A Note on Spectral Triples on the Quantum Disk

A Note on Spectral Triples on the Quantum Disk

Slawomir KLIMEK †, Matt MCBRIDE ‡ and John Wilson PEOPLES †

S. Klimek, M. McBride and J.W. Peoples

† Department of Mathematical Sciences, Indiana University-Purdue University
Indianapolis,
† 402 N. Blackford St., Indianapolis, IN 46202, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Department of Mathematics and Statistics, Mississippi State University,
‡ 175 President’s Cir., Mississippi State, MS 39762, USA
<EMAIL_ADDRESS>
Received April 03, 2019, in final form May 24, 2019; Published online May 28,
2019

By modifying the ideas from our previous paper [SIGMA 13 (2017), 075, 26
pages, arXiv:1705.04005], we construct spectral triples from implementations
of covariant derivations on the quantum disk.

invariant and covariant derivations; spectral triple; quantum disk

46L87; 46L89; 58B34; 58J42

##  1 Introduction

Spectral triples are a key tool in noncommutative geometry [2], since they
allow using analytical methods in studying quantum spaces. In this note we
show how, by changing the concept of a Hilbert space implementation of an
unbounded derivation, one can use the techniques of our papers [9, 10] to
construct meaningful, geometrical spectral triples for the quantum disk, the
Toeplitz $C^{*}$-algebra of the unilateral shift.

Our previous paper [9] classified unbounded derivations, covariant with
respect to a natural rotation, and their implementations in Hilbert spaces
obtained from the GNS construction with respect to invariant states.
Surprisingly, no implementation of a covariant derivation in any GNS Hilbert
space for a faithful, normal, invariant state turned out to have compact
parametrices for a large class of boundary conditions. However, if we relax
the concept of an implementation by allowing operators to act between
different Hilbert spaces, then it turns out, as demonstrated in this paper,
that there is an interesting class of examples of spectral triples that can be
constructed this way.

In [1, 8] our earlier attempts at constructing Dirac-type operators on the
quantum disk using APS-type boundary conditions to eliminate infinite
dimensional kernel/cokernel did not lead to examples of spectral triples. The
class of operators considered in [1, 8], designed to mimic the classical
Atiyah–Patodi–Singer theory, does not have bounded commutators with
representations of the algebra. Additionally, as pointed out in [7], there are
fundamental reasons why APS boundary are not compatible with spectral triples
even in classical geometry for algebras of functions which are non-constant on
the boundary, as the corresponding domains of the Dirac-type operators are not
preserved by the representations of the algebra. Fortunately, for similar yet
quite different Dirac-type operators considered in this paper, coming from
implementations of derivations, there is enough flexibility that the size of
the kernel can be controlled by the growth conditions of the coefficients, as
observed previously in [9, 10]. Such an option does not seem to have an
obvious analog for the classical Dirac operator.

Other examples of spectral triples of the Toeplitz algebra, in GNS Hilbert
spaces of non-normal states, were also constructed in [3], [4, Section 4.2],
[5] and in [6]. Those authors were working with irreducible representations of
the Toeplitz algebra, unlike the representations considered in this paper.

We review the notation and basic concepts from [9] below and, in a number of
places, we use the results contained in that reference.

##  2 Quantum disk

Let $\\{E_{k}\\}_{k=0}^{\infty}$ be the canonical basis for
$\ell^{2}({\mathbb{Z}}_{\geq 0})$ and $U$ be the unilateral shift defined by

$\displaystyle UE_{k}=E_{k+1}.$

Note that $U$ is an isometry, i.e., $U^{*}U=I$. Consider the Toeplitz algebra
$A=C^{*}(U)$, the $C^{*}$-algebra generated by $U$. This algebra is called the
quantum disk. We also use the diagonal label operator

$\displaystyle{\mathbb{K}}E_{k}=kE_{k}.$

It follows that for $a\colon{\mathbb{Z}}_{\geq 0}\to{\mathbb{C}}$, we have

$\displaystyle a({\mathbb{K}})E_{k}=a(k)E_{k}.$

These are precisely the operators which are diagonal with respect to
$\\{E_{k}\\}$. The operators $({\mathbb{K}},U)$ serve as noncommutative polar
coordinates, and they satisfy the following relation

$\displaystyle{\mathbb{K}}U=U({\mathbb{K}}+I).$

Let $c$ be the set of $a(k)$, as above, which are convergent as $k\to\infty$
and let $c_{00}^{+}$ be the set of all eventually constant functions, i.e.,
functions $a(k)$ such that there exists $k_{0}$ where $a(k)$ is constant for
$k\geq k_{0}$.

Consider, for future reference, the following dense $*$-subalgebra of $A$

$\displaystyle\mathcal{A}=\bigg{\\{}a=\sum_{n\geq
0}U^{n}a_{n}({\mathbb{K}})+\sum_{n<0}a_{n}({\mathbb{K}})(U^{*})^{-n}\colon
a_{n}(k)\in c_{00}^{+},\ \textrm{finite sums}\bigg{\\}}.$

By Proposition 3.1 in [9], $\mathcal{A}=\operatorname{Pol}(U,U^{*})$.

##  3 Derivations on quantum disk

Let $\rho_{\theta}\colon A\to A$, $0\leq\theta<2\pi$, be a one parameter group
of automorphisms of $A$ defined by

$\displaystyle\rho_{\theta}(a)={\rm e}^{{\rm i}\theta{\mathbb{K}}}a{\rm
e}^{-{\rm i}\theta{\mathbb{K}}}.$

Since $\rho_{\theta}(U)={\rm e}^{{\rm i}\theta}U$ and
$\rho_{\theta}(U^{*})={\rm e}^{-{\rm i}\theta}U^{*}$, the automorphisms
$\rho_{\theta}$ are well defined on $A$ and they preserve $\mathcal{A}$. By
Proposition 4.2 in [9], any densly-defined derivation $d\colon\mathcal{A}\to
A$, covariant with respect to $\rho_{\theta}$

$\displaystyle\rho_{\theta}(d(a))={\rm e}^{{\rm i}\theta}d(\rho_{\theta}(a)),$

is of the following form

$\displaystyle d(a)=[U\beta({\mathbb{K}}),a],$

where $\beta(k+1)-\beta(k)\in c$. We use notation

$\displaystyle\lim_{k\to\infty}\beta(k+1)-\beta(k):=\beta_{\infty},$

and below we only consider covariant derivations with $\beta_{\infty}\neq 0$.

##  4 Covariant implementations on quantum disk

We will begin by introducing the following family of states $\tau_{w}\colon
A\to{\mathbb{C}}$ on $A$, defined by

$\displaystyle\tau_{w}(a)=\operatorname{tr}(w({\mathbb{K}})a),$

where $w(k)>0$ for all $k\in{\mathbb{Z}}_{\geq 0}$ and

$\displaystyle\sum_{k=0}^{\infty}{w(k)}=1.$

As a result of Proposition 5.4 in [9], $\tau_{w}$ are precisely the
$\rho_{\theta}$-invariant, normal, faithful states on $A$. Let $H_{w}$ be the
Hilbert space obtained by Gelfand–Naimark–Segal (GNS) construction on $A$
using state $\tau_{w}$. Since the state is faithful, $H_{w}$ is the completion
of $A$ with respect to the inner product given by

$\displaystyle(a,b)_{w}=\tau(w({\mathbb{K}})a^{*}b).$

A simple calculation leads to the following precise description: $H_{w}$ is
the Hilbert space consisting of infinite series of operators

$\displaystyle f=\sum_{n\geq
0}U^{n}f_{n}({\mathbb{K}})+\sum_{n<0}f_{n}({\mathbb{K}})(U^{*})^{-n}$ (4.1)

satisfying

$\displaystyle\|f\|_{w}^{2}=\sum_{n\geq 0}\sum_{k\geq
0}w(k)|f_{n}(k)|^{2}+\sum_{n<0}\sum_{k\geq 0}w(k-n)|f_{n}(k)|^{2}<\infty.$

It is important to notice that $\mathcal{A}\subseteq H_{w}$ and that
$\mathcal{A}$ is dense in $H_{w}$. The space of all formal series (4.1) with
arbitrary coefficients $f_{n}(k)$ will be denoted by $\mathcal{F}$.

The GNS representation map $\pi_{w}\colon A\to B(H_{w})$ is given by left-hand
multiplication

$\displaystyle\pi_{w}(a)f=af.$

Define a one parameter group of unitary operators $U_{\theta}^{w}\colon
H_{w}\to H_{w}$ via the formula

$\displaystyle U_{\theta}^{w}f=\sum\limits_{n\geq 0}U^{n}{\rm e}^{{\rm
i}n\theta}f_{n}({\mathbb{K}})+\sum\limits_{n<0}{\rm e}^{{\rm
i}n\theta}f_{n}({\mathbb{K}})(U^{*})^{-n}.$

It is easily seen that they are implementing $\rho_{\theta}$, as we have

$\displaystyle\pi_{w}(\rho_{\theta}(a))=U_{\theta}^{w}\pi_{w}(a)\big{(}U_{\theta}^{w}\big{)}^{-1}.$

Consider an additional weight, $w^{\prime}(k)$, possibly different from
$w(k)$, satisfying the same conditions. An operator $D\colon
H_{w}\supseteq\mathcal{A}\to H_{w^{\prime}}$ is called a covariant
implementation of a covariant derivation $d$ if for every $a\in\mathcal{A}$,
and for every $f\in\mathcal{A}$ considered as an element of both $H_{w}$ and
$H_{w^{\prime}}$, we have

$\displaystyle D\pi_{w}(a)f-\pi_{w^{\prime}}(a)Df=\pi_{w^{\prime}}(d(a))f,$

and, additionally, $D$ satisfies

$\displaystyle U_{\theta}^{w^{\prime}}D\big{(}U_{\theta}^{w}\big{)}^{-1}f={\rm
e}^{{\rm i}\theta}Df.$

Allowing for implementations between different Hilbert spaces is the key
difference between this paper and [9].

Exactly the same argument as in Proposition 6.1 in [9] shows that any
implementation $D$ is of the form

$\displaystyle Df=U\beta({\mathbb{K}})f-fU\alpha({\mathbb{K}}),$ (4.2)

where $\alpha(k)$ is a sequence such that

$\displaystyle\sum_{k=0}^{\infty}|\beta(k)-\alpha(k)|^{2}w^{\prime}(k)<\infty.$
(4.3)

The assumption $\beta_{\infty}\neq 0$ implies that $\beta(k)$ has at most
finitely many zeros. Without loss of generality, we may assume that
$\beta(k)\neq 0$ as this can be obtained by a bounded perturbation. Arguments
in [9] show that if $\alpha(k)$ has infinitely many zeros, then
$\operatorname{Ker}(D)$ has infinite dimension, and so $D$ cannot define a
spectral triple. Hence we assume that for every $k$ we have

$\displaystyle\alpha(k),\beta(k)\neq 0.$ (4.4)

Additionally, as in [9], we write

$\displaystyle\alpha(k)=\beta(k)\frac{\mu(k+1)}{\mu(k)},\qquad\text{where}\quad\mu(0)=1.$
(4.5)

Notice that the operator $D$, given by the formula (4.2), is a well-defined
linear map on $\mathcal{F}$ and $D\colon\mathcal{F}\to\mathcal{F}$, where
again $\mathcal{F}$ is the space of all formal series defined by equation
(4.1). This allows us to describe the maximal domain of $D$ in $H_{w}$ as

$\displaystyle\textrm{dom}(D)=\left\\{f\in H_{w}\subseteq\mathcal{F}\colon
Df\in H_{w^{\prime}}\right\\}.$

Arguing exactly as in the paragraph preceding Proposition 5 of [1], at least
for the choices of parameters at the end of the paper, the operator $D$ has a
natural radial/angular decomposition in full analogy with the classical
$d$-bar operator on the disk.

The following is the main technical result of this paper.

######  Theorem 4.1.

Assume the following

$\displaystyle\left|\frac{\beta(k)\cdots\beta(k+n)}{\beta(j)\cdots\beta(j+n)}\right|\leq{\rm
const}\qquad\textrm{for all}\quad k\leq j,\quad n\in{\mathbb{Z}}_{\geq 0},$
(4.6)
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{1}{(\max(j,k)+1)^{2}}\left|\frac{\mu(j)}{\mu(k)}\right|^{2}\frac{w(k)}{w^{\prime}(j)}<\infty,$
(4.7)

there exists $N$ such that

$\displaystyle\sum_{k=0}^{\infty}\frac{(1+k)^{2n}}{|\mu(k)|^{2}}w(k)<\infty$
(4.8)

for $n<N$ and infinite for $n\geq N$. Assume additionally that
$\beta_{\infty}\neq 0$, and (4.3), (4.4), and (4.5) hold. Then $D$ on
$\operatorname{dom}(D)$ has a compact parametrix. In fact, there exists a
compact operator $Q\colon H_{w^{\prime}}\to H_{w}$, satisfying $DQ=I$ and
$QD=I-C$ where C is a compact operator.

###### Proof.

We begin by expressing the action of $D$ in terms of the Fourier decomposition

$\displaystyle Df=\sum_{n\geq
0}U^{n+1}(D_{n}f_{n})({\mathbb{K}})+\sum_{n<0}(D_{n}f_{n})({\mathbb{K}})(U^{*})^{-n-1},$

where $D_{n}$ are given by the following formulas: for $n\geq 0$ and
$f\in\textrm{dom}(D_{n})=\big{\\{}f\in\ell^{2}_{w}\colon
D_{n}f\in\ell^{2}_{w^{\prime}}\big{\\}}$,

$\displaystyle
D_{n}f(k)=\beta(k+n)f(k)-\beta(k)\frac{\mu(k+1)}{\mu(k)}f(k+1),$

and for $n<0$ and
$f\in\operatorname{dom}(D_{n})=\big{\\{}f\in\ell^{2}_{w_{n}}\colon
D_{n}f\in\ell^{2}_{w^{\prime}_{n+1}}\big{\\}}$,

$\displaystyle
D_{n}f(k)=\beta(k-n-1)\frac{\mu(k-n)}{\mu(k-n-1)}f(k)-\beta(k-1)f(k-1),$

where we used the following notation

$\displaystyle\ell^{2}_{w}=\left\\{f(k)\colon\sum_{k=0}^{\infty}|f(k)|^{2}w(k)<\infty\right\\},$

and

$\displaystyle\ell^{2}_{w_{n}}=\left\\{f(k)\colon\sum_{k=0}^{\infty}|f(k)|^{2}w(k-n)<\infty\right\\}.$

Naturally, two cases ($n\geq 0$ and $n<0$) arise.

Case 1 ($n\geq 0$): We begin by looking at $n\geq N$ from condition (4.8). The
formal kernel of $D_{n}$ is one-dimensional and spanned by the following
vector

$\displaystyle h^{(n)}(k)=\frac{\beta(k)\cdots\beta(k+n-1)}{\mu(k)}.$

By formula (4.8), we have $\big{\|}h^{(n)}\big{\|}_{\ell^{2}_{w}}=\infty$, so
the kernel of $D_{n}$ in $H_{w}$ is trivial.

Consider the following operator
$Q_{n}\colon\ell^{2}_{w^{\prime}}\to\ell^{2}_{w}$

$\displaystyle
Q_{n}g=\sum_{j=k}^{\infty}\frac{\beta(k)\cdots\beta(k+n-1)}{\beta(j)\cdots\beta(j+n)}\frac{\mu(j)}{\mu(k)}g(j).$

We have the following formula for the Hilbert–Schmidt norm of $Q_{n}$

$\displaystyle\|Q_{n}\|_{\rm
HS}^{2}=\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}\left|\frac{\beta(k)\cdots\beta(k+n-1)}{\beta(j)\cdots\beta(j+n)}\right|^{2}\left|\frac{\mu(j)}{\mu(k)}\right|^{2}\frac{w(k)}{w^{\prime}(j)}.$

Therefore, using assumption (4.6), we can estimate as follows

$\displaystyle\|Q_{n}\|_{\rm
HS}^{2}\leq\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}\left|\frac{\beta(k)\cdots\beta(k+n-1)}{\beta(j)\cdots\beta(j+n-1)}\right|^{2}\frac{1}{|\beta(j+n)|^{2}}\left|\frac{\mu(j)}{\mu(k)}\right|^{2}\frac{w(k)}{w^{\prime}(j)}\leq$
$\displaystyle\hphantom{\|Q_{n}\|_{\rm
HS}^{2}}{}\leq\textrm{const}\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}\frac{1}{(1+j+n)^{2}}\left|\frac{\mu(j)}{\mu(k)}\right|^{2}\frac{w(k)}{w^{\prime}(j)}.$

Consequently, we have $\|Q_{n}\|_{\rm HS}^{2}<\infty$ by (4.7) and
$\|Q_{n}\|_{\rm HS}\to 0$ as $n\to\infty$ by Lebesgue’s dominated convergence
theorem.

An easy calculation shows that $D_{n}Q_{n}g=g$ for every
$g\in\ell^{2}_{w^{\prime}}$. Consequently we have:
$\operatorname{Ran}(Q_{n})\subseteq\operatorname{dom}(D_{n})$ and moreover
$\operatorname{Ran}(D_{n})=\ell^{2}_{w^{\prime}}$. Given any
$f\in\operatorname{dom}(D_{n})$ the difference $Q_{n}D_{n}f-f$ is in the
domain of $D_{n}$, and

$\displaystyle D_{n}(Q_{n}D_{n}f-f)=0.$

But the kernel of $D_{n}$ is trivial so $Q_{n}D_{n}f=f$, and by continuity
$Q_{n}D_{n}=I$, therefore $Q_{n}$ is the inverse of $D_{n}$.

######  Remark 4.2.

Notice that in the $n\geq N$ case the maximal domain
$\operatorname{dom}(D_{n})$ is equal the minimal domain
$\operatorname{dom}_{\min}(D_{n})$, defined as the closure of $c_{00}$ with
respect to the graph norm. Here $c_{00}$ is the set of sequences that are
eventually zero. Given any $f\in\operatorname{dom}(D_{n})$ we have that
$D_{n}f\in\ell^{2}_{w^{\prime}}$ and we choose a sequence
$\big{\\{}\tilde{f}_{j}\big{\\}}$, $\tilde{f}_{j}\in c_{00}$ converging to
$D_{n}f$. Consider the sequence $f_{j}=Q_{n}\tilde{f}_{j}$, which is in
$c_{00}$ as easily seen from the formula for $Q_{n}$. Then we have $f_{j}\to
f$ by continuity of $Q_{n}$ and $D_{n}f_{j}\to D_{n}f$, so that $f$ in the
closure of $c_{00}$.

We will now consider $0\leq n<N$. We define the following operator

$\displaystyle\tilde{Q}_{n}g(k)=\sum_{j=k}^{\infty}\frac{\beta(k)\cdots\beta(k+n-1)}{\beta(j)\cdots\beta(j+n)}\frac{\mu(j)}{\mu(k)}g(j).$

By the results in the case $n\geq N$, previously discussed, $\tilde{Q}_{n}$ is
a Hilbert–Schmidt operator from $\ell^{2}_{w^{\prime}}$ to $\ell^{2}_{w}$. We
will verify that the operator

$\displaystyle
Q_{n}g(k)=\tilde{Q}_{n}g(k)-\frac{\tilde{Q}_{n}g(0)}{\beta(0)\cdots\beta(n-1)}h^{(n)}(k)$

is a parametrix of $D_{n}$. The second term in the above expression is a rank
1 operator, so $Q_{n}$ is still a Hilbert–Schmidt operator. Additionally

$\displaystyle
D_{n}Q_{n}g=D_{n}\tilde{Q}_{n}g-\frac{\tilde{Q}_{n}g(0)}{\beta(0)\cdots\beta(n-1)}D_{n}\big{(}h^{(n)}\big{)}=g$

by results in the previous case.

Given $f\in$ dom$(D_{n})$ we have

$\displaystyle
Q_{n}D_{n}f(k)=\lim_{L\to\infty}\sum_{j=k}^{L}\frac{\beta(k)\cdots\beta(k+n-1)}{\beta(j)\cdots\beta(j+n)}\frac{\mu(j)}{\mu(k)}\left(\beta(j+n)f(j)-\beta(j)\frac{\mu(j+1)}{\mu(j)}f(j+1)\right)$
$\displaystyle\hphantom{Q_{n}D_{n}f(k)=}{}-\sum_{j=0}^{L}\frac{\mu(j)}{\beta(j)\cdots\beta(j+n)}\left(\beta(j+n)f(j)-\beta(j)\frac{\mu(j+1)}{\mu(j)}f(j+1)\right)h^{(n)}(k).$

Simplifying the telescoping sum yields

$\displaystyle
Q_{n}D_{n}f(k)=f(k)-\frac{f(0)}{\beta(0)\cdots\beta(n-1)}h^{(n)}(k).$

Clearly, by the imposed conditions,

$\displaystyle C_{n}f(k):=\frac{f(0)}{\beta(0)\cdots\beta(n-1)}h^{(n)}(k)$

is a Hilbert–Schmidt operator, and $Q_{n}D_{n}=I-C_{n}$, proving that $Q_{n}$
is a parametrix of $D_{n}$.

Case 2 ($n<0$): Clearly $\operatorname{Ker}(D_{n})=0$ and $D_{n}$ is
invertible with inverse

$\displaystyle
Q_{n}g(k)=\sum_{j=0}^{k}\frac{\beta(j)\cdots\beta(j-n-2)}{\beta(k)\cdots\beta(k-n-1)}\frac{\mu(j-n-1)}{\mu(k-n)}g(j).$

The above is done by direct calculation and can be verified by checking
$Q_{n}D_{n}=I$ and $Q_{n}D_{n}=I$, see additionally Lemma 7.7 in [9]. We will
now show that $Q_{n}$ is a Hilbert–Schmidt operator. Consider the
Hilbert–Schmidt norm

$\displaystyle\|Q_{n}\|_{\rm
HS}^{2}=\sum_{k=0}^{\infty}\sum_{j=0}^{k}\left|\frac{\beta(j)\cdots\beta(j-n-2)}{\beta(k)\cdots\beta(k-n-1)}\right|^{2}\left|\frac{\mu(j-n-1)}{\mu(k-n)}\right|^{2}\frac{w(k-n)}{w^{\prime}(j-n-1)}.$

Changing the indices $k-n\to k^{\prime}$ and $j-n-1\to j^{\prime}$, the order
of summation, and estimating as above yields

$\displaystyle\|Q_{n}\|_{\rm
HS}^{2}\leq\sum_{j=-n-1}^{\infty}\sum_{k=j+1}^{\infty}\frac{1}{(1+k)^{2}}\left|\frac{\mu(j)}{\mu(k)}\right|^{2}\frac{w(k)}{w^{\prime}(j)}<\infty.$

It follows that $\|Q_{n}\|_{\rm HS}^{2}\to 0$ as $-n\to\infty$ since this is
the tail of an absolutely convergent series.

Consequently, in all cases, $D_{n}$ has Hilbert–Schmidt, and thus compact,
parametrices $Q_{n}$, with Hilbert–Schmidt norms approaching zero. Thus, it
follows that $D$ has a compact parametrix. This completes the theorem. ∎

The main significance of this result is outlined in the following theorem.
First, we introduce some notation related to spectral triples as considered in
[9].

Let $H=H_{w^{\prime}}\bigoplus H_{w}$, with grading
$\Gamma\big{|}_{H_{w^{\prime}}}=1$ and $\Gamma\big{|}_{H_{w}}=-1$. Define a
representation $\pi\colon A\to B(H)$ of $A$ in $H$ by the formula

$\displaystyle\pi(a)=(\pi_{w^{\prime}}(a),\pi_{w}(a)),$

and also define a quantum analog of a Dirac operator on the unit disk by

$\displaystyle{\mathcal{D}}=\left[\begin{matrix}0&D\\\
D^{*}&0\end{matrix}\right],$

so that $\pi(a)$ are even and ${\mathcal{D}}$ is odd with respect to grading
$\Gamma$.

######  Theorem 4.3.

With the above notation, $(\mathcal{A},H,{\mathcal{D}})$ forms an even
spectral triple over $A$.

###### Proof.

By Theorem 4.1, we have that $D$ is has a compact parametrix and so
${\mathcal{D}}$ has compact parametrices by the results in the appendix of
[9]. Clearly $\pi(a)$ preserves the domain of $D$ and, since $D$ is an
implementation of a derivation $d\colon\mathcal{A}\to A$, the commutator
$[{\mathcal{D}},\pi(a)]$ is bounded for all $a\in\mathcal{A}$. This completes
the proof. ∎

We conclude this paper by giving explicit examples of parameters $w(k)$,
$w^{\prime}(k)$, $\beta(k)$, and $\mu(k)$ that satisfy the conditions of the
above theorems.

Assume $3<a<2b-1<c$, nonnegative $N\geq\frac{c-2b-1}{2}$, $\beta(k)=1+k$, and
consider the following sequences:

$\displaystyle
w^{\prime}(k)=\frac{w^{\prime}(0)}{(1+k)^{a}},\qquad\mu(k)=\frac{1}{(1+k)^{b}},\qquad
w(k)=\frac{w(0)}{(1+k)^{c}},$

where $w^{\prime}(0)$ and $w(0)$ are such that
$\sum\limits_{k=0}^{\infty}w^{\prime}(k)=1=\sum\limits_{k=0}^{\infty}w(k).$
Then a straightforward calculation shows that they satisfy the necessary
conditions for $(\mathcal{A},H,{\mathcal{D}})$ to be an even spectral triple.
Moreover, there is a choice of parameters $a$, $b$, $c$ such that $N$ can be
equal to zero.

### Acknowledgments

We would like to thank an anonymous referee for pointing out a critical issue
with the initial version of this paper.

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# Pipeline Provenance for Analysis, Evaluation, Trust or Reproducibility

Michael A. C. Johnson Max Planck Institute for Radio Astronomy  German
Aerospace Center (DLR), Institute of Data Science  University of Manchester
Hans-Rainer Klöckner Max Planck Institute for Radio Astronomy  Albina
Muzafarova BETTA Security GmbH  Kristen Lackeos Max Planck Institute for Radio
Astronomy  David J. Champion Max Planck Institute for Radio Astronomy  Marta
Dembska German Aerospace Center (DLR), Institute of Data Science  Sirko
Schindler German Aerospace Center (DLR), Institute of Data Science  Marcus
Paradies Technical University of Ilmenau

###### Abstract

Data volumes and rates of research infrastructures will continue to increase
in the upcoming years and impact how we interact with their final data
products. Little of the processed data can be directly investigated and most
of it will be automatically processed with as little user interaction as
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We present PRAETOR111Pipeline pRovenance for Analysis, Evaluation, Trust Or
Reproducibility, a software suite that enables automated generation,
modelling, and analysis of provenance information of Python pipelines.
Furthermore, the evaluation of the pipeline performance, based upon a user
defined quality matrix in the provenance, enables the first step of machine
learning processes, where such information can be fed into dedicated
optimisation procedures.

Astronomical software (1855) — Software documentation(1869) —
Interdisciplinary astronomy(804)

##  1 Introduction

In a general sense, provenance information documents the history of a thing
whereas in case of data products, this concept has been adopted to document
the entire production history of the item itself. However, collecting data,
generating science-ready datasets and scientific results is a complex
procedure, including a broad range of performance measurements of the
infrastructure, expert knowledge, and domain-specific aspects in the data
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PRAETOR has been developed having astronomical use cases in mind (Johnson et
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analysing and validating the scientific results. Such workflows are mostly
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regulated processing and optimisation of production lines for science-ready
data products.

PRAETOR has been developed to collect provenance of workflows (in the
following we refer to pipelines as automated workflows), a user interface (UI)
that allows browsing through the provenance data, a database solution to query
and analyse the captured provenance, and the ability to concatenate provenance
information of individual pipeline executions.

##  2 Software

In the development of PRAETOR, we have investigated a way to capture
provenance from Python based pipelines and developed diagnostic tools to
evaluate the pipeline by utilising its provenance information. To fully
understand the inner workings of the astronomical pipeline described above,
one needs information on its input parameters, the original dataset, the
observatory metadata, the individual function calls and their parameters, and
the relation between all of these pieces of information. Such a set of
information is a complex structure and extracting these details is a
challenging task.

In order to document this information, we extended the standard language for
recording provenance (PROV) and its data model for things, processes, and
those responsible for processes as entities, activities, and agents,
respectively (Belhajjame et al., 2013). However, the main extensions to PROV
were in the form of attributes and were defined to represent specific
components that are common within Python pipelines such as: function names,
Python modules and versions thereof, and memory consumption of individual
processes. For analysis purposes, we implemented a quality metric attribute
which can be attached to any component within the pipeline as well as to the
pipeline itself. Based on the astronomical use case analysis (Johnson et al.,
2021), we have identified a number of items to extend the PROV model for
PRAETOR, but their implementation will be left to future work. Many of these
extensions are relations between different objects within the provenance, e.g.
whether an object was used as data or as parameter, or whether a process was
responsible for loading a specific object. However, the motivation for using a
minimally adapted PROV in the current release of PRAETOR was compatibility
with existing tools designed for PROV, such as the ProvToolBox and prov2neo.
In addition, the generated PRAETOR-based provenance is designed to be
interoperable with tools developed for other PROV-based provenance models,
such as those from IVOA (Servillat et al., 2020).

Extracting provenance information and the relation of the individual
operations can be an overwhelming task, in particular, if the pipelines are
unknown and treated as "black boxes". In order to obtain a first overview of
the available information a UI has been developed to browse through the
provenance data. Once a deeper understanding of the available information has
been acquired, queries and analysis on the captured provenance can be done via
database operations.

The UI provides a general overview of the used software packages, tools,
files, the processing time, basic memory consumption, and individual
functions. Information of the individual functions can be accessed on a
dedicated page that provides details on invocations of functions in the
pipeline and their in- and out-put parameters. To investigate the sequence of
function calls within the pipeline itself adjacent activities can be followed.

Apart from being a complex structure, provenance data can also grow to
substantial sizes and efficient processing can be a limiting issue. Therefore
the PRAETOR package provides a framework for uploading provenance to two
different kinds of database structures - graph databases Neo4j and triple
stores RDF/fuseki. A set of queries are available which extract key
information into pandas dataframes, such as: function invocations, inputs, and
outputs.

##  3 A first glimpse

The software suite has three main components: provenance generation/capturing,
provenance analysis and the UI. The generation and analysis components are
packaged within the same Python package, whereas the UI is deployed as a
series of Docker containers. The Python package is available on the pypi-hub
and can be installed using e.g. pip install praetor in either a virtual
environment or container alongside an existing pipeline. The information
captured by the package includes any imported modules, functions, file access,
and other core functionality to Python. Which of these pieces of information
is included can be defined in the praetor_settings_user.py settings file. The
UI can be installed by cloning the gitlab repository and following the
relevant installation instructions to start up its Docker environment. Once
running, the UI will be available via the local host in an web browser and the
queries are based on to the triple store queries within the PRAETOR analysis
package.

Various tutorials are available including a full n2n-example that explains how
to build a Singularity container and generate provenance of an example
pipeline. The example pipeline calls various functions, each having different
input and output parameters and are called either within the pipeline itself
or via a module. A subset of the provenance of the example pipeline is shown
in Figure LABEL:fig:ui. This information can also be obtained using the
database framework of PRAETOR. For this, we suggest to do the installation
within a virtual environment like Conda as explained in the n2n-example.

##  4 Conclusion

Provenance is of timely importance, by documenting the pathway through
scientific processing and by publishing FAIR (Findable, Accessible,
Interoperable, Reusable) data products (Wilkinson et al., 2016). We have
presented PRAETOR - a software suite for automated generation and analysis of
provenance information from Python based pipelines. We explained the rationale
for capturing provenance in a PROV extension and the decision to divide the
software into three stand-alone packages to operate on any Python pipeline. A
showcase of the analysis of generated provenance via the user interface and
the database framework has been presented.

The provenance information of multiple executions of a pipeline can be used to
characterise the input and output parameters. Pipelines that automatically
produce science-ready data sets, often have the problem of data
irreversibility, where the raw input data and intermediate results are not
stored for repeat analysis. PRAETOR provides the tools to evaluate such
pipelines and is the basis to any AI and neural network optimisation.

We acknowledge the support of the German Federal Ministry for Economic Affairs
and Energy on the basis of a decision of the German Bundestag under the
project number 50OO1905.

## References

  * Belhajjame et al. (2013) Belhajjame, K., B’Far, R., Cheney, J., et al. 2013, W3C Recommendation, 14, 15
  * Johnson et al. (2021) Johnson, M., Paradies, M., Dembska, M., et al. 2021, in TaPP 2021
  * Servillat et al. (2020) Servillat, M., Riebe, K., Boisson, C., et al. 2020
  * Wilkinson et al. (2016) Wilkinson, M. D., Dumontier, M., Aalbersberg, I. J., et al. 2016, Scientific Data, 3, doi: 10.1038/sdata.2016.18

Figure 1: Memory consumption of the processing of the pipeline over time.
Figure 2: plane_function call, showing the in- and output information and the
occurrence of that function in the pipeline.