# Unisolvence of unsymmetric random Kansa collocation by Gaussians and other
analytic RBF vanishing at infinity

Alvise Sommariva and Marco Vianello
University of Padova

(May 2024)

###### Abstract

We give a short proof of almost sure invertibility of unsymmetric random Kansa
collocation matrices by a class of analytic RBF vanishing at infinity, for the
Poisson equation with Dirichlet boundary conditions. Such a class includes
popular Positive Definite instances such as Gaussians, Generalized Inverse
MultiQuadrics and Matérn RBF. The proof works on general domains in any
dimension, with any distribution of boundary collocation points and any
continuous random distribution of internal collocation points.

2020 AMS subject classification: 65D12, 65N35. Keywords: Poisson equation,
unsymmetric Kansa collocation method, Radial Basis Functions, Gaussians,
Generalized Inverse MultiQuadrics, Matérn RBF, unisolvence.

##  1 Introduction

The unsymmetric Kansa method for meshless discretization of PDEs, in both the
square (standard collocation) and rectangular (overtesting) formulations, has
a long and successful history in applications; see, with no pretence of
exhaustivity, [4, 5, 6, 10, 12, 14, 19, 22] with the references therein.
Nevertheless, existence of sufficient conditions ensuring invertibility of
unsymmetric Kansa collocation matrices is still a substantially open problem,
as it has been for more than 20 years after the breakthrough paper by Hon and
Schaback [11], who showed that there exist point configurations leading to
singularity of the collocation matrices (though these are very special and
“rare”cases). The lack of clear well-posedness conditions for the discrete
problem has been considered one of the main drawbacks of unsymmetric Kansa
collocation, despite its manifest effectiveness in many applications.

In two recent papers, a new approach to prove unisolvence in the case of
random collocation has been proposed, for Thin-Plate Splines, MultiQuadrics
and Inverse MultiQuadrics. The key property is that each basis function is
analytic but presents some singularity either in the real (TPS,[8]) or in the
complex domain (MQ and IMQ, [3]), which is not a singularity for the other
basis elements. A similar approach has been used also for polynomial-free
random interpolation by general Polyharmonic Splines [1, 20] and Generalized
Multiquadrics [21]. However, it does not work for example in the case of
Gaussians, which are entire functions.

In this paper we study unisolvence of unsymmetric random Kansa collocation for
the Poisson equation by a class of analytic RBF with scaled radial function

$\phi_{\varepsilon}(r)=\phi(\varepsilon r)$

which is $C^{2}$ in $[0,+\infty)$ and analytic in $(0,+\infty)$, such that
each RBF and its Laplacian vanish at $\infty$ but the Laplacian does not
vanish at the RBF center. As we shall see, such a class includes popular
Positive Definite instances such as Gaussians, Generalized Inverse
MultiQuadrics and Matérn RBF. The scale $\varepsilon>0$ represents the so-
called _shape parameter_ associated with RBF and as known is relevant in order
to control the trade-off between conditioning and accuracy; cf. e.g. [10, 13].

We consider in particular the Poisson equation with Dirichlet boundary
conditions (cf. e.g. [9])

$\left\\{\begin{array}[]{l}\Delta u(P)=f(P)\;,\;P\in\Omega\;,\\\
u(P)=g(P)\;,\;P\in\partial\Omega\;,\end{array}\right.$ (1)

where $\Omega\subset\mathbb{R}^{d}$ is a bounded domain (connected open set),
$P=(x_{1},\dots,x_{d})$ and $\Delta=\partial^{2}/\partial
x_{1}^{2}+\dots+\partial^{2}/\partial x_{d}^{2}$ is the Laplacian. In [8],
where we studied Kansa discretization by Thin-Plate Splines in
$\mathbb{R}^{2}$, we assumed that the boundary is a curve possessing an
analytic parametrization. Differently here, as in [3] (which concerns
MultiQuadrics and Inverse MultiQuadrics), we do not make any restrictive
assumption on $\partial\Omega$, except for the usual ones that guarantee well-
posedness and regularity of the solution (cf. e.g. [9]).

We recall that unsymmetric Kansa collocation consists in seeking a function

$u_{N}(P)=\sum_{j=1}^{n}{c_{j}\,\phi_{j}(P)}+\sum_{k=1}^{m}{d_{k}\,\psi_{k}(P)}\;,\;\;N=n+m\;,$
(2)

where

$\phi_{j}(P)=\phi_{\varepsilon}(\|P-P_{j}\|_{2})\;,\;\;\\{P_{1},\dots,P_{n}\\}\subset\Omega\;,$
(3)
$\psi_{k}(P)=\phi_{\varepsilon}(\|P-Q_{k}\|_{2})\;,\;\;\\{Q_{1},\dots,Q_{m}\\}\subset\partial\Omega\;,$
(4)

such that

$\left\\{\begin{array}[]{l}\Delta u_{N}(P_{i})=f(P_{i})\;,\;i=1,\ldots,n\\\
u_{N}(Q_{h})=g(Q_{h})\;,\;h=1,\ldots,m\;.\end{array}\right.$ (5)

Kansa collocation can be rewritten in matrix form as

$\left(\begin{array}[]{cc}\Delta\Phi&\Delta\Psi\\\ \\\
\Phi&\Psi\end{array}\right)\left(\begin{array}[]{c}\mathbf{c}\\\ \\\
\mathbf{d}\end{array}\right)=\left(\begin{array}[]{c}\mathbf{f}\\\ \\\
\mathbf{g}\end{array}\right)$ (6)

where the $N\times N$ block matrix is

$K_{N}=K_{N}(\\{P_{i}\\},\\{Q_{h}\\})=\left(\begin{array}[]{cc}\Delta\Phi&\Delta\Psi\\\
\\\
\Phi&\Psi\end{array}\right)=\left(\begin{array}[]{cc}(\Delta\phi_{j}(P_{i}))&(\Delta\psi_{k}(P_{i}))\\\
\\\ (\phi_{j}(Q_{h}))&(\psi_{k}(Q_{h}))\end{array}\right)$

and $\textbf{f}=\\{f(P_{i})\\}$, $\textbf{g}=\\{g(Q_{h})\\}$, $1\leq i,j\leq
n$, $1\leq h,k\leq m$.

##  2 Unisolvence of random Kansa collocation

We can now state and prove our main result on unisolvence of random Kansa
collocation by a class of analytic RBF vanishing at infinity. Thus will be
followed by a Corollary on the application to Gaussians and other Positive
Definite RBF. In the sequel we shall denote by $H(D)$ the space of analytic
functions in the open connected set $D\subset\mathbb{R}^{d}$.

###### Theorem 1

Let $\phi:[0,+\infty)\to\mathbb{R}$ be a radial function such that:

  * $(i)$

$\phi\in C^{2}([0,+\infty))\cap H((0,+\infty))$, $\lim_{r\to\infty}\phi(r)=0$;

  * $(ii)$

$\ell(r)=\phi^{\prime\prime}(r)+\phi^{\prime}(r)/r$ is continuous at $r=0$,
$\ell(0)\neq 0$, $\lim_{r\to\infty}\ell(r)=0$;

  * $(iii)$

the RBF interpolation matrix $V_{m}=\phi_{\varepsilon}(\|Q_{h}-Q_{k}\|)$,
$1\leq h,k\leq m$, is nonsingular for every set of distinct points
$\\{Q_{1},\dots,Q_{m}\\}\subset\mathbb{R}^{d}$.

Moreover, $K_{n}$ be the Kansa collocation matrix by
$\phi_{\varepsilon}(r)=\phi(\varepsilon r)$ defined above for equation (1) in
$\Omega\subset\mathbb{R}^{d}$, $d\geq 2$, where $\\{Q_{h}\\}$ is any fixed set
of $m$ distinct points on $\partial\Omega$, and $\\{P_{i}\\}$ is a sequence of
i.i.d. (independent and identically distributed) random points in $\Omega$
with respect to any probability density $\sigma\in L^{1}_{+}(\Omega)$.

Then for every $m\geq 1$ and for every $n\geq 0$ the matrix $K_{n}$ is a.s.
(almost surely) nonsingular.

Proof. Let us fix $\varepsilon>0$ and define
$\phi_{A}(P)=\phi_{\varepsilon}(\|P-A\|_{2})$. Then we have
$\phi_{A}(B)=\phi_{B}(A)$ and $\phi_{A}(A)=\phi(0)$. Moreover (cf. [10,
Appendix D])

$\Delta\phi_{A}(P)=\varepsilon^{2}(\phi^{\prime\prime}(\varepsilon\|P-A\|_{2})+\phi^{\prime}(\varepsilon\|P-A\|_{2})/(\varepsilon\|P-A\|_{2}))=\varepsilon^{2}\ell(\varepsilon\|P-A\|_{2})$

so that $\Delta\phi_{A}(B)=\Delta\phi_{B}(A)$ and
$\Delta\phi_{A}(A)=\varepsilon^{2}\ell(0)$. Observe in addition that
$\phi_{A}(\cdot),\Delta\phi_{A}(\cdot)\in H(\mathbb{R}^{d}\setminus\\{A\\})$
by analiticity of $\phi$ and $\sqrt{\cdot}$ in $\mathbb{R}^{+}$, and by
$(i),(ii)$

$\lim_{P\to\infty}\phi_{A}(P)=0=\lim_{P\to\infty}\Delta\phi_{A}(P)\;.$ (7)

The proof proceeds by induction on $n$. The assertion is certainly true for
$n=0$ by $(iii)$, since $K_{0}=V_{m}$. Let us assume that the assertion is
true for a fixed $n\geq 0$ and define the matrix

$K(P)=\left(\begin{array}[]{ccc}\Delta\Phi&\Delta\Psi&(\Delta\vec{\phi}(P))^{t}\\\
\\\ \Phi&\Psi&(\vec{\psi}(P))^{t}\\\ \\\
\Delta\vec{\phi}(P)&\Delta\vec{\psi}(P)&\varepsilon^{2}\ell(0)\end{array}\right)$

where $\vec{\phi}(P)=(\phi_{1}(P),\dots,\phi_{n}(P))$ and
$\vec{\psi}(P)=(\psi_{1}(P),\dots,\psi_{m}(P))$. Observe that
$|\mbox{det}(K_{n+1})|=|\mbox{det}(K(P_{n+1}))|$, since
$\psi_{k}(P_{n+1})=\phi_{n+1}(Q_{k})$,
$\Delta\phi_{j}(P_{i})=\Delta\phi_{i}(P_{j})$, and thus $K(P_{n+1})$
corresponds to a permutation of rows and columns of $K_{n+1}$. Now in view of
(7) we have that

$\lim_{P\to\infty}K(P)=L=\left(\begin{array}[]{cc}K_{n}&\vec{0}^{\,t}\\\ \\\
\vec{0}&\varepsilon^{2}\ell(0)\end{array}\right)$

where the limit matrix $L$ is a.s. invertible, since
$\mbox{det}(L)=\varepsilon^{2}\ell(0)\mbox{det}(K_{n})$ and a.s.
$\mbox{det}(K_{n})\neq 0$ by inductive hypothesis while $\ell(0)\neq 0$ by
$(ii)$.

This means that the function

$F(P)=\mbox{det}(K(P))$

which is in $H(\mathbb{R}^{d}\setminus\\{P_{1},\dots,P_{n}\\})$, a.s. does not
vanish in a neighborhood of $\infty$ in $\mathbb{R}^{d}$, and hence is not
identically zero in the open connected set
$\mathbb{R}^{d}\setminus\\{P_{1},\dots,P_{n}\\}$.

In view of basic result in the theory of analytic functions, the zero set of
$F$ in the open connected set
$\mathbb{R}^{d}\setminus\\{P_{1},\dots,P_{n}\\}$, say $Z(F)$, is then a null
set, i.e. it has zero Lebesgue measure (cf. [15] for an elementary proof).
Consequently, the zero set of $F$ in $\Omega$, say $Z_{\Omega}(F)$ is also a
null set since

$Z_{\Omega}(F)=(Z_{\Omega}(F)\cap\\{P_{1},\dots,P_{n}\\})\cup(Z(F)\cap\Omega)$

where the first is a finite set and the second a subset of a null set, and
hence is a null set also for $\sigma(x)dx$, i.e. the probability that
$P_{n+1}$ falls in $Z_{\Omega}(F)$ is zero.

To conclude, considering the probability of the corresponding events

$\mbox{prob}\\{\mbox{det}(K_{n+1})=0\\}=\mbox{prob}\\{F(P_{n+1})=0\\}$
$=\mbox{prob}\\{F\equiv 0\\}+\mbox{prob}\\{F\not\equiv 0\;\&\;P_{n+1}\in
Z_{\Omega}(F)\\}=0+0=0\;,$

and the inductive step is completed. $\square$

###### Corollary 1

Under the assumptions of Theorem 1 on the boundary and the internal
collocation points, the Kansa collocation matrix for equation (1) by Gaussian,
Generalized Inverse MultiQuadric and Matérn RBF is almost surely nonsingular.

Proof. It is sufficient to check properties $(i)-(iii)$ of Theorem 1. For all
instances $(iii)$ holds since $\phi(\|x\|_{2})$ is a strictly Positive
Definite RBF and $V_{m}$ is a positive definite matrix (cf. e.g. [10]).

  * •

Gaussians: $\phi(r)=e^{-r^{2}}$ and $\ell(r)=4e^{-r^{2}}(r^{2}-1)$ are entire
functions which vanish at $+\infty$, $\ell(0)=4$;

  * •

Generalized Inverse MultiQuadrics: $\phi(r)=(1+r^{2})^{\beta}$, $\beta<0$, and
$\ell(r)=4\beta(1+r^{2})^{\beta-2}(1+\beta r^{2})$ are analytic in
$\mathbb{R}$ and vanish at $\pm\infty$, $\ell(0)=4\beta$; this class includes
the standard IMQ ($\beta=-1/2$) and the IQ (Inverse Quadratic, $\beta=-1$);

  * •

Matérn: $\phi(r)=\frac{2^{1-\nu}}{\Gamma(\nu)}\,r^{\nu}K_{\nu}(r)$, where
$K_{\nu}$ is the modified Bessel function of the second kind or Macdonald
function of order $\nu>0$ (cf. e.g. [17]). These radial functions has been
used for Kansa collocation e.g. in [16] with half-integer parameter
$\nu=k+1/2$ with $k\in\mathbb{N}$. Now, let us take in general
$\nu\in\mathbb{R}$, $\nu>1$. It is known that $K_{\nu}$ is analytic in
$\mathbb{C}\setminus(-\infty,0]$, and $K_{\nu}(r)\sim\sqrt{\pi/(2r)}\,e^{-r}$
as $r\to+\infty$ while $r^{\nu}K_{\nu}(r)\sim 2^{\nu-1}\Gamma(\nu)$ as $r\to
0^{+}$. Moreover $(r^{\nu}K_{\nu}(r))^{\prime}=-r^{\nu}K_{\nu-1}(r)$ so that
$\phi^{\prime}(r)=-\frac{2^{1-\nu}}{\Gamma(\nu)}\,r^{\nu}K_{\nu-1}(r)$ and
$\phi^{\prime\prime}(r)=-\frac{2^{1-\nu}}{\Gamma(\nu)}(r\cdot
r^{\nu-1}K_{\nu-1}(r))^{\prime}=-\frac{2^{1-\nu}}{\Gamma(\nu)}\,(r^{\nu-1}K_{\nu-1}(r)-r^{\nu}K_{\nu-2}(r))$,
from which one easily gets

$\ell(r)=-\frac{2^{1-\nu}}{\Gamma(\nu)}\,(2r^{\nu-1}K_{\nu-1}(r)-r^{\nu}K_{\nu-2}(r))\;.$

From the relations above we can finally compute
$\lim_{r\to+\infty}\phi(r)=0=\lim_{r\to+\infty}\ell(r)$ and $\lim_{r\to
0^{+}}\ell(r)=-\frac{\Gamma(\nu-1)}{\Gamma(\nu)}=\frac{1}{1-\nu}\neq 0\;$.
$\square$

To our knowledge, this is the first set of sufficient conditions for
unisolvence of unsymmetric Kansa collocation by Gaussian and Matérn RBF. In
the case of classical Inverse MultiQuadrics ($\beta=-1/2$) an alternative but
more complicated unisolvence proof for random collocation has been recently
provided in [3].

Acknowledgements.

Work partially supported by the DOR funds of the University of Padova, and by
the INdAM-GNCS 2024 Projects “Kernel and polynomial methods for approximation
and integration: theory and application software”.

This research has been accomplished within the RITA “Research ITalian network
on Approximation” and the SIMAI Activity Group ANA&A, and the UMI Group TAA
“Approximation Theory and Applications”.

## References

  * [1] L. Bos, A. Sommariva, M. Vianello, A note on polynomial-free unisolvence of polyharmonic splines at random points, Appl. Math. Lett., revision required, preprint at: https://www.math.unipd.it/~marcov/pdf/unisolvence.pdf.
  * [2] R. Cavoretto, A. De Rossi, Adaptive procedures for meshfree RBF unsymmetric and symmetric collocation methods, Appl. Math. Comput. 382 (2020), 125354.
  * [3] R. Cavoretto, A. De Rossi, F. Dell’Accio, A. Sommariva, M. Vianello, Nonsingularity of unsymmetric Kansa matrices: random collocation by MultiQuadrics and Inverse MultiQuadrics, preprint at: https://www.math.unipd.it/~marcov/pdf/KansaMQ.pdf.
  * [4] M. Chen, L. Ling, Extrinsic Meshless Collocation Methods for PDEs on Manifolds, SIAM J. Numer. Anal. 58 (2020), 988–1007.
  * [5] W. Chen, Z.-J. Fu, C.S. Chen, Different Formulations of the Kansa Method: Domain Discretization, in W. Chen et al., Recent Advances in Radial Basis Function Collocation Methods, SpringerBriefs in Applied Sciences and Technology, 2014.
  * [6] K.C. Cheung, L. Ling, R. Schaback, $H^{2}$-Convergence of Least-Squares Kernel Collocation Methods, SIAM J. Numer. Anal. 56 (2018), 614–633.
  * [7] F. Dell’Accio, A. Sommariva, M. Vianello, Random sampling and unisolvent interpolation by almost everywhere analytic functions, Appl. Math. Lett. 145 (2023), 108734.
  * [8] F. Dell’Accio, A. Sommariva, M. Vianello, Unisolvence of random Kansa collocation by Thin-Plate Splines for the Poisson equation, Eng. Anal. Bound. Elem., to appear.
  * [9] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, AMS, 1998.
  * [10] G.E. Fasshauer, Meshfree Approximation Methods with Matlab, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific, 2007.
  * [11] Y.C. Hon, R. Schaback, On unsymmetric collocation by radial basis functions, Appl. Math. Comput. 119 (2001), 177–-186.
  * [12] E.J. Kansa, Application of Hardy’s multiquadric interpolation to hydrodynamics, in Proc. 1986 Simul. Conf., Vol. 4, pp. 111–-117.
  * [13] E. Larsson, R. Schaback, Scaling of radial basis functions, IMA Journal of Numerical Analysis (2023), drad035.
  * [14] L. Ling, R. Opfer, R. Schaback, Results on meshless collocation techniques, Eng. Anal. Bound. Elem. 30 (2006), 247–253.
  * [15] B.S. Mityagin, The Zero Set of a Real Analytic Function, Math. Notes 107 (2020), 529–530.
  * [16] C.T. Mouat, R.K. Beatson, RBF Collocation, Report UCDMS2002/3, Dept. of Mathematics and Statistics, University of Canterbury, https://www.math.canterbury.ac.nz/research/ucdms2002n3.pdf.
  * [17] F.W.J. Olver, L.C. Maximon, NIST Digital Library of Mathematical Functions: Chapter 10 Bessel Functions, https://dlmf.nist.gov/10.
  * [18] R. Schaback, All well-posed problems have uniformly convergent and stable discretizations, Numer. Math. 132 (2016), 597–630
  * [19] R. Schaback, H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer. 15 (2006) 543–639.
  * [20] A. Sommariva, M. Vianello, Polynomial-free unisolvence of polyharmonic splines with odd exponent by random sampling, Dolomites Res. Notes. Approx. (DRNA), to appear.
  * [21] A. Sommariva, M. Vianello, Random sampling and polynomial-free interpolation by Generalized MultiQuadrics, preprint at: https://www.math.unipd.it/~marcov/pdf/GMQ.pdf.
  * [22] H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., vol. 17, Cambridge Univ. Press, Cambridge, 2005.


# ICIP 2022 Challenge on Parasitic Egg Detection and Classification in
Microscopic Images: Dataset, Methods and Results

Nantheera Anantrasirichai1, Thanarat H. Chalidabhongse2,
Duangdao Palasuwan3, Korranat Naruenatthanaset2, Thananop Kobchaisawat2,
Nuntiporn Nunthanasup3, Kanyarat Boonpeng3,
Xudong Ma1 and Alin Achim1111This work was supported by the Newton Fund
Institutional Links, British Council [grant number 623714323]; the National
Research Council of Thailand [grant number N31A640268]; the Chulalongkorn
University Technology Center’s Innovation Fund.
1Visual Information Laboratory, University of Bristol, UK,
2Department of Computer Engineering, Chulalongkorn University, Thailand,
3Oxidation in Red Cell Disorders Research Unit,
Department of Clinical Microscopy, Chulalongkorn University, Thailand

###### Abstract

Manual examination of faecal smear samples to identify the existence of
parasitic eggs is very time-consuming and can only be done by specialists.
Therefore, an automated system is required to tackle this problem since it can
relate to serious intestinal parasitic infections. This paper reviews the ICIP
2022 Challenge on parasitic egg detection and classification in microscopic
images. We describe a new dataset for this application, which is the largest
dataset of its kind. The methods used by participants in the challenge are
summarised and discussed along with their results.

##  1 Introduction

Intestinal parasitic infections remain among the leading causes of morbidity
worldwide, especially in tropical and sub-tropical areas with high temperate
climates. According to WHO, approximately 1.5 billion people, or 24% of the
world’s population, were infected with soil-transmitted helminth infections
(STH), and more than 800 million children worldwide required preventive
chemotherapy for STH in 2020222https://www.who.int/news-room/fact-
sheets/detail/soil-transmitted-helminth-infections. Most infections can cause
diarrhea and other symptoms, such as malnutrition and anaemia, particularly in
children, who may suffer from growth failure. Most infected persons can also
shed cysts or eggs in their living environment, and unwittingly cause
transmission of parasites to other individuals. An improvement in personal
hygiene, better sanitation and a widespread health education campaign have
reduced helminthic infections, but protozoa are still present even in
asymptomatic hosts leading to chronic diseases. In developing countries,
intestinal protozoa and STH have been recognised as one of the most
significant causes of illness. Diagnosis of intestinal parasites is usually
based on direct examination in the laboratory. Unfortunately, this time-
consuming method (30 min/sample) shows low sensitivity and requires an
experienced and skilled medical laboratory technologist, being thus
impractical for on-site use. This means an automate routine faecal examination
for parasitic diseases is essential.

Table 1: Size and resolution ranges of 11 parasitic egg types in our dataset Type | size ($\mu$m) | width (pixels)
---|---|---
Ascaris lumbricoides | 60$\times$85 | 131-439
Capillaria philippinensis | 20-22$\times$36-45 | 56-234
Enterobius vermicularis | 20-30$\times$50-60 | 76-272
Fasciolopsis buski | 80-85$\times$130-140 | 170-806
Hookworm egg | 36-40$\times$64-76 | 114-410
Hymenolepis diminuta | 60-80 | 132-461
Hymenolepis nana | 30-47 | 95-300
Opisthorchis viverrine | 11-12$\times$22-32 | 41-186
Paragonimus spp | 77-80 | 116-477
Taenia spp. egg | 30-35 | 85-244
Trichuris trichiura | 22-23$\times$50-54 | 76-395

This challenge aims to encourage and highlight novel strategies with a focus
on robustness and accuracy in data-driven technologies to automatically detect
parasitic eggs and also identify the egg type in compound microscopy images.
We provide a new large dataset containing 11 parasitic egg types. The dataset
is well-balanced between classes, with varying sizes from very small,
22$\times$11 $\mu$m (Opisthorchis viverrine), to very large, 140$\times$85
$\mu$m (Fasciolopsis buski), as shown in Table 1. Our dataset exhibits real
and synthetic degradation due to out-of-focus, noise, motion blur, colour, and
light imbalance. A wide variety of image quality and appearances create a
large range of different image characteristics. This aims to enhance the
robustness of the detection models.

The challenge has gathered experts in the fields of image processing, medical
imaging and computer vision, which should not be limited to the domain of
microscopic imaging as knowledge transfer can benefit the growth of the
community. Our dataset has been hitherto downloaded from more than 20
countries in 4 continents. The outcome of the challenge could further improve
and assist diagnosis in real clinical use, and even automate detection and
identification of intestinal parasite eggs, which can be used by non-experts.

##  2 Related work

Existing datasets: All available microscopic datasets for parasitic egg
detection and classification are generally small in both numbers of categories
and images. For example, the Parasitology Laboratory of Kansas State
University provides 5 types of parasitic eggs with only a total of 33 images
(https://www.k-state.edu/parasitology/). The Japan Advanced Institute of
Science and Technology offers 8 types with 213 images in total [1]. A 4-class
dataset of 162 images captured with 10$\times$ magnification is reported in
[2]. The larger dataset in [3] containing 3,327 1000$\times$ magnified
microscopic images has only three parasitic egg types.

Existing detection and classification methods: Traditional methods of parasite
egg detection involve pre-processing, feature extraction and classification
with support vector machine (SVM) [4, 5] or artificial neural networks (ANN)
[6, 4]. The performances are however limited because the methods cannot deal
with complex datasets. Deep learning has recently shown its ability and high
efficiency to tackle problems in object detection [7]. The state-of-the-art
methods of parasite egg detection and classification are therefore based on
this technology, especially the convolutional neural networks (CNNs). The
early work employs seven convolutional layers trained from scratch [3]. Later,
a transfer learning strategy using a pretrained AlexNet [8] and a pretrained
ResNet50 [9] was employed to classify parasitic egg types, and the positions
of the eggs were identified using sliding window technique [2]. Some methods
follow semantic segmentation using UNet architecture [10] or a fully
convolutional network (FCN) [11]. These techniques however require full pixel-
wise annotation. More modern object detection approaches based on a Faster-
RCNN [12] are employed in [1] and [13], achieving promising results.

##  3 Dataset for the Challenge

Our dataset, called Chula-ParasiteEgg-11, contains 11 types of parasitic eggs
from faecal smear samples. With a total of 13,750 microscopic images, it has
become the largest dataset of its kind. In this challenge, the training and
testing datasets are randomly divided resulting 1,000 images/class for
training and approximately 250 images/class for testing. Some testing images
contain two or more types of parasitic eggs and some testing images do not
have any eggs. The labels take the form of bounding boxes. The visual quality
of some images is degraded to replicate the real scenario, in which the images
are acquired with speed or automated system in different lighting conditions.

Samples from the dataset are shown in Fig. 1. Their average size ranges
approximately in the interval of 15-140 $\mu$m. The details of the size and
diameter of each parasitic egg type are listed in Table 1. Training and
testing datasets are available on the IEEE Dataport [14].

Parasite specimens and acquisition: The stool samples of parasitic eggs used
for the experiments in this study were prepared in laboratory conditions at
Department of Clinical Microscopy, Faculty of Allied Health Sciences,
Chulalongkorn University. The specimens were fixed with 10% formalin prior to
storing at 4°C. Stool samples were examined microscopically by direct simple
smear. After the samples were prepared, they were immediately examined
visually using a light microscope with 40$\times$ magnification. Three
researchers checked the egg types and the majority vote was used as the label.
The bounding boxes indicating the position of the egg were the average amongst
the majority.

A total of 1,250 images of each parasitic egg type was acquired from several
different devices, including Canon EOS 70D camera body with Olympus BX53
microscopes, DS-Fi2 Nikon camera body with Nikon Eclipse Ni microscopes,
Samsung Galaxy J7 Prime phone, iPhone 12 and 13 with either 10$\times$ times
eyepieces lens of Nikon Eclipse Ni or Olympus BX53 devices. These result in
different resolutions, different lighting and setting conditions.

Image degradation: The visual quality of the microscopic images captured above
are degraded so that i) the detection competition is more challenging, ii) the
robustness of the detection models is enhanced, and iii) the possible images
captured by motorised stage microscope are replicated. The quality degradation
procedure is as follows.

  * •

Crop each side by a random value from 0 to 30%,

  * •

Apply Gaussian blur with a standard deviation between 0.0 - 3.0,

  * •

Apply motion blur (10% of total images) with a random kernel size of 3- 35
pixels and orientation of 0-360 degree,

  * •

Apply Gaussian noise with a standard deviation between 0.0 - 25.5 (to each
colour channel separately),

  * •

Apply Poisson noise with a lambda (the expected rate of occurrences) between
0.0 - 5.0,

  * •

Adjust image saturation by adding a value in a range of -25 - +25 to the S
channel of the HSV colour space,

  * •

Adjust contrast using gamma correction with a gamma value between 0.5 - 2.0.

Figure 1: Examples of 11 parasitic egg images after quality degradation. Note
that these cropped images are not true scaling as for better visualisation.
The last row shows variation of the background and debris.

After the challenge competition, we also made the original-quality images
available for research in fields such as image restoration, image enhancement
or super-resolution.

##  4 Methods

All methods in this competition exploited deep learning techniques. Most of
them were developed based on the state of the art in object detection,
including YOLOv5 [15, 16], Fast-RCNN [13, 17], EfficientDet [18], Cascade
R-CNN [15, 19, 20], CBNetV2 [21, 22], CenterNet2 [23], Task-aligned One-stage
object Detection (TOOD) [24, 17], and RetinaNet [25, 26]. These methods are
convolutional neural networks with various backbone architectures, where the
most popular architecture is based on ResNet blocks [24, 17, 23, 22, 27, 25,
28]. Most methods employed Feature Pyramid Network (FPN) to extract multi-
scale features [24, 18, 25, 22], whilst some methods employed Transformer-
based architectures as feature extractors [19, 29, 30, 20, 26]. The models
were pretrained with COCO dataset [24, 23, 21, 22, 27, 29, 20, 30, 25], or
ImageNet-1K [19]. These methods have proved that using pre-trained weights
significantly improves the detection performance. The final models of some
teams were fused from multiple models obtained by cross validation and fine-
tuning parameters [21, 17, 27, 24]. Further improvement has been achieved with
small object detection mechanisms [17, 22] and pseudo labelling strategy [21,
24].

The top-performance frameworks were developed with CBNetV2, YOLOv5, Cascade
R-CNN, and FPNs. Multiple models were fused and pseudo labelling was applied.
The results shown in Fig. 2 and Table 2 were reproduced directly from their
submitted models without retraining with the hidden data.

##  5 Results

###  5.1 Evaluation criteria

All training images were released along with 2,200 unknown-labelled testing
images. The participants uploaded their results of the testing dataset to the
website, where a Mean Intersection-over-Union (mIoU) was the main criteria for
evaluation. Each type of parasitic eggs was assessed and assigned an
Intersection-over-Union (IoU) score, then the average of their IoU scores of
all parasitic egg types were calculated, finally resulting in the mIoU score.
Submissions were ranked according to the criteria and shown on the
leaderboard. The top-performing teams were asked to submit their code and
models for final checking and tested on 550 hidden testing images for the
final validation. We also computed the F1-score (IoU$\geq$0.5) for each egg
type to evaluate the performance using the precision and recall. The
mAP@[0.5:0.95] shows the average of mAP (mean Average Precision) over
different IoU thresholds, ranging from 0.5 to 0.95, step 0.05. If a further
comparison is needed for tie-breaking, the computational speed will be used as
the last criterion.

Figure 2: (Top) Combined F1-score and mIoU values. (Bottom) Combined F1-score and mAP values. Blue circles are the results from the 2200-test set we gave to participants. Red triangles are the results from the 550-hidden set. Table 2: Average top-5 detection results for each parasitic egg type (FNR=false negative rate) Type | mIoU | F1score | Precision | Recall | FNR(%)
---|---|---|---|---|---
A.L. | 0.82 | 0.90 | 0.94 | 0.87 | 11.96
C.P. | 0.77 | 0.90 | 0.86 | 0.95 | 3.94
E.V. | 0.81 | 0.92 | 0.89 | 0.96 | 3.25
F.B. | 0.89 | 0.97 | 0.97 | 0.98 | 2.26
H.E. | 0.92 | 0.99 | 0.98 | 0.99 | 0.61
H.D. | 0.82 | 0.91 | 0.85 | 1.00 | 0.25
H.N. | 0.83 | 0.93 | 0.94 | 0.92 | 7.30
O.V. | 0.81 | 0.93 | 0.89 | 0.97 | 2.50
P.S. | 0.80 | 0.90 | 0.86 | 0.97 | 2.27
T.S. | 0.93 | 0.98 | 0.97 | 1.00 | 0.25
T.T. | 0.90 | 0.97 | 0.98 | 0.96 | 4.25

###  5.2 Result comparison

There were 82 teams registered but only 30 teams submitted their results. Fig.
2 shows the mIoU vs. F1-score and the mAP vs. F1-score of each team. The
winner achieved a mIoU of 0.942 and F1-score of 0.995 on the hidden data. The
two runner-ups achieved close performance, where the mIoUs are 0.932 and
0.930, and F1-score are 0.989 and 0.987, respectively. The mAPs of these top
three are 0.925, 0.905 and 0.896. However, the performances of individual
class detection are quite different, particularly on the Capillaria
philippinensis eggs, where the first and the second teams achieve a mIoU of
0.86 and 0.69, respectively. Fig. 2 bottom reveals that some teams achieve
better mAPs but worse F1-scores than other teams. This is because they achieve
high precision and low recall, implying high fault negatives (miss detection).
Testing with the hidden data, the average performances of the top teams in
term of mIoU, F1-score and mAP dropped by 0.82%, 0.02% and 1.88%,
respectively, compared to testing with the disclosed test data.

Table 2 shows the average performance of the top-five methods on detecting
each type of parasitic eggs. The Capillaria philippinensis egg is the most
difficult to detect – lowest values of mIoU and F1-score. The detected boxes
are usually larger than the actual ground truth. The large eggs, i.e. Ascaris
lumbricoides and Hymenolepis nana, are missed easier than other types (high
false negative rate (FNR or miss rate)). This occurs when the high level of
blur deteriorates detail and texture inside the eggs, leading to
misclassification between them. The Taenia spp. eggs are the easiest to
detect, even though their size is not large. This is because their shape and
texture are different from debris in the background.

##  6 Conclusions

This paper reviews the ICIP 2022 Challenge on parasitic egg detection and
classification in microscopic images on the data and the methods used. A new
dataset comprising 11 parasitic egg types was captured with several
microscopes and lenses, generating a total of 13,750 images. The quality of
each image was randomly degraded with noise, Gaussian blur, motion blur and
different levels of brightness, and colour adjustment. All methods used in
this competition exploited deep learning technologies, where the architectures
were based on state of the art object detection methods. Through this
challenge, the performance of parasitic egg detection and classification
reaches 0.942, 0.995 and 0.925 in terms of mIoU, F1-score and mAP,
respectively.

## References

  * [1] N. Viet, D. ThanhTuyen, and T. Hoang,  “Parasite worm egg automatic detection in microscopy stool image based on Faster R-CNN,”  Proceedings of the 3rd International Conference on Machine Learning and Soft Computing - ICMLSC, pp. 197–202, 2019.
  * [2] T. Suwannaphong, S. Chavana, S. Tongsom, D. Palasuwan, T. H Chalidabhongse, and N. Anantrasirichai,  “Parasitic egg detection and classification in low-cost microscopic images using transfer learning,”  arXiv preprint arXiv:2107.00968, 2021.
  * [3] J.A. Quinn, R. Nakasi, P.K. Mugagga, P. Byanyima, W. Lubega, and A. Andama,  “Deep convolutional neural networks for microscopy-based point of care diagnostics,”  Machine Learning for Health Care Conf, vol. 56, pp. 271–281, 2016\.
  * [4] K. Ray, S. Shil, S. Saharia, N. Sarma, and N. Karabasanavar,  “Detection and identification of parasite eggs from microscopic images of fecal samples,”  Computational Intelligence in Pattern Recognition, pp. 45–55, 2020\.
  * [5] D. Avci and A. Varol,  “An expert diagnosis system for classification of human parasite eggs based on multi-class svm,”  Expert Systems with Applications, vol. 36, pp. 43–48, 2009.
  * [6] Y. Yang, D. Park, H. Kim, M. Choi, and J. Chai,  “Automatic identification of human helminth eggs on microscopic fecal specimens using digital image processing and an artificial neural network,”  IEEE Transactions on Biomedical Engineering, vol. 48, pp. 718–730, 2001.
  * [7] N. Anantrasirichai and D. Bull,  “Artificial intelligence in the creative industries: A review,”  Artificial intelligence review, 2021.
  * [8] A. Krizhevsky, I. Sutskever, and G. Hinton,  “Imagenet classification with deep convolutional neural networks,”  Communications of the ACM, vol. 60, pp. 84–90, 2017.
  * [9] K. He, X. Zhang, S.Ren., and J. Sun,  “Deep residual learning for image recognition,”  Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
  * [10] Y. Li, R. Zheng, Y. Wu, K. Chu, Q. Xu, M. Sun, and Z. J. Smith,  “A low-cost, automated parasite diagnostic system via a portable, robotic microscope and deep learning,”  Journal of biophotonics, vol. 12, no. 9, pp. e201800410, 2019.
  * [11] P. Najgebauer, R. Grycuk, L. Rutkowski, R. Scherer, and A. Siwocha,  “Microscopic sample segmentation by fully convolutional network for parasite detection,”  in Artificial Intelligence and Soft Computing, Cham, 2019, pp. 164–171, Springer International Publishing.
  * [12] S. Ren, K. He, R. Girshick, and J. Sun,  “Faster R-CNN: Towards real-time object detection with region proposal networks,”  in Advances in Neural Information Processing Systems, 2015, vol. 28.
  * [13] P. Mayo, N. Anantrasirichai, T. H Chalidabhongse, D. Palasuwan, and A. Achim,  “Detection of parasitic eggs from microscopy images and the emergence of a new dataset,”  arXiv preprint arXiv:2203.02940, 2022.
  * [14] D. Palasuwan, K. Naruenatthanaset, T. Kobchaisawat, T H Chalidabhongse, Nunthanasup N, K. Boonpeng, and N. Anantrasirichai,  “Parasitic egg detection and classification in microscopic images,” 2022\.
  * [15] Y. Wang, Z. He, S. Huang, and H. Du,  “Parasitic egg detection and classification via multi-technique,”  submitting to ICIP2022.
  * [16] Yohanssen Pratama,  “Parasitic egg detection and classification by utilizing yolo algorithm with deep latent space image restoration and grabcut augmentation,”  submitting to ICIP2022.
  * [17] A. Turec̆ková,  “Report to ICIP challenge on parasitic egg detection and classification in microscopic images,”  2022\.
  * [18] N. AlDahoul, H. A. Karim, S. Limson Kee, and M. J. T. Tan,  “Localization and classification of parasitic eggs in microscpic images using an efficientdet detector,”  in IEEE International Conference on Image Processing, 2022.
  * [19] Y. Pang and W. Sun,  “Report for parasitic egg detection and classification in microscopic images,”  2022\.
  * [20] X. Lu, J. Tang, and Y. Huang,  “ambilight team,”  2022\.
  * [21] M. Li and S. Liu,  “NEGU team,”  2022\.
  * [22] S. Muralidhara and N. Hegde,  “Improved parasitic egg detection and classification in low-cost microscopic images using super resolution pre-processing and CBNetV2,”  submitting to ICIP2022.
  * [23] K. Tango, T. Ohkawa, R. Furuta, and Y. Sato,  “Background mixup data augmentation for hand and object-in-contact detection,”  in arXiv:2202.13941v2, submitting to ICIP2022.
  * [24] Z. H. Aung, K. Srithaworn, and T. Achakulvisut,  “Multitask learning via pseudo-label generation and ensemble prediction for parasitic egg cell detection: IEEE ICIP Challenge 2022,”  in IEEE International Conference on Image Processing, 2022.
  * [25] K. Pho, H. Lam, T. Le, and A. Yoshitaka,  “Attention-driven retinanet for parasitic egg detection,”  submitting to ICIP2022.
  * [26] N. Hongcharoen,  “Palmy team,”  2022\.
  * [27] J. Ruiz-Santaquiteria, A. Pedraza, N. Vallez, and A. Velasco,  “Parasitic egg detection with a deep learning ensemble,”  submitting to ICIP2022.
  * [28] N. Bandara, S. Chandanayake, D. Premathilake, and C. Ekenayake1,  “Rethinking object detection in terms of classification and localization through parallel deep learning models,”  2022\.
  * [29] A. Pedraza, J. Ruiz-Santaquiteria, O. Deniz, and G. Bueno,  “Parasitic egg detection and classification with transformer-based architectures,”  submitting to ICIP2022.
  * [30] F-J Tsai, T-J Hsu, C-H Chang, and P-H Shao,  “Report to ICIP challenge on parasitic egg detection and classification in microscopic images,”  2022\.


$J\pi_{r}(\mathfrak{A}_{\mathcal{E}})J=\pi_{l}(\mathfrak{A}_{\mathcal{E}})$.
Moreover,

$\mathcal{J}\tilde{L}(\pi_{l}(a))\xi=\mathcal{J}L(a)\xi=R(Ja)\mathcal{J}\xi=\tilde{R}(J\pi_{r}(Ja)^{\ast}J)\mathcal{J}\xi=\tilde{R}(\pi_{l}(a)^{\ast})\mathcal{J}\xi$

for all $\xi\in\mathcal{H}$. Thus
$\tilde{R}(x)=\mathcal{J}\tilde{L}(x)^{\ast}\mathcal{J}$ for all
$x\in\pi_{l}(\mathfrak{A}_{\mathcal{E}})$. From this it is easy to see that
$\tilde{L}$ extends to a normal representation if and only if $\tilde{R}$
does.

(ii)$\implies$(iii), (iii)$\implies$(iv) are clear from the definitions.

(iv)$\implies$(iii): By polarization, $\Gamma(a,b)$ is $\sigma$-weakly
continuous for all $a,b\in\mathfrak{A}_{\mathcal{E}}$. If $\xi=R(b)\delta(a)$,
$\eta=R(d)\delta(c)$ with $a,b,c,d\in\mathfrak{A}_{\mathcal{E}}$, then

$\displaystyle\langle\hat{\Gamma}(\xi,\eta),J\pi_{r}(a)^{\ast}J\rangle_{A^{\ast},A}$
$\displaystyle=\langle R(b)\delta(a),R(a)R(d)\delta(c)\rangle$
$\displaystyle=\langle\delta(a),R(b^{\flat}ad)\delta(c)\rangle$
$\displaystyle=\langle\Gamma(a,c),J\pi_{r}(b^{\flat}ad)^{\ast}J\rangle_{A^{\ast},A}.$

Thus $\hat{\Gamma}(\xi,\eta)$ is $\sigma$-weakly continuous. By
sesquilinearity, this extends to $\xi,\eta$ in $\mathcal{H}$. For arbitrary
$\xi,\eta\in\overline{\mathcal{H}}$ take sequences $(\xi_{k})$, $(\eta_{k})$
in $\mathcal{H}$ such that $\xi_{k}\to\xi$, $\eta_{k}\to\eta$. Since

$\lVert\hat{\Gamma}(\xi,\eta)-\hat{\Gamma}(\xi_{k},\eta_{k})\rVert\leq\lVert\xi-\xi_{k}\rVert\lVert\eta_{k}\rVert+\lVert\eta_{k}-\eta\rVert\lVert\xi\rVert$

and the norm limit of $\sigma$-weakly continuous functionals is
$\sigma$-weakly continuous, we obtain that $\hat{\Gamma}(\xi,\eta)$ is
$\sigma$-weakly continuous.

(iii)$\implies$(ii): For $\xi,\eta\in\overline{\mathcal{H}}$ let
$\omega_{\xi,\eta}$ denote the $\sigma$-weakly continuous extension of
$\hat{\Gamma}(\xi,\eta)$ to $\mathcal{M}^{\mathrm{op}}$. By the Kaplansky
density theorem,
$\lVert\omega_{\xi,\eta}\rVert=\lVert\hat{\Gamma}(\xi,\eta)\rVert\leq\lVert\xi\rVert\lVert\eta\rVert$.
Thus for every $x\in\mathcal{M}^{\mathrm{op}}$ there exists $\tilde{R}(x)\in
B(\overline{\mathcal{H}})$ with $\lVert\tilde{R}(x)\rVert\leq\lVert x\rVert$
such that $\omega_{\xi,\eta}(x)=\langle\xi,\tilde{R}(x)\eta\rangle$. Clearly
$\tilde{R}$ is linear, weakly continuous on the unit ball and extends the map
from (ii). Since the restriction of $\tilde{R}$ on
$J\pi_{r}(\mathfrak{A}_{\mathcal{E}})J$ is a $\ast$-homomorphism (for the
multiplication in $\mathcal{M}^{\mathrm{op}}$) and it is weakly continuous, it
follows that $\tilde{R}$ is a $\ast$-homomorphism on
$\mathcal{M}^{\mathrm{op}}$ by a density argument. As $R$ is non-degenerate,
$\tilde{R}$ is unital. Thus $\tilde{R}$ is a normal representation of
$\mathcal{M}^{\mathrm{op}}$ on $\overline{\mathcal{H}}$. ∎

######  Remark 7.3.

According to Remark 7.1, if $a\in\mathfrak{A}_{\mathcal{E}}$, then $\Gamma(a)$
can be expressed in terms of the form $\mathcal{E}$ without reference to
$\mathcal{H}$ and $\delta$. Thus (iv) in the previous theorem gives a
criterion for the normality of $\mathcal{H}$ that can be checked without the
need for an explicit construction of $\mathcal{H}$ and $\delta$.

######  Definition 7.4.

We say the quantum Dirichlet form $\mathcal{E}$ or the associated quantum
Markov semigroup $(P_{t})$ is _$\Gamma$ -regular_ if one of the equivalent
conditions from Theorem 7.2 holds.

######  Remark 7.5.

$\Gamma$-regularity also has a probabilistic counterpart in terms of Markov
dilations [JRS], but we will not go into details here.

As a consequence of the construction from Subsection 4.2 we obtain that the
generator of a $\Gamma$-regular GNS-symmetric quantum Dirichlet form is of the
form $\delta^{\ast}\bar{\delta}$, where $\delta$ is a twisted derivation with
values in a noncommutative $L_{2}$ space.

######  Corollary 7.6.

If $\mathcal{E}$ is a $\Gamma$-regular GNS-symmetric quantum Dirichlet form
with generator $\mathcal{L}^{(2)}$, then there exists a von Neumann algebra
$\hat{\mathcal{M}}\supset\mathcal{M}$, a weight $\hat{\varphi}$ on
$\hat{\mathcal{M}}$, a faithful normal conditional expectation
$E\colon\hat{\mathcal{M}}\to\mathcal{M}$ such that $\varphi\circ
E=\hat{\varphi}$, and a closable linear map
$\delta\colon\mathfrak{A}_{\mathcal{E}}\to
L_{2}(\hat{\mathcal{M}},\hat{\varphi})$ such that

$\delta(\Lambda_{\varphi}(xy))=x\delta(\Lambda_{\varphi}(y))+\delta(\Lambda_{\varphi}(x))\sigma^{\varphi}_{i/2}(y)$

for all $x,y\in\Lambda_{\varphi}^{-1}(\mathfrak{A}_{\mathcal{E}})$ and

$\mathcal{L}^{(2)}=\delta^{\ast}\bar{\delta}.$

######  Remark 7.7.

The twisted product rule in the previous corollary can also be expressed in
terms of the embeddings $\Delta_{\varphi}^{\alpha}\Lambda_{\varphi}$ for
$\alpha\in[0,\frac{1}{2}]$ as follows:

$\delta(\Delta_{\varphi}^{\alpha}\Lambda_{\varphi}(xy))=\sigma^{\varphi}_{-i\alpha}(x)\delta(\Delta_{\varphi}^{\alpha}\Lambda_{\varphi}(y))+\delta(\Delta_{\varphi}^{\alpha}\Lambda_{\varphi}(x))\sigma^{\varphi}_{i(1/2-\alpha)}(y).$

######  Remark 7.8.

In the case when $\varphi$ is a trace, the existence of a derivation as in
Subsection 4.2 for $\Gamma$-regular quantum Dirichlet forms was established in
[JRS]. The triple $(\mathfrak{A}_{\mathcal{E}},\hat{\mathcal{M}},\delta)$ is
called a _derivation triple_ in this setting [BGJ22, LJL20].

##  8\. Examples

In this final section we give examples of quantum Dirichlet forms for which we
can describe the associated Tomita bimodule and derivation more explicitly. We
start by discussing how tracially symmetric quantum Markov semigroups and
quantum Markov semigroups on finite-dimensional von Neumann algebras fit into
the picture in the first two subsections, before we treat new examples with
depolarizing semigroups, the Ornstein–Uhlenbeck semigroup on free Araki–Woods
factors and translation-invariant quantum Markov semigroups on compact quantum
groups in the last three subsections.

###  8.1. Tracially symmetric quantum Markov semigroups

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and $\tau$ a normal
semi-finite faithful trace on $\mathcal{M}$. In this case the modular group
$\sigma^{\tau}$ is trivial and the involutions ♯ and ♭ on
$\mathfrak{A}_{\tau}$ coincide with $J_{\tau}$.

If $\mathcal{E}$ is a quantum Dirichlet form on $L_{2}(\mathcal{M},\tau)$,
then it is automatically GNS-symmetric, and
$\mathfrak{A}_{\mathcal{E}}=D(\mathcal{E})\cap\Lambda_{\tau}(\mathfrak{n}_{\tau})$.
Let $\mathcal{H}$ be the associated Tomita bimodule. Since

$\mathcal{U}_{t}(\delta(a)b)=\delta(\sigma^{\tau}_{t}(a))\sigma^{\tau}_{t}(b)=\delta(a)b$

and elements of the form $\delta(a)b$ with $a,b\in\mathfrak{A}_{\mathcal{E}}$
linearly span $\mathcal{H}$, the group $(\mathcal{U}_{t})$ is trivial.

In other words, we have a Hilbert bimodule $\overline{\mathcal{H}}$ over
$\mathfrak{A}_{\mathcal{E}}$ with contractive and non-degenerate left and
right action, an anti-unitary involution $\mathcal{J}$ on
$\overline{\mathcal{H}}$ that intertwines the left and right action and a
derivation $\delta\colon\mathfrak{A}_{\mathcal{E}}\to\overline{\mathcal{H}}$
such that $\mathcal{J}\circ\delta=\delta\circ J$ and

$\mathcal{E}(a,b)=\langle\delta(a),\delta(b)\rangle$

for all $a,b\in\mathfrak{A}_{\mathcal{E}}$. This is the same data as one gets
from the construction in [CS03a] for tracially symmetric quantum Markov
semigroups.

###  8.2. QMS on finite-dimensional von Neumann algebras

Let $\mathcal{M}\subset M_{n}(\mathbb{C})$ be a unital ∗-subalgebra,
necessarily weakly closed, and $h\in\mathcal{M}_{+}$ with
$\operatorname{tr}(h)=1$. Write $\varphi$ for the state induced by $h$, that
is, $\varphi(x)=\operatorname{tr}(xh)$. We assume that $\varphi$ is faithful,
or, equivalently, that $h$ is invertible.

By Alicki’s theorem (see [Ali76, Theorem 3], [CM17, Theorem 3.1] for the case
of matrix algebras and [Wir22, Corollary 5.4] for the extension to arbitrary
finite-dimensional von Neumann algebras) a continuous semigroup $(P_{t})$ on
$\mathcal{M}$ is a $\varphi$-symmetric QMS if and only if there exist
$\omega_{1},\dots,\omega_{m}\in\mathbb{R}$ and $v_{1},\dots,v_{m}\in
M_{n}(\mathbb{C})$ satisfying

  * •

$\operatorname{tr}(v_{j})=0$ for $1\leq j\leq m$,

  * •

$\operatorname{tr}(v_{j}^{\ast}v_{k})=\delta_{jk}\operatorname{tr}(v_{j}^{\ast}v_{j})$
for $1\leq j,k\leq m$,

  * •

$\\{v_{j}\mid 1\leq j\leq m\\}=\\{v_{j}^{\ast}\mid 1\leq j\leq m\\}$,

  * •

$hv_{j}h^{-1}=e^{-\omega_{j}}v_{j}$ for $1\leq j\leq m$

such that the generator $\mathcal{L}$ of $(P_{t})$ is given by

$\mathcal{L}=\sum_{j=1}^{m}(e^{-\omega_{j}/2}v_{j}^{\ast}[v_{j},\cdot\,]+e^{\omega_{j}/2}[\,\cdot\,,v_{j}]v_{j}^{\ast}).$

Note that in general the matrices $v_{j}$ cannot be chosen in $\mathcal{M}$.

In this case, the differential structure from Theorem 6.8 coincides with the
differential structure studied in [CM17,Wir22]. Let us first fix some
notation. The space $L_{2}(\mathcal{M},\varphi)$ can be identified with
$\mathcal{M}$ with the inner product $\langle
x,y\rangle_{h}=\operatorname{tr}(x^{\ast}yh)$. Clearly, this inner product can
be extended to $M_{n}(\mathbb{C})$, and we write $H$ for the resulting Hilbert
space.

With these identifications, the Tomita algebra $\mathfrak{A}_{\varphi}$ is
$L_{2}(\mathcal{M},\varphi)$ with the product from $\mathcal{M}$, the
involution $a^{\sharp}=a^{\ast}$ and $U_{z}a=h^{iz}ah^{-iz}$. Since the
quantum Dirichlet form $\mathcal{E}$ associated with $(P_{t})$ is defined
everywhere, we have $\mathfrak{A}_{\mathcal{E}}=\mathfrak{A}_{\varphi}$.

Let $\mathcal{H}$ be the linear span of $\\{([v_{j},a]b)_{j}\mid a,b\in
L_{2}(\mathcal{M},\varphi)\\}$ inside $H^{\oplus m}$. If we endow $H^{\oplus
m}$ with the componentwise left and right actions of
$L_{2}(\mathcal{M},\varphi)$, then $\mathcal{H}$ is a sub-bimodule: If
$a,b,c\in L_{2}(\mathcal{M},\varphi)$, then $([v_{j},a]bc)_{j}\in\mathcal{H}$
and

$(a[v_{j},b]c)_{j}=([v_{j},ab]c-[v_{j},a]bc)_{j}\in\mathcal{H}.$

The space $\mathcal{H}$ becomes a Tomita bimodule over
$L_{2}(\mathcal{M},\varphi)$ with

$\displaystyle\mathcal{U}_{z}\xi$
$\displaystyle=(e^{i\omega_{j}z}h^{iz}\xi_{j}h^{-iz})_{j},$
$\displaystyle\mathcal{J}([v_{j},a]b)_{j}$
$\displaystyle=(h^{1/2}(b^{\ast}[v_{j^{\ast}},a]^{\ast})h^{1/2})_{j},$

where $j^{\ast}\in\\{1,\dots,m\\}$ is the unique index such that
$v_{j}^{\ast}=v_{j^{\ast}}$.

The map $\delta\colon L_{2}(\mathcal{M},\varphi)\to\mathcal{H}$ given by
$\delta(a)=(ie^{-\omega_{j}/4}[v_{j},a])_{j}$ is a symmetric derivation and

$\displaystyle\langle\delta(a),\delta(b)\rangle$
$\displaystyle=\sum_{j=1}^{m}\operatorname{tr}((ie^{-\omega_{j}/4}[v_{j},a])^{\ast}(ie^{-\omega_{j}/4}[v_{j},b])h)$
$\displaystyle=\sum_{j=1}^{m}e^{-\omega_{j}/2}\operatorname{tr}(a^{\ast}v_{j}^{\ast}[v_{j},b]h-a^{\ast}[v_{j},b]\sigma
v_{j}^{\ast})$
$\displaystyle=\sum_{j=1}^{m}\operatorname{tr}(e^{-\omega_{j}/2}a^{\ast}v_{j}^{\ast}[v_{j},b]h-e^{\omega_{j}/2}a^{\ast}[v_{j},b]v_{j}^{\ast}h)$
$\displaystyle=\langle a,\mathcal{L}(b)\rangle.$

###  8.3. Depolarizing semigroups

Let $\mathcal{M}$ be a von Neumann algebra and $\varphi$ a normal faithful
state on $\mathcal{M}$. The depolarizing semigroup (with respect to $\varphi$)
is given by $P_{t}(x)=e^{-t}x+(1-e^{-t})\varphi(x)$. It is evident that
$(P_{t})$ is a GNS-symmetric quantum Markov semigroup on $\mathcal{M}$.

The associated Tomita algebra is simply $\mathfrak{A}_{\varphi}$ and the
associated Tomita bimodule is a subbimodule of
$\mathfrak{A}_{\varphi}\odot\mathfrak{A}_{\varphi}$ with the usual inner
product, $L=\pi_{l}\otimes\pi_{l}$, $R=\pi_{r}\otimes\pi_{r}$,
$\mathcal{U}_{z}=U_{z}\otimes U_{z}$ and $\mathcal{J}(a\otimes
b)=J_{\varphi}b\otimes J_{\varphi}a$. The associated derivation is then given
by $\delta(a)=i(a\otimes\Omega_{\varphi}-\Omega_{\varphi}\otimes a)$, where
$\Omega_{\varphi}$ denotes the cyclic vector in $L_{2}(\mathcal{M},\varphi)$
representing $\varphi$.

###  8.4. Free Araki–Woods factors

Free Araki–Woods factors were introduced by Shlyakhtenko in the context of
free probability [Shl97]. We give a slightly modified, but equivalent
definition here, which stresses the analogy to the definition of algebras
$\Gamma_{\mathcal{M}}(\mathcal{H})$ from Subsection 4.2 (see [Vae05] for
example). Let $H$ be a Hilbert space, $I\colon H\to H$ an anti-unitary
involution and $(V_{t})$ a strongly continuous unitary group on $H$ such that
$V_{t}I=IV_{t}$ for all $t\in\mathbb{R}$. Let $A$ be the unique positive self-
adjoint non-singular operator on $H$ such that $V_{t}=A^{it}$ for all
$t\in\mathbb{R}$, and let $T=IA^{-1/2}$.

On the full Fock space

$\mathcal{F}(H)=\mathbb{C}\oplus\bigoplus_{n=1}^{\infty}H^{\otimes n}$

define the creation and annihilation operators $a(\xi)$ and $a^{\ast}(\xi)$
for $\xi\in H$ as in Subsection 4.2. Let $s(\xi)=a(\xi)+a^{\ast}(\xi)$. The
free Araki–Woods factor $\Gamma(H,(V_{t}))^{\prime\prime}$ is the von Neumann
algebra generated by the operators $s(\xi)$ for $\xi\in D(T)$ with $T\xi=\xi$.

If we view $H$ as a Tomita $\mathbb{C}$-$\mathbb{C}$-correspondence with
conjugation $I$ and unitary group $(V_{-t})$, then
$\Gamma(H,(V_{t}))^{\prime\prime}$ is the same as the free Gaussian algebra
$\Gamma_{\mathbb{C}}(H)$ from Subsection 4.2. The resulting von Neumann
algebras can be viewed as operator-valued versions of Shlyakhtenko’s free
Araki–Woods factors [Shl97], which are a special case of our construction when
$\mathfrak{A}=\mathbb{C}$, so that they might be of independent interest in
operator-valued free probability theory. In this article, the construction is
used to extend the construction of derivation triples from [JRS] to GNS-
symmetric quantum Markov semigroups.

Write $\Omega$ for the vector $(1,0,\dots)\in\mathcal{F}(H)$ and $\varphi$ for
the associated vector state on $\Gamma(H,(V_{t}))^{\prime\prime}$. The state
$\varphi$ is called the free quasi-free state. The vector $\Omega$ is cyclic
and separating for $\Gamma(H,(V_{t}))^{\prime\prime}$, which means that the
GNS Hilbert space associated with $\varphi$ is $\mathcal{F}(H)$ with the
defining action of $\Gamma(H,(V_{t}))^{\prime\prime}$. The associated modular
group acts as $\Delta_{\varphi}^{it}|_{H^{\otimes n}}=(V_{-t})^{\otimes n}$
and the associated modular conjugation acts as $J_{\varphi}|_{H^{\otimes
n}}=I^{\otimes n}\tau_{n}$, where $\tau_{n}$ is the reversal map given by
$\tau_{n}(\xi_{1}\otimes\dots\otimes\xi_{n})=\xi_{n}\otimes\dots\otimes\xi_{1}$.

The number operator $N$ on $\mathcal{F}(H)$ is the positive self-adjoint
operator on $\mathcal{F}(H)$ with domain

$D(N)=\\{\xi\in\mathcal{F}(H)\mid\sum_{n=0}^{\infty}n^{2}\lVert\xi_{n}\rVert^{2}<\infty\\}$

that acts on $H^{\otimes n}$ as multiplication by $n$. It generates a
symmetric QMS on $\mathcal{F}(H)$, which clearly commutes with
$(\Delta_{\varphi}^{it})$. Let $(P_{t})$ denote the associated GNS-symmetric
QMS on $\Gamma(H,(V_{t}))^{\prime\prime}$ and $\mathcal{E}$ the associated
quantum Dirichlet form.

The QMS $(P_{t})$ can equivalently be described as
$\Gamma^{\prime\prime}(e^{-t}\mathrm{id}_{H})$, where $\Gamma^{\prime\prime}$
is Shlyakhtenko’s second quantization functor (see also [Hia03] or [BM21],
where this QMS is called the Ornstein–Uhlenbeck semigroup on
$\Gamma(H,(V_{t}))^{\prime\prime}$).

Let $\mathfrak{A}=\operatorname{lin}\bigcup_{n}D(T^{\otimes n})$. Inductively
one can show that every $\eta\in\mathfrak{A}$ there exists $x$ in the algebra
generated by $\\{s(\xi)\mid\xi\in D(T),\,T\xi=\xi\\}$ such that
$x\Omega=\eta$, and $x$ is unique since $\Omega$ is separating for
$\Gamma(H,(V_{t}))$. This is the Wick word decomposition of $\eta$ and denoted
by $x=W(\eta)$. In particular, $\mathfrak{A}$ is a Tomita subalgebra of
$\mathfrak{A}_{\mathcal{E}}$ and the multiplication is given by
$\xi\eta=W(\xi)W(\eta)\Omega$. Moreover, $\mathfrak{A}$ is a core for
$\mathcal{E}$ by Lemma 3.4. For this reason we can restrict our attention to
$\mathfrak{A}$ instead of $\mathfrak{A}_{\mathcal{E}}$.

Let us first describe the Tomita bimodule $\mathcal{H}$. On $H\oplus H$ we
have the anti-linear involution $I\oplus I$ and the strongly continuous
unitary group $(V_{t}\oplus V_{t})$, which commutes with $I\oplus I$. Thus we
can construct $\Gamma(H\oplus H,(V_{t}\oplus V_{t}))^{\prime\prime}$ as
before, and the GNS Hilbert space with respect to the free quasi-free state
$\psi$ on $\Gamma(H\oplus H,(V_{t}\oplus V_{t}))^{\prime\prime}$ is
canonically identified with $\mathcal{F}(H\oplus H)$. By a slight abuse of
notation, we also write $\Omega$ for the vacuum vector in $\mathcal{F}(H\oplus
H)$.

In particular, the group generated by the modular operator and modular
conjugation operator act as $\Delta_{\psi}^{it}|_{(H\oplus H)^{\otimes
n}}=(V_{-t}\oplus V_{-t})^{\otimes n}$ and $J_{\psi}|_{(H\oplus H)^{\otimes
n}}=(I\oplus I)^{\otimes n}\Sigma_{n}$, where $\tau_{n}$ is the reversal map
as defined above. Let $\mathcal{H}=\operatorname{lin}\bigcup_{n}D((T\oplus
T)^{\otimes n})$, which is contained in $\mathfrak{A}_{\psi}$ under these
identifications.

As another application of the second quantization theorem, the map that sends
$s(\xi)\mapsto s((\xi,0))$ for $\xi\in D(T)$ extends to a normal unital state-
preserving $\ast$-homomorphism $\Phi$ from $\Gamma(H,(V_{t}))^{\prime\prime}$
to $\Gamma(H\oplus H,(V_{t}\oplus V_{t}))^{\prime\prime}$. The defining
property of $\Phi$ implies that $\Phi(W(\xi))=W(\xi,0)$ for
$\xi\in\mathfrak{A}$.

Define left and right actions of $\mathfrak{A}$ on $\mathcal{F}(H\oplus H)$ by

$\displaystyle L(\xi)x\Omega$ $\displaystyle=\Phi(W(\xi))x\Omega$
$\displaystyle R(\xi)x\Omega$ $\displaystyle=x\Phi(W(\xi))\Omega$

for $\xi\in\mathfrak{A}$ and $x\in\Gamma(H\oplus H,(V_{t}\oplus
V_{t}))^{\prime\prime}$.

Clearly, $L(\xi)$ and $R(\xi)$ map $\mathcal{H}$ into $\mathcal{H}$ for
$x\Omega\in\mathfrak{A}$. Together with the restrictions of
$(\Delta_{\psi}^{it})$ and $J_{\psi}$, this makes $\mathcal{H}$ into a Tomita
bimodule over $\mathfrak{A}$.

Now let us describe the derivation $\delta$. Let $\Sigma\colon H\oplus H\to
H\oplus H$ be the flip map, that is, $\Sigma(\xi,\eta)=(\eta,\xi)$. The family
$(\alpha_{t})_{t}=(\mathcal{F}(e^{it\Sigma})\,\cdot\,\mathcal{F}(e^{-it\Sigma}))_{t}$
is a weak∗ continuous group of $\ast$-automorphisms on $B(\mathcal{F}(H\oplus
H))$, and by the second quantization lemma, it maps $\Gamma(H\oplus
H,(V_{t}\oplus V_{t}))^{\prime\prime}$ into itself.

Let $D(\partial)$ be the set of all $x\in\Gamma(H\oplus H,(V_{t}\oplus
V_{t}))^{\prime\prime}$ for which $\lim_{t\to 0}\frac{1}{t}(\alpha_{t}(x)-x)$
exists in the strong∗-topology. Since $(\alpha_{t})$ consists of
$\ast$-automorphisms, $D(\partial)$ is a $\ast$-subalgebra of $\Gamma(H\oplus
H,(V_{t}\oplus V_{t}))^{\prime\prime}$ and the operator

$\partial\colon D(\partial)\to\Gamma(H\oplus H,(V_{t}\oplus
V_{t}))^{\prime\prime},\,x\mapsto\frac{1}{i}\lim_{t\searrow
0}\frac{1}{t}(\alpha_{t}(x)-x)$

is a $\ast$-derivation. Moreover, since $\Sigma$ commutes with
$\Delta_{\psi}^{it}$, the derivation $\partial$ commutes with $\sigma^{\psi}$.

Let

$\delta\colon\mathfrak{A}\to\mathcal{H},\,\xi\mapsto\partial(\Phi(W(\xi)))\Omega=\partial(W(\xi,0))\Omega.$

We have

$\displaystyle\delta(\xi\eta)$
$\displaystyle=\partial(\Phi(W(\xi)W(\eta)))\Omega$
$\displaystyle=\Phi(W(\xi))\partial(\Phi(W(\eta)))\Omega+\partial(\Phi(W(\xi)))\Phi(W(\eta))\Omega$
$\displaystyle=L(\xi)\delta(\eta)+R(\eta)\delta(\xi).$

Since $\partial$ commutes with the modular group, we have

$\displaystyle\delta(\Delta_{\varphi}^{it}\xi)=\partial(W(\Delta_{\varphi}^{it}\xi,0))\Omega=\partial(\sigma^{\psi}_{t}(W(\xi,0)))\Omega=\sigma^{\psi}_{t}(\partial(W(\xi,0)))\Omega=\Delta_{\psi}^{it}\delta(\xi).$

Similarly,

$\displaystyle\delta(J_{\varphi}\xi)=\partial(W(J_{\varphi}\xi,0))\Omega=\partial(\sigma^{\psi}_{-i/2}(W(\xi,0))^{\ast})\Omega=\sigma^{\psi}_{-i/2}(\partial(W(\xi,0)))^{\ast}\Omega=J_{\psi}\delta(\xi).$

Thus $\delta$ is a symmetric derivation.

For $\xi\in D(T)^{\odot n}$ we have

$\displaystyle\delta(\xi)$ $\displaystyle=\frac{1}{i}\lim_{t\searrow
0}\frac{1}{t}(\mathcal{F}(e^{it\Sigma})W(\xi,0)\mathcal{F}(e^{-it\Sigma})\Omega-W(\xi,0)\Omega)$
$\displaystyle=\frac{1}{i}\lim_{t\searrow
0}\frac{1}{t}((e^{it\Sigma})^{\otimes n}(\xi,0)-(\xi,0))$
$\displaystyle=\sum_{k=1}^{n}(\mathrm{id}^{k-1}\otimes\Sigma\otimes\mathrm{id}^{n-k})(\xi,0).$

Thus, if $\xi\in D(T)^{\odot m}$ and $\eta\in D(T)^{\odot n}$, then

$\displaystyle\langle\delta(\xi),\delta(\eta)\rangle_{\mathcal{H}}$
$\displaystyle=\delta_{m,n}\sum_{k=1}^{n}\langle(\mathrm{id}^{k-1}\otimes\Sigma\otimes\mathrm{id}^{n-k})(\xi,0),(\mathrm{id}^{k-1}\otimes\Sigma\otimes\mathrm{id}^{n-k})(\eta,0)\rangle$
$\displaystyle=n\delta_{m,n}\langle\xi,\eta\rangle$
$\displaystyle=\mathcal{E}(\xi,\eta).$

###  8.5. Compact quantum groups

Our last example are translation-invariant quantum Markov semigroups on
compact quantum groups. We refer to [Wor98] as a reference for the general
theory of compact quantum groups and to [CFK14] for the result on quantum
Markov semigroups on compact quantum groups we need. Note that some of the
common notation for compact quantum groups conflicts with the notation used in
Tomita–Takesaki theory. For example, $S$ and $\varphi$ are used for different
objects than in the rest of the article.

A compact quantum group is a pair $(A,\Delta)$ consisting of a unital
$C^{\ast}$-algebra $A$ and a unital $\ast$-homomorphism $\Delta\colon A\to
A\otimes_{\min}A$ such that

$(\Delta\otimes\mathrm{id})\Delta=(\mathrm{id}\otimes\Delta)\Delta$

and the sets

$\operatorname{lin}(A\otimes 1)\Delta(A),\,\operatorname{lin}(1\otimes
A)\Delta(A)$

are dense in $A\otimes_{\min}A$.

We will use the sumless Sweedler notation $\Delta(a)=a_{(1)}\otimes a_{(2)}$
for the comultiplication.

On a compact quantum group $A$ there exists a unique state $h$, called the
Haar state, such that

$(h\otimes\mathrm{id})\Delta(a)=(\mathrm{id}\otimes h)\Delta(a)=h(a)1$

for all $a\in A$.

Let $(\pi_{h},L_{2}(A,h),\Omega_{h})$ be the cyclic representation associated
with $h$, and let $\mathcal{M}=\pi_{h}(A)^{\prime\prime}$. We also denote the
vector state $\langle\Omega_{h},\cdot\,\Omega_{h}\rangle$ on $\mathcal{M}$ by
$h$.

An $n$-dimensional unitary corepresentation of $(A,\Delta)$ is a unitary $u\in
M_{n}(A)$ such that

$\Delta(u_{jk})=\sum_{l=1}^{n}u_{jl}\otimes u_{lk}.$

The linear span of the matrix coefficients $u_{jk}$, where $u$ runs over all
unitary corepresentations of $(A,\Delta)$, forms a dense unital
$\ast$-subalgebra $\mathcal{A}$ of $A$.

A unitary corepresentation $u$ is called irreducible if $\\{u\\}^{\prime}\cap
M_{n}(\mathbb{C})=\mathbb{C}1$. Two unitary corepresentations $u$ and $v$ are
called equivalent if there exists a unitary matrix $U\in M_{n}(\mathbb{C})$
such that $v=U^{\ast}uU$. If $(u^{(\alpha)})_{\alpha\in I}$ is a complete set
of representatives of the equivalence classes of unitary corepresentations of
$(A,\Delta)$ and $n_{\alpha}$ is the dimension of the corepresentation
$u^{(\alpha)}$, then $\\{u^{(\alpha)}_{jk}\mid\alpha\in I,\,1\leq j,k\leq
n_{\alpha}\\}$ is a basis of $\mathcal{A}$.

With the counit $\varepsilon\colon\mathcal{A}\to\mathbb{C}$ and the antipode
$S\colon\mathcal{A}\to\mathcal{A}$ given by

$\displaystyle\varepsilon(u^{(\alpha)}_{jk})$ $\displaystyle=\delta_{jk}$
$\displaystyle S(u^{(\alpha)}_{jk})$
$\displaystyle=(u^{(\alpha)}_{kj})^{\ast},$

the $\ast$-algebra $\mathcal{A}$ becomes a Hopf $\ast$-algebra.

For $f\colon\mathcal{A}\to\mathbb{C}$ linear and $a\in\mathcal{A}$ the
convolution is defined by $f\ast a=(\mathrm{id}\otimes f)\Delta(a)$, or, in
Sweedler notation, $f\ast a=f(a_{(2)})a_{(1)}$. Convolution by the counit is
the identity map. Similarly, if $f,g\in\mathcal{A}^{\ast}$, then $f\ast
g=(f\otimes g)\Delta$.

Let $\varphi\in\mathcal{A}^{\ast}$ be a hermitian functional such that
$\varphi(1)=0$, $\varphi(a^{\ast}a)\geq 0$ if $a\in\ker\varepsilon$ and
$\varphi\circ S=\varphi$ and define the associated convolution semigroup of
states by

$\varphi_{t}=\varepsilon+\sum_{k=1}^{\infty}\frac{t^{k}}{k!}\varphi^{\ast k}.$

This induces to a semigroup $(P_{t})$ on $\mathcal{A}$ given by
$P_{t}=(\mathrm{id}\otimes\varphi_{t})\Delta$. The semigroup $(P_{t})$ extends
to a GNS-symmetric quantum Markov semigroup on $\mathcal{M}$, which is
translation-invariant, that is, $\Delta P_{t}=(\mathrm{id}\otimes
P_{t})\Delta$ for all $t\geq 0$. Conversely, any translation-invariant GNS-
symmetric quantum Markov semigroup on $\mathcal{M}$ is of this form [CFK14,
Theorem 3.4, Corollary 4.6]. Let $\mathcal{E}$ denote the associated GNS-
symmetric quantum Dirichlet form on $L_{2}(A,h)$.

The generating functional $\varphi$ gives rise to a _Schürmann triple_
$((\pi,H),\eta,\varphi)$, where $\pi$ is a contractive unital
$\ast$-representation of $\mathcal{A}$ on the pre-Hilbert space $H$ and
$\eta\colon\mathcal{A}\to H$ is a surjective linear map such that

$\displaystyle\eta(ab)$ $\displaystyle=\pi(a)\eta(b)+\varepsilon(b)\eta(a)$
$\displaystyle\langle\eta(a),\eta(b)\rangle$
$\displaystyle=\varphi((a-\varepsilon(a))^{\ast}(b-\varepsilon(b)))$

for all $a,b\in\mathcal{A}$.

The bimodule and derivation associated with the quantum Dirichlet form
$\mathcal{E}$ have been essentially (without the group $(\mathcal{U}_{z})$ and
the conjugation $\mathcal{J}$) described in [CFK14, Section 8]. Since
$\mathcal{A}\Omega_{h}$ is a Tomita subalgebra of $\mathfrak{A}_{h}$ by
[Wor98, Theorem 1.4] and core for $\mathcal{E}$ by Lemma 3.4, we can restrict
our attention to $\mathcal{A}\Omega_{h}$.

Let

$\displaystyle\mathcal{H}$
$\displaystyle=\operatorname{lin}\\{a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)})\mid
a,b\in\mathcal{A}\\},$ $\displaystyle
L(a\Omega_{h})(b\Omega_{h}\otimes\eta(c))$
$\displaystyle=ab\Omega_{h}\otimes\pi(a)\eta(c),$ $\displaystyle
R(a\Omega_{h})(b\Omega_{h}\otimes\eta(c))$
$\displaystyle=ba\Omega_{h}\otimes\eta(c),$ $\displaystyle\delta(a\Omega_{h})$
$\displaystyle=a_{(1)}\Omega_{h}\otimes\eta(a_{(2)})$

for $a,b,c\in\mathcal{A}$. Note that
$\mathcal{H}=\operatorname{lin}\\{R(b\Omega_{h})\delta(a\Omega_{h})\mid
a,b\in\mathcal{A}\\}$.

It was shown in [CFK14, Proposition 8.1] that
$\delta(\xi\cdot\eta)=L(\xi)\delta(\eta)+R(\eta)\delta(\xi)$ and
$\mathcal{E}(\xi,\zeta)=\langle\delta(\xi),\delta(\zeta)\rangle_{\mathcal{H}}$
for $\xi,\zeta\in\mathcal{A}\Omega_{h}$. Clearly, $R(\xi)$ maps $\mathcal{H}$
into itself, and the product ensures that the same is the case for $L(\xi)$.
In other words,

$L(a\Omega_{h})(b_{(1)}c\Omega_{h}\otimes\eta(b_{(2)}))=(ab)_{(1)}c\Omega_{h}\otimes\eta((ab)_{(2)})-a_{(1)}bc\Omega_{h}\otimes\eta(a_{(2)}).$

The group $(\mathcal{U}_{z})$ can be described as follows. Let $\sigma$ be the
modular group associated with $h$. Define

$\mathcal{U}_{z}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)}))=\sigma_{z}(a)_{(1)}\sigma_{z}(b)\otimes\eta(\sigma_{z}(a)_{(2)}).$

Clearly $\mathcal{U}_{z}\delta(a\Omega_{h})=\delta(\sigma_{z}(a)\Omega_{h})$
and $\mathcal{U}_{z}R(a\Omega_{h})=R(\sigma_{z}(a)\Omega_{h})\mathcal{U}_{z}$,
which also implies
$\mathcal{U}_{z}L(a\Omega_{h})=L(\sigma_{z}(a)\Omega_{h})\mathcal{U}_{z}$ by
the product rule. Moreover,

$\displaystyle\quad\;\langle\mathcal{U}_{z}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)})),c_{(1)}d\Omega_{h}\otimes\eta(c_{(2)})\rangle$
$\displaystyle=h(\sigma_{z}(b)^{\ast}\sigma_{z}(a)^{\ast}_{(1)}c_{(1)}d)\varphi((\sigma_{z}(a)_{(2)}-\varepsilon(\sigma_{z}(a)_{(2)}))^{\ast}(c_{(2)}-\varepsilon(c_{(2)})))$
$\displaystyle=h(\sigma_{z}(b)^{\ast}\sigma_{z}(a)^{\ast}_{(1)}c_{(1)}d)(\varphi(\sigma_{z}(a)_{(2)}^{\ast}c_{(2)})-\overline{\varepsilon(\sigma_{z}(a)_{(2)})}\varphi(c_{(2)})-\varepsilon(c_{(2)})\overline{\varphi(\sigma_{z}(a)_{(2)})}),$

where we used $\varphi(1)=0$ in the last step. Now let us analyze the summands
separately. For the first summand we have

$\displaystyle
h(\sigma_{z}(b)^{\ast}\sigma_{z}(a)^{\ast}_{(1)}c_{(1)}\varphi(\sigma_{z}(a)_{(2)}^{\ast}c_{(2)})d)$
$\displaystyle=h(\sigma_{z}(b)^{\ast}\varphi\ast(\sigma_{z}(a)^{\ast}c)d)$
$\displaystyle=h(\sigma_{-\bar{z}}(\sigma_{\bar{z}}(b^{\ast})\varphi\ast(\sigma_{\bar{z}}(a^{\ast})c)d)$
$\displaystyle=h(b^{\ast}\varphi\ast(a^{\ast}\sigma_{-\bar{z}}(c))\sigma_{-\bar{z}}(d))$

since the modular group commutes with convolution by $\varphi$ by GNS-symmetry
[compare CFK]. For the second and third summand,

$\displaystyle
h(\sigma_{z}(b)^{\ast}\overline{\varepsilon(\sigma_{z}(a)_{(2)})}\sigma_{z}(a)^{\ast}_{(1)}\varphi(c_{(2)})c_{(1)}d)$
$\displaystyle=h(\sigma_{\bar{z}}(b^{\ast})(\varepsilon\ast\sigma_{\bar{z}}(a^{\ast}))(\varphi\ast
c)d)$
$\displaystyle=h(b^{\ast}a^{\ast}(\varphi\ast\sigma_{-\bar{z}}(c))\sigma_{-\bar{z}}(d))$

and

$\displaystyle
h(\sigma_{z}(b)^{\ast}\overline{\varphi(\sigma_{z}(a)_{(2)})}\sigma_{z}(a)^{\ast}_{(1)}\varepsilon(c_{(2)})c_{(1)}d)$
$\displaystyle=h(\sigma_{\bar{z}}(b^{\ast})(\varphi\ast\sigma_{\bar{z}}(a^{\ast}))(\varepsilon\ast
c)d)$ $\displaystyle=h(b^{\ast}(\varphi\ast
a^{\ast})\sigma_{-\bar{z}}(c)\sigma_{-\bar{z}}(d))$

by the same reasoning.

Reversing these computations, we get

$\displaystyle\langle\mathcal{U}_{z}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)})),c_{(1)}d\Omega_{h}\otimes\eta(c_{(2)})\rangle$
$\displaystyle\qquad=\langle
a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)}),\mathcal{U}_{-\bar{z}}(c_{(1)}d\Omega_{h}\otimes\eta(c_{(2)}))\rangle.$

As for the conjugation operator $\mathcal{J}$, let

$\mathcal{J}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)}))=\sigma_{i/2}(b)^{\ast}\sigma_{i/2}(a)^{\ast}_{(1)}\otimes\pi(\sigma_{i/2}(b)^{\ast})\eta(\sigma_{i/2}(a)^{\ast}_{(2)}).$

It is not hard to see that
$\mathcal{J}R(a\Omega_{h})=L(\sigma_{i/2}(a)^{\ast}\Omega_{h})\mathcal{J}$,
$\mathcal{J}\delta(a\Omega_{h})=\delta(\sigma_{i/2}(a)^{\ast}\Omega_{h})$ and
$\mathcal{U}_{z}\mathcal{J}=\mathcal{J}\mathcal{U}_{\bar{z}}$.

Furthermore,

$\displaystyle\mathcal{J}^{2}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)}))$
$\displaystyle=\mathcal{J}(\sigma_{i/2}(b)^{\ast}\sigma_{i/2}(a)^{\ast}_{(1)}\otimes\pi(\sigma_{i/2}(b)^{\ast})\eta(\sigma_{i/2}(a)^{\ast}_{(2)}))$
$\displaystyle=\mathcal{J}(L(\sigma_{i/2}(b)^{\ast}\Omega_{h})\delta(\sigma_{i/2}(a)^{\ast}\Omega_{h}))$
$\displaystyle=\mathcal{J}(\delta(\sigma_{i/2}(ab)^{\ast}\Omega_{h})-R(\sigma_{i/2}(a)^{\ast}\Omega_{h})\delta(\sigma_{i/2}(b)^{\ast}\Omega_{h}))$
$\displaystyle=\delta(ab\Omega_{h})-L(a\Omega_{h})\delta(b\Omega_{h})$
$\displaystyle=R(b\Omega_{h})\delta(a\Omega_{h})$
$\displaystyle=a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)}),$

so that $\mathcal{J}$ is an involution, and

$\displaystyle\langle\mathcal{J}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)})),c_{(1)}d\Omega_{h}\otimes\eta(c_{(2)})\rangle$
$\displaystyle\qquad=h(\sigma_{i/2}(a)\sigma_{i/2}(b)c_{(1)}d)\varphi((\sigma_{i/2}(ab)_{(2)}-\varepsilon(\sigma_{i/2}(a)_{(2)})\sigma_{i/2}(b))(c_{(2)}-\varepsilon(c_{(2)}))).$

Considering all three non-zero summands separately as in the case of
$\mathcal{U}_{z}$, one obtains

$\displaystyle\langle\mathcal{J}(a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)})),c_{(1)}d\Omega_{h}\otimes\eta(c_{(2)})\rangle$
$\displaystyle\qquad=\langle
a_{(1)}b\Omega_{h}\otimes\eta(a_{(2)}),\mathcal{J}(c_{(1)}d\Omega_{h}\otimes\eta(c_{(2)}))\rangle.$

Thus $\mathcal{J}$ and $(\mathcal{U}_{z})$ make $\mathcal{H}$ into a Tomita
bimodule over $\mathcal{A}\Omega_{h}$ and
$\delta\colon\mathcal{A}\Omega_{h}\to\mathcal{H}$ is a symmetric derivation
such that $D(\bar{\delta})=D(\mathcal{E})$ and
$\langle\bar{\delta}(\xi),\bar{\delta}(\eta)\rangle=\mathcal{E}(\xi,\eta)$.

## References

  * [AGS14] Luigi Ambrosio, Nicola Gigli and Guiseppe Savaré  “Metric measure spaces with Riemannian Ricci curvature bounded from below”  In _Duke Math. J._ 163.7, 2014, pp. 1405–1490  DOI: 10.1215/00127094-2681605
  * [AH77] Sergio Albeverio and Raphael Høegh-Krohn  “Dirichlet forms and Markov semigroups on $C^{*}$-algebras”  In _Comm. Math. Phys._ 56.2, 1977, pp. 173–187  URL: http://projecteuclid.org.libraryproxy.ist.ac.at/euclid.cmp/1103901144
  * [Ali76] Robert Alicki  “On the detailed balance condition for non-Hamiltonian systems”  In _Rep. Math. Phys._ 10.2, 1976, pp. 249–258  DOI: 10.1016/0034-4877(76)90046-X
  * [Bar17] Ivan Bardet  “Estimating the decoherence time using non-commutative Functional Inequalities”, 2017  arXiv:1710.01039 [quant-ph]
  * [BGJ22] Michael Brannan, Li Gao and Marius Junge  “Complete logarithmic Sobolev inequalities via Ricci curvature bounded below”  In _Adv. Math._ 394, 2022, pp. Paper No. 10812960  DOI: 10.1016/j.aim.2021.108129
  * [BK19] Fabrice Baudoin and Daniel J. Kelleher  “Differential one-forms on Dirichlet spaces and Bakry-Émery estimates on metric graphs”  In _Trans. Amer. Math. Soc._ 371.5, 2019, pp. 3145–3178  DOI: 10.1090/tran/7362
  * [BM21] Panchugopal Bikram and Rajeeb R. Mohanta  “Contractivity properties of Ornstein-Uhlenbeck semigroup for mixed $q$-Araki-Woods von Neumann algebras”  In _J. Math. Phys._ 62.11, 2021, pp. Paper No. 11350219  DOI: 10.1063/5.0057987
  * [Cas21] Martijn Caspers  “Gradient forms and strong solidity of free quantum groups”  In _Math. Ann._ 379.1-2, 2021, pp. 271–324  DOI: 10.1007/s00208-020-02109-y
  * [CE79] Erik Christensen and David E. Evans  “Cohomology of operator algebras and quantum dynamical semigroups”  In _J. London Math. Soc. (2)_ 20.2, 1979, pp. 358–368  DOI: 10.1112/jlms/s2-20.2.358
  * [CFK14] Fabio Cipriani, Uwe Franz and Anna Kula  “Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory”  In _J. Funct. Anal._ 266.5, 2014, pp. 2789–2844  DOI: 10.1016/j.jfa.2013.11.026
  * [Cip97] Fabio Cipriani  “Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras”  In _J. Funct. Anal._ 147.2, 1997, pp. 259–300  DOI: 10.1006/jfan.1996.3063
  * [CIW21] Martijn Caspers, Yusuke Isono and Mateusz Wasilewski  “$L_{2}$-cohomology, derivations, and quantum Markov semi-groups on $q$-Gaussian algebras”  In _Int. Math. Res. Not. IMRN_ , 2021, pp. 6405–6441  DOI: 10.1093/imrn/rnaa044
  * [CM17] Eric A. Carlen and Jan Maas  “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance”  In _J. Funct. Anal._ 273.5, 2017, pp. 1810–1869  DOI: 10.1016/j.jfa.2017.05.003
  * [CM20] Eric A. Carlen and Jan Maas  “Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems”  In _J. Stat. Phys._ 178.2, 2020, pp. 319–378  DOI: 10.1007/s10955-019-02434-w
  * [CS03] F. Cipriani and Jean-Luc Sauvageot  “Noncommutative potential theory and the sign of the curvature operator in Riemannian geometry”  In _Geom. Funct. Anal._ 13.3, 2003, pp. 521–545  DOI: 10.1007/s00039-003-0421-z
  * [CS03a] Fabio Cipriani and Jean-Luc Sauvageot  “Derivations as square roots of Dirichlet forms”  In _J. Funct. Anal._ 201.1, 2003, pp. 78–120  DOI: 10.1016/S0022-1236(03)00085-5
  * [CS15] Martijn Caspers and Adam Skalski  “The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms”  In _Comm. Math. Phys._ 336.3, 2015, pp. 1637–1664  DOI: 10.1007/s00220-015-2302-3
  * [CS17] Fabio Cipriani and Jean-Luc Sauvageot  “Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras”  In _Adv. Math._ 322, 2017, pp. 308–340  DOI: 10.1016/j.aim.2017.10.017
  * [Dab10] Yoann Dabrowski  “A non-commutative Path Space approach to stationary free Stochastic Differential Equations”, 2010  DOI: 10.48550/ARXIV.1006.4351
  * [DI16] Yoann Dabrowski and Adrian Ioana  “Unbounded derivations, free dilations, and indecomposability results for $\rm II_{1}$ factors”  In _Trans. Amer. Math. Soc._ 368.7, 2016, pp. 4525–4560  DOI: 10.1090/tran/6470
  * [DL92] E. Davies and J. Lindsay  “Noncommutative symmetric Markov semigroups”  In _Math. Z._ 210.3, 1992, pp. 379–411  DOI: 10.1007/BF02571804
  * [DL93] E. Davies and J. Lindsay  “Superderivations and symmetric Markov semigroups”  In _Comm. Math. Phys._ 157.2, 1993, pp. 359–370  URL: http://projecteuclid.org.libraryproxy.ist.ac.at/euclid.cmp/1104253944
  * [EL77] David E. Evans and John T. Lewis  “Dilations of irreversible evolutions in algebraic quantum theory”  In _Communications Dublin Inst. Advanced Studies. Ser. A_ , 1977, pp. v+104
  * [FOT11] Masatoshi Fukushima, Yoichi Oshima and Masayoshi Takeda  “Dirichlet forms and symmetric Markov processes” 19, De Gruyter Studies in Mathematics  Walter de Gruyter & Co., Berlin, 2011, pp. x+489
  * [GKS76] Vittorio Gorini, Andrzej Kossakowski and E… Sudarshan  “Completely positive dynamical semigroups of N-level systems”  In _Journal of Mathematical Physics_ 17.5, 1976, pp. 821–825  DOI: 10.1063/1.522979
  * [GL95] Stanisław Goldstein and J. Lindsay  “KMS-symmetric Markov semigroups”  In _Math. Z._ 219.4, 1995, pp. 591–608  DOI: 10.1007/BF02572383
  * [GL99] Stanisław Goldstein and J. Lindsay  “Markov semigroups KMS-symmetric for a weight”  In _Math. Ann._ 313.1, 1999, pp. 39–67  DOI: 10.1007/s002080050249
  * [GR21] Li Gao and Cambyse Rouzé  “Complete entropic inequalities for quantum Markov chains”, 2021  DOI: 10.48550/ARXIV.2102.04146
  * [Gri99] Alexander Grigor’yan  “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds”  In _Bull. Amer. Math. Soc. (N.S.)_ 36.2, 1999, pp. 135–249  DOI: 10.1090/S0273-0979-99-00776-4
  * [GY19] Luca Giorgetti and Wei Yuan  “Realization of rigid $C^{\ast}$-tensor categories via Tomita bimodules”  In _J. Operator Theory_ 81.2, 2019, pp. 433–479  DOI: 10.7900/jot
  * [Haa75] Uffe Haagerup  “The standard form of von Neumann algebras”  In _Math. Scand._ 37.2, 1975, pp. 271–283  DOI: 10.7146/math.scand.a-11606
  * [Hia03] Fumio Hiai  “$q$-deformed Araki-Woods algebras”  In _Operator algebras and mathematical physics (Constanţa, 2001)_ Theta, Bucharest, 2003, pp. 169–202
  * [HT13] Michael Hinz and Alexander Teplyaev  “Vector analysis on fractals and applications”  In _Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics_ 601, Contemp. Math.  Amer. Math. Soc., Providence, RI, 2013, pp. 147–163  DOI: 10.1090/conm/601/11960
  * [JMP14] Marius Junge, Tao Mei and Javier Parcet  “An invitation to harmonic analysis associated with semigroups of operators”  In _Harmonic analysis and partial differential equations_ 612, Contemp. Math.  Amer. Math. Soc., Providence, RI, 2014, pp. 107–122  DOI: 10.1090/conm/612/12227
  * [JRS] Marius Junge, Éric Ricard and Dimitri Shlyakhtenko  “Noncommutative Diffusion Semigroups and Free Probability”  Unpublished preprint
  * [JX07] Marius Junge and Quanhua Xu  “Noncommutative maximal ergodic theorems”  In _J. Amer. Math. Soc._ 20.2, 2007, pp. 385–439  DOI: 10.1090/S0894-0347-06-00533-9
  * [JZ15] Marius Junge and Qiang Zeng  “Noncommutative martingale deviation and Poincaré type inequalities with applications”  In _Probab. Theory Related Fields_ 161.3-4, 2015, pp. 449–507  DOI: 10.1007/s00440-014-0552-1
  * [Kus89] Shigeo Kusuoka  “Dirichlet forms on fractals and products of random matrices”  In _Publ. Res. Inst. Math. Sci._ 25.4, 1989, pp. 659–680  DOI: 10.2977/prims/1195173187
  * [Lin76] Göran Lindblad  “On the generators of quantum dynamical semigroups”  In _Comm. Math. Phys._ 48.2, 1976, pp. 119–130  URL: http://projecteuclid.org/euclid.cmp/1103899849
  * [LJL20] Haojian Li, Marius Junge and Nicholas LaRacuente  “Graph Hörmander Systems”  arXiv, 2020  DOI: 10.48550/ARXIV.2006.14578
  * [MM17] Markus Mittnenzweig and Alexander Mielke  “An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models”  In _J. Stat. Phys._ 167.2, 2017, pp. 205–233  DOI: 10.1007/s10955-017-1756-4
  * [OOT17] Rui Okayasu, Narutaka Ozawa and Reiji Tomatsu  “Haagerup approximation property via bimodules”  In _Math. Scand._ 121.1, 2017, pp. 75–91  DOI: 10.7146/math.scand.a-25970
  * [Pas73] William L. Paschke  “Inner product modules over $B^{\ast}$-algebras”  In _Trans. Amer. Math. Soc._ 182, 1973, pp. 443–468  DOI: 10.2307/1996542
  * [Per74] Michael D. Perlman  “Jensen’s inequality for a convex vector-valued function on an infinite-dimensional space”  In _J. Multivariate Anal._ 4, 1974, pp. 52–65  DOI: 10.1016/0047-259X(74)90005-0
  * [Pet09] Jesse Peterson  “A 1-cohomology characterization of property (T) in von Neumann algebras”  In _Pacific J. Math._ 243.1, 2009, pp. 181–199  DOI: 10.2140/pjm.2009.243.181
  * [Sau96] Jean-Luc Sauvageot  “Semi-groupe de la chaleur transverse sur la $C^{*}$-algèbre d’un feuilletage riemannien”  In _J. Funct. Anal._ 142.2, 1996, pp. 511–538  DOI: 10.1006/jfan.1996.0158
  * [Sch02] Jürgen Schweizer  “Hilbert $C^{\ast}$-modules with a predual”  In _Journal of Operator Theory_ 48.3 Theta Foundation, 2002, pp. 621–632  URL: http://www.jstor.org/stable/24715589
  * [Shl97] Dimitri Shlyakhtenko  “Free quasi-free states”  In _Pacific J. Math._ 177.2, 1997, pp. 329–368  DOI: 10.2140/pjm.1997.177.329
  * [Shl99] Dimitri Shlyakhtenko  “$A$-valued semicircular systems”  In _J. Funct. Anal._ 166.1, 1999, pp. 1–47  DOI: 10.1006/jfan.1999.3424
  * [Ske01] Michael Skeide  “Hilbert modules and applications in quantum probability”, 2001  URL: http://web.unimol.it/skeide/_MS/downloads/habil.pdf
  * [SV19] Adam Skalski and Ami Viselter  “Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms”  In _J. Math. Pures Appl. (9)_ 124, 2019, pp. 59–105  DOI: 10.1016/j.matpur.2018.04.007
  * [Tak03] Masamichi Takesaki  “Theory of operator algebras. II” Operator Algebras and Non-commutative Geometry, 6 125, Encyclopaedia of Mathematical Sciences  Springer-Verlag, Berlin, 2003, pp. xxii+518  DOI: 10.1007/978-3-662-10451-4
  * [Vae05] Stefaan Vaes  “États quasi-libres libres et facteurs de type III (d’après D. Shlyakhtenko)” Séminaire Bourbaki. Vol. 2003/2004  In _Astérisque_ , 2005, pp. Exp. No. 937ix329–350
  * [Ver22] Matthijs Vernooij  “On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups”  arXiv, 2022  DOI: 10.48550/ARXIV.2203.12307
  * [Wir20] Melchior Wirth  “A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy”  arXiv, 2020  DOI: 10.48550/ARXIV.1808.05419
  * [Wir22] Melchior Wirth  “Christensen–Evans theorem and extensions of GNS-symmetric quantum Markov semigroups”  arXiv, 2022  DOI: 10.48550/ARXIV.2203.00341
  * [Wor98] Stanisław Lech Woronowicz  “Compact quantum groups”  In _Symétries quantiques (Les Houches, 1995)_ North-Holland, Amsterdam, 1998, pp. 845–884
  * [WZ21] Melchior Wirth and Haonan Zhang  “Complete gradient estimates of quantum Markov semigroups”  In _Comm. Math. Phys._ 387.2, 2021, pp. 761–791  DOI: 10.1007/s00220-021-04199-4