proposition 4.4]
that one gets a completely Dirichlet form on the GNS Hilbert space
$L^{2}(M_{\varphi},\varphi)$ of the _centralizer_ $M_{\varphi}$ of $\varphi$,
which is of course a tracial von Neumann algebra.

Our methods allow to extend this result to the “fully” non-tracial setting in
that we obtain a modular completely Dirichlet form on the GNS Hilbert space of
$M$ on not just of the centralizer. Note however that Nelson’s definition of
the map $\delta$ between $L^{2}$ spaces seems slightly different, owing to the
use of $M\overline{\otimes}M^{\mathrm{op}}$ instead of $M\overline{\otimes}M$.

The last example concerns group von Neumann algebras. The case of discrete
groups was treated in [CS03, Section 10.2], but to cover general locally
compact groups, possibly non-unimodular, one needs the theory for non-tracial
reference weights as developed here.

######  Example 5.5.

Let $G$ be a locally compact group with left Haar measure $\mu$ and modular
function $\Delta_{G}$. As discussed in [Tak03, Proposition VII.3.1], the space
$C_{c}(G)$ of compactly supported continuous function on $G$ with the $L^{2}$
inner product, the convolution product, the involution
$f^{\sharp}(g)=\Delta_{G}(g)^{-1}\overline{f(g^{-1})}$ and the complex one-
parameter group $U_{z}f(g)=\Delta_{G}(g)^{iz}f(g)$ forms a Tomita algebra. We
write $\lambda$ and $\rho$ for the associated left and right action of
$C_{c}(G)$ on $L^{2}(G)$ and $\mathfrak{A}_{G}$ for the associated full left
Hilbert algebra.

Let $\pi$ be a strongly continuous orthogonal representation of $G$ on the
real Hilbert space $H$. A continuous map $b\colon G\to H$ is called
$1$-cocycle if $b(gh)=b(g)+\pi(g)b(h)$ for all $g,h\in G$. We extend $\pi$ to
a unitary representation of $G$ on the complexification $H^{\mathbb{C}}$ of
$H$ and write $\xi\mapsto\bar{\xi}$ for the anti-unitary involution induced by
$H\subset H^{\mathbb{C}}$.

On $C_{c}(G;H^{\mathbb{C}})$ define a left and right action of $C_{c}(G)$ by

$\displaystyle(f\ast\xi)(g)$
$\displaystyle=\int_{G}f(h)\pi(h)\xi(h^{-1}g)\,d\mu(h)$ $\displaystyle(\xi\ast
f)(g)$ $\displaystyle=\int_{G}f(h^{-1}g)\xi(h)\,d\mu(h),$

an anti-unitary involution by
$(\mathscr{J}\xi)(g)=-\Delta_{G}(g)^{-1/2}\pi(g)\overline{\xi(g^{-1})}$ and a
complex one-parameter group by
$\mathscr{U}_{z}\xi(g)=\Delta_{G}(g)^{iz}\xi(g)$. One can check that
$C_{c}(G;H^{\mathbb{C}})$ with this operations is a Tomita bimodule over
$C_{c}(G)$.

Let

$\delta\colon C_{c}(G)\to C_{c}(G;H^{\mathbb{C}}),\,\delta(f)(g)=f(g)b(g).$

Using the cocycle property of $b$, one gets

$\displaystyle\delta(f_{1}\ast f_{2})(g)$
$\displaystyle=\int_{G}f_{1}(h)f_{2}(h^{-1}g)\,d\mu(h)\,b(g)$
$\displaystyle=\int_{G}f_{1}(h)f_{2}(h^{-1}g)(\pi(h)b(h^{-1}g)+b(h))\,d\mu(h)$
$\displaystyle=(f_{1}\ast\delta(f_{2}))(g)+(\delta(f_{1})\ast f_{2})(g).$

It is readily verified that $\delta$ also satisfies $\delta\circ
J=\mathscr{J}\circ\delta$ and $\delta\circ U_{z}=\mathscr{U}_{z}\circ\delta$
for all $z\in\mathbb{C}$. Hence $\delta$ is a symmetric derivation. As a
multiplication operator, it is clearly closable.

Therefore,

$\mathscr{E}\colon L^{2}(G,\mu)\to[0,\infty],\,\mathscr{E}(f)=\int_{G}\lvert
f(g)\rvert^{2}\lVert b(g)\rVert^{2}\,d\mu(g)$

is a modular completely Dirichlet form with respect to $\mathfrak{A}_{G}$. The
associated quantum dynamical semigroup on $L(G)$ is given by

$P_{t}\left(\int_{G}\hat{x}(g)\lambda(g)\,d\mu(g)\right)=\int_{G}e^{-t\lVert
b(g)\rVert^{2}}\hat{x}(g)\lambda(g)\,d\mu(g).$

In this case, complete positivity of $P_{t}$ also follows directly from
Schönberg’s theorem as $g\mapsto\lVert b(g)\rVert^{2}$ is a conditionally
negative definite function on $G$.

## References

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