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# Convergence of Tsirelson convolution systems of probability spaces

Remus Floricel  University of Regina, Department of Mathematics, Regina, SK,
Canada<EMAIL_ADDRESS>and  Patrick Melanson  University of Regina,
Department of Mathematics, Regina, SK, Canada<EMAIL_ADDRESS>
###### Abstract.

We associate two specific projective systems of probability spaces with any
Tsirelson convolution system. If the projective limits of these systems exist,
then we call the convolution system convergent and $K$-convergent,
respectively. It is shown that convergent convolution systems give rise to
continuous products of probability spaces, while $K$-convergent convolution
systems lead to flow systems. We investigate the relationship between
convergence and $K$-convergence, as well as their connections to two-parameter
product systems of Hilbert spaces.

###### Key words and phrases:

convolution systems, continuous product of probability spaces, flow systems,
subproduct and product system of Hilbert spaces

###### 2020 Mathematics Subject Classification:

Primary 28A33, 60A10, 60B15, 46L53, 46L55

##  1\. Introduction

Motivated by Arveson’s theory of product systems [Arveson89, Arveson-book], B.
Tsirelson initiated the study of two-parameter product systems of Hilbert
spaces in [Tsi03, Tsi04] (see also [TV]) through probabilistic methods,
including the analysis of stochastic processes and flows. Considering that
convolution semigroups of measures can naturally be used to construct
stochastic processes via Kolmogorov’s extension theorem, which in turn lead to
product systems, one of Tsirelson’s initial steps in his investigation was to
introduce the concept of a convolution system of standard probability spaces
as a generalization of convolution semigroups. However, unlike convolution
semigroups of measures, convolution systems may not always give rise to a
stochastic flow. This limitation primarily stems from the fact that the
category of probability spaces generally does not admit projective limits,
making any attempt to generalize Kolmogorov’s extension theorem quite
challenging [Bochner, Chol, Rao71]. When a convolution system leads to a
stochastic flow, or a flow system, as it is called in [Tsi04, Sec. 3b], the
flow system can then be directly used to construct a continuous product of
probability spaces (CPPS), largely due to the specific properties of the
standard spaces used in this construction (see [Tsi04, Sec. 3c]). This, in
turn, leads to a two-parameter product system of Hilbert spaces by
transitioning to the corresponding $L^{2}$-spaces.

The purpose of this note is to show that continuous products of (not
necessarily standard) probability spaces over a linearly ordered set can be
obtained from some convolution systems, even if they may not admit flow
systems. It also explores the interplay between convolution systems that admit
CPPS and those that admit flow systems. This is achieved by associating two
projective systems of probability spaces with a given convolution system (see
Proposition 3.1 and Proposition 4.1). If the projective limits of these
systems exist, they will be referred to as convergent and $K$-convergent,
respectively. In these cases, they give rise to a continuous product of
probability spaces and a flow system, respectively

We show in Theorem 3.3 that the product system of $L^{2}$-Hilbert spaces of
the projective CPPS associated with a convergent convolution system can be
described in terms of the subproduct system of $L^{2}$-Hilbert spaces of the
convolution system through an inductive limit construction. We also show in
Proposition 4.2 that any $K$-convergent system is convergent and find in
Theorem 4.5 the necessary and sufficient conditions for a convergent system to
be $K$-convergent.

At this end, we note that some of the techniques employed in this paper can be
regarded as the projective counterparts to the inductive techniques used in
[FK] within the context of $C^{*}$-algebras.

######  Notation 1.1.

All the notation used in this article is standard. The category
$\mathbf{Prob}$ of probability spaces consists of probability spaces
$(\Omega,\mathcal{F},\mu)$, where $\mathcal{F}$ is a $\sigma$-field of subsets
of $\Omega$ and $\mu$ is a probability measure on $\mathcal{F}$, with
morphisms
$T:(\Omega,\mathcal{F},\mu)\to(\Omega^{\prime},\mathcal{F}^{\prime},\mu^{\prime})$
given by measurable and measure-preserving transformations
$T:\Omega\to\Omega^{\prime}$. Any such morphism $T$ induces the Koopman
isometry $U_{T}:L^{2}(\mu^{\prime})\to L^{2}(\mu)$ by
$U_{T}(f)=f\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}T$
for $f\in L^{2}(\mu^{\prime})$. As usual, relations between sets and functions
are to be interpreted up to a set of measure zero.

##  2\. Background

###  2.1. Convolution systems of probability spaces and subproduct systems of
Hilbert spaces

In this subsection, we present the main objects of study in this article, all
of which were formally introduced by B. Tsirelson in [Tsi03, Tsi04]. It is
important to note that Tsirelson’s work assumes all probabilistic spaces to be
standard. To maintain the highest level of generality in this note, and in
line with the development of an ”uncountable” measure theory as proposed in
[Tao], we entirely drop this assumption. Additionally, we will work with
convolution systems over arbitrary linearly ordered sets instead of
$\mathbb{R}$, as suggested in [Tsi04]. Similarly, for the same reasons, we do
not automatically assume the separability of Hilbert spaces, which is an
important requirement in Arveson’s theory of product systems.

######  Definition 2.1.

(i) A convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
over a linearly ordered set $(\mathbb{S},\leq)$ consists of a family
$\\{(\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t})\,|\,s,\,t\in\mathbb{S},\,s<t\\}$
of probability spaces and a family
$\\{T_{r,s,t}\,|\,r,\,s,\,t\in\mathbb{S},\,r<s<t\\}$ of morphisms
$T_{r,s,t}:(\Omega_{r,s},\mathcal{F}_{r,s},\mu_{r,s})\times(\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t})\to(\Omega_{r,t},\mathcal{F}_{r,t},\mu_{r,t})$,
called the multiplication of the system, which is associative in the sense
that the diagram

(1)
${\Omega_{r,s}\times\Omega_{s,t}\times\Omega_{t,u}}$${\Omega_{r,t}\times\Omega_{t,u}}$${\Omega_{r,s}\times\Omega_{s,u}}$${\Omega_{r,u},}$$\scriptstyle{T_{r,s,t}\times\operatorname{id}_{\Omega_{t,u}}}$$\scriptstyle{\operatorname{id}_{\Omega_{r,s}}\times
T_{s,t,u}}$$\scriptstyle{T_{r,t,u}}$$\scriptstyle{T_{r,s,u}}$

commutes for all $r<s<t<u$ in $\mathbb{S}$.

(ii) A continuous product of probability spaces, abbreviated as CPPS, is a
convolution system whose multiplication consists of isomorphisms of
probability spaces.

(iii) A morphism of convolution systems, from
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
to
$\mathscr{S^{\prime}}=\\{\Omega_{s,t}^{\prime},\mathcal{F}_{s,t}^{\prime},\mu_{s,t},T_{r,s,t}^{\prime}\\}_{\mathbb{S}}$,
is a family $\theta=\\{\theta_{s,t}\\}_{s<t}$ of morphisms of probability
spaces
$\theta_{s,t}:(\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t})\to(\mathcal{F}_{s,t}^{\prime},\Sigma_{s,t}^{\prime},\mu_{s,t}^{\prime})$,
which makes the diagram

(2)
${\Omega_{r,s}\times\Omega_{s,t}}$${\Omega^{\prime}_{r,s}\times\Omega^{\prime}_{s,t}}$${\Omega_{r,t}}$${\Omega_{r,t}^{\prime}}$$\scriptstyle{\theta_{r,s}\times\theta_{s,t}}$$\scriptstyle{T_{r,s,t}}$$\scriptstyle{T_{r,s,t}^{\prime}}$$\scriptstyle{\theta_{r,t}}$

commutative, for all $r<s<t$ in $\mathbb{S}$

(iv) A flow system $((\Omega,\mathcal{F},P),\\{X_{s,t}\\})_{\mathscr{S}}$ over
a convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
consists of a probability space $(\Omega,\mathcal{F},P)$ and a family
$\\{X_{s,t}\,|\,s,\,t\in\mathbb{S},\,s<t\\}$ of morphisms of probability
spaces
$X_{s,t}:(\Omega,\mathcal{F},P)\to(\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t})$
that satisfies the following conditions:

  1. (1)

$\mathcal{F}$ is the joint of the $\sigma$-fields
$\mathcal{F}_{s,t}^{*}=X_{s,t}^{-1}(\mathcal{F}_{s,t})$, i.e., the
$\sigma$-field generated by the field
$\mathcal{F}_{0}=\bigcup_{s<t}\mathcal{F}_{s,t}^{*}$;

  2. (2)

$X_{t_{1},t_{2}}$, $X_{t_{1},t_{2}}$,…., $X_{t_{1},t_{2}}$ are independent,
for all $t_{1}<t_{2}<\cdots<t_{n}$ in $\mathbb{S}$;

  3. (3)

$X_{r,t}(\omega)=T_{r,s,t}(X_{r,s}(\omega),X_{s,t}(\omega))$,
$\omega\in\Omega$, for all $r<s<t$ in $\mathbb{S}$.

######  Example 2.1.

i) Let $(S,\mathcal{G})$ be a measurable semigroup, consisting of a semigroup
$S$ and a $\sigma$-field $\mathcal{G}$ on $S$ such that the semigroup
operation $(s,t)\mapsto st$ is $\mathcal{G}$-measurable. The convolution
$\mu\ast\nu$ of two probability measures $\mu$ and $\nu$ on $\mathcal{G}$ is
defined as the push-forward of the product measure $\mu\times\nu$ by the
semigroup operation. Suppose that $\mu$ is an indempotent probability measure
on $\mathcal{G}$, i.e., $\mu\ast\mu=\mu$. Then it gives rise to a trivial
convolution system
$\mathscr{S}_{\mu}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t}=\mu,T_{r,s,t}\\}_{\mathbb{S}}$
over any linearly ordered set $(\mathbb{S},\leq)$, where $\Omega_{s,t}=S$,
$\mathcal{F}_{s,t}=\mathcal{G}$ and $T_{r,s,t}$ is given by the semigroup
operation, for all $r<s<t$ in $\mathbb{S}$. We recall that the structure of
idempotent regular Borel probability measures on locally compact Hausdorff
second-countable topological semigroups is well-understood (see e.g. [HM, Th.
2.8]).

ii) More generally, let $\\{\mu_{t}\\}_{t\in\mathbb{R}}$ be a one-parameter
convolution semigroup of probability measures over a measurable semigroup
$(S,\mathcal{G})$. Then
$\mathscr{S}_{\\{\mu_{t}\\}}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{R}}$
is a convolution system over the set of all real numbers, where
$\Omega_{s,t}=S$, $\mathcal{F}_{s,t}=\mathcal{G}$, $\mu_{s,t}=\mu_{s-t}$, and
$T_{r,s,t}$ is given by the semigroup operation, for all real numbers $r<s<t$.

######  Observation 2.1.

Tsirelson’s method for constructing CPPSs from convolution systems of standard
probability spaces [Tsi04, Sect. 3c] does not directly involve the given
convolution system, but rather the flow system associated with it, if one
exists. Also, Tsirelson showed in [Tsi04] that a convolution system over
$\mathbb{S}=(\mathbb{R},\leq)$ admits a flow system if and only if it is
separable, in the sense of Definition 3b4 in [Tsi04]. For example, taking
$S=(\mathbb{Z}/m\mathbb{Z},+)$ and $\mu$ being the uniform distribution on
$\mathbb{Z}/m\mathbb{Z}$, the convolution system constructed as in Example 2.1
(i), based on this data, is not separable. On the other hand, taking
$S=(\mathbb{R},+)$, the system constructed in Example 2.1 (ii) is separable.

Convolution systems and CPPSs can be regarded as the probabilistic
counterparts to the notions of two-parameter subproduct systems and product
systems of Hilbert spaces, respectively. However, the former appear to be more
versatile than the latter. The concept of two-parameter product systems of
Hilbert spaces was first introduced by Tsirelson in [Tsi03] under the name
“local continuous product of Hilbert spaces”, obtained by excluding unbounded
intervals from the definition of a continuous product of Hilbert spaces.
Meanwhile, two-parameter subproduct systems were formally introduced in [FK]
under the name “Tsirelson subproduct system”, although the concept was well-
known long before their formal introduction.

######  Definition 2.2.

A two-parameter subproduct system of Hilbert spaces, or subproduct system for
short, $\mathscr{H}=\\{H_{s,t},U_{r,s,t}\\}_{\mathbb{S}}$ over a linearly
ordered set $(\mathbb{S},\leq)$ consists of a family
$\\{H_{s,t}\,|\,s,\,t\in\mathbb{S},\,s<t\\}$ of Hilbert spaces $H_{s,t}$ and a
family $\\{U_{r,s,t}\,|\,r,\,s,\,t\in\mathbb{S},\,r<s<t\\}$ of isometries
$U_{r,s,t}:H_{r,t}\to H_{r,s}\otimes H_{s,t}$ that satisfy the co-
associativity law

(3) $\displaystyle\left(1_{H_{r,s}}\otimes
U_{s,t,u}\right)U_{r,s,u}=\left(U_{r,s,t}\otimes 1_{H_{t,u}}\right)U_{r,t,u},$

for all $r<s<t<u$ in $\mathbb{S}$.

If the operators $U_{r,s,t}$ are all unitary operators, then
$\mathscr{H}=\\{H_{s,t},U_{r,s,t}\\}_{\mathbb{S}}$ will be referred to as a
two-parameter product system of Hilbert spaces, or product system for short.

The concept of morphism/isomorphism of subproduct systems is naturally
defined, similarly to that of morphism/isomorphism of convolution systems.

######  Observation 2.2.

Any convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
gives rise to the subproduct system

$L^{2}(\mathscr{S})=\\{L^{2}(\mu_{s,t}),\,U_{T_{r,s,t}}\\}_{\mathbb{S}},$

where $U_{T_{r,s,t}}$ is the Koopman isometry induced by $T_{r,s,t}$. If
$\mathscr{S}$ is a CPPS, then $L^{2}(\mathscr{S})$ is a product system.
Moreover, any morphism $\theta=\\{\theta_{s,t}\\}_{0<s<t}$ of convolution
systems, from $\mathscr{S}$ to $\mathscr{S}^{\prime}$, induces the morphism
$U_{\theta}=\\{U_{\theta_{s,t}}\\}_{0<s<t}$ of subproduct systems, from
$L^{2}(\mathscr{S}^{\prime})$ to $L^{2}(\mathscr{S})$.

###  2.2. Projective systems of probability spaces

In this sub-section, we briefly recall the notion of a projective system and
projective limit of probability spaces, which is central to the purpose of
this paper. For a comprehensive discussion of the subject, we refer the reader
to [Bochner, Rao71, Chol].

A projective system of probability spaces over a directed set $(I,\leq)$ is a
pair $\\{(\Omega_{i},\mathcal{F}_{i},\mu_{i})\\},T_{i,j}\\}$ consisting of a
family $\\{(\Omega_{i},,\mathcal{F}_{i},\mu_{i})\\}_{i\in I}$ of probability
spaces and a family $\\{T_{i,j}\\}_{i,\,j\in I,\,i\leq j}$ of morphisms of
probability spaces
$T_{i,j}:(\Omega_{j},\mathcal{F}_{j},\mu_{j})\to(\Omega_{i},\mathcal{F}_{i},\mu_{i})$,
$i\leq j$, satisfying the compatibility condition $T_{i,j}T_{j,k}=T_{i,k}$,
for all $i\leq j\leq k$, and $T_{i,i}=\operatorname{id}_{\Omega_{i}}$.

Let
$\Omega=\mathop{\oalign{lim\cr$\longleftarrow$\cr}}\\{\Omega_{i},T_{i,j}\\}$
be the projective limit of the projective system $\\{\Omega_{i},T_{i,j}\\}$
over $(I,\leq)$ in the category Set of sets, i.e., $\Omega$ is the subset of
the cartesian product $\Omega^{I}=\bigtimes_{i\in I}\Omega_{i}$ consisting of
those elements $\omega=\\{\omega_{i}\\}_{i\in I}$, called threads, such that
for each $i\leq j$ in $I$, $\omega_{i}=T_{i,j}(\omega_{j})$. We notice that
the set $\Omega$ can be empty, even in cases where the mappings $T_{i,j}$ are
surjective [Henkin, Waterhouse]. If $\Omega$ is non-empty and the coordinate
projections

$T_{i}:\Omega\ni\\{\omega_{j}\\}_{j\in I}\mapsto\omega_{i}\in\Omega_{i}$

are all surjective, then, using terminology from [Bochner], we say that the
Set-projective system $\\{\Omega_{i},T_{i,j}\\}$ is simply maximal. In this in
case, the mappings $T_{i,j}$ must also be all surjective.

Assuming further that the system $\\{\Omega_{i},T_{i,j}\\}$ is simply maximal,
we consider the field $\mathcal{F}_{0}=\bigcup_{i\in
I}T_{i}^{-1}(\mathcal{F}_{i})$ and the joint $\mathcal{F}=\bigvee_{i\in
I}T_{i}^{-1}(\mathcal{F}_{i})$ of the $\sigma$-fields
$T_{i}^{-1}(\mathcal{F}_{i})$. Let $\mu:\mathcal{F}_{0}\to[0,1]$ be the
finitely additive set function, defined uniquely by the equation

(4) $\displaystyle\mu(T_{i}^{-1}(A))=\mu_{i}(A),\;A\in\mathcal{F}_{i}.$

Generally, $\mu$ may fail to be $\sigma$-additive and thus cannot be extended
to $\mathcal{F}$.

######  Definition 2.3.

A simply maximal projective system of probability spaces
$\\{(\Omega_{i},\mathcal{F}_{i},\mu_{i}),T_{i,j}\\}$ is said to be convergent
if $\mu$ is $\sigma$-additive on $\mathcal{F}_{0}$.

If this is the case, then we shall use the same notation, $\mu$, to denote the
(unique) $\sigma$-additive extension of $\mu$ to $\mathcal{F}$. The resulting
probability space

$(\Omega,\mathcal{F},\mu):=\mathop{\oalign{lim\cr$\longleftarrow$\cr}}_{I}\\{(\Omega_{i},\mathcal{F}_{i},\mu_{i}),T_{i,j}\\}$

is called the projective limit of
$\\{(\Omega_{i},\mathcal{F}_{i},\mu_{i}),T_{i,j}\\}$.

Numerous criteria for the existence of the projective limit have been
identified (see [Rao71, Chol, Frol] and the references therein); in this
paper, we employ a simple yet effective one for our purposes. If there exists
an “upper bound”, consisting on a probability space $(N,\mathcal{N},\nu)$ and
a family $\\{S_{i}\\}_{i\in I}$ of epimorphisms
$S_{i}:(N,\mathcal{N},\nu)\to(\Omega_{i},\mathcal{F}_{i},\mu_{i})$ such that
$\mathcal{N}=\bigvee_{i\in I}S_{i}^{-1}(\mathcal{F}_{i})$ and
$T_{i,j}S_{j}=S_{i}$, for all $i,\,j\in I$ with $i\leq j$ then the set-
function $\mu$ in (4) must be $\sigma$-additive, and thus the projective limit
exists. From this perspective, the projective limit is the least upper bound
$(N,\mathcal{N},\nu)$ with the aforementioned properties. Consequently, it is
unique (up to isomorphism) and possesses the property that
$(\Omega,\mathcal{F},\mu)=\mathop{\oalign{lim\cr$\longleftarrow$\cr}}_{I_{0}}\\{(\Omega_{i},\mathcal{F}_{i},\mu_{i}),T_{i,j}\\}$,
for any cofinal subset $I_{0}$ of $(I,\leq)$.

###  2.3. Finite Partitions

Let $(\mathbb{S},\leq)$ be a linearly ordered set. For any two elements
$s,\,t\in\mathbb{S}$, $s<t$, consider the partially ordered set
$(\mathcal{K}_{s,t},\subseteq)$ of all finite partitions of the interval
$[s,t]$, ordered by inclusion. We notice that if
$I=\\{s=\iota_{0}<\iota_{1}<\iota_{2}<\,\dots<\iota_{m}<\iota_{m+1}=t\\}\in\mathcal{K}_{s,t}$,
then any refinement $J\in\mathcal{K}_{s,t}$ of $I$, i.e., $I\subseteq J$, can
be written as

(5) $\displaystyle J=I_{0}\cup I_{1}\cup\cdots\cup I_{m},$

where $I_{i}=\\{j\in J,\iota_{i}\leq
j\leq\iota_{i+1}\\}=\\{\iota_{i}=\iota_{i_{0}}<\iota_{i_{1}}<\dots<\iota_{i_{n_{I_{i}}}}<\iota_{i+1}\\}\in\mathcal{K}_{\iota_{i},\iota_{i+1}},$
for some $n_{I_{i}}\in\mathbb{N}$ depending on the partition $I_{i}$, for all
$0\leq i\leq m$.

We also consider the set $\mathcal{K}=\bigcup_{s<t}\mathcal{K}_{s,t}$ of all
finite subsets $I\subset\mathbb{S}$, $|I|\geq 2$, ordered by inclusion as
well. As above, if $I\in\mathcal{K}_{s,t}$ and $J\in\mathcal{K}$ is a
refinement of $I$, then $J$ can be written as

(6) $\displaystyle J={I}_{L}\cup\widetilde{I}\cup{I}_{R},$

where ${I}_{L}=\\{j\in J\mid j\leq\iota_{0}\\}$, ${I}_{R}=\\{j\in J\mid
j\geq\iota_{m+1}\\}$ and $\widetilde{I}=I_{0}\cup\cdots\cup
I_{m}\in\mathcal{K}_{s,t}$, with $I_{k}$ defined as above. Note that
$\widetilde{I}$ is a refinement of $I$.

##  3\. Convergent convolution systems and associated CPPSs

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a convolution system. For any two elements $s,\,t\in\mathbb{S}$, $s<t$, and
any partition $I\in\mathcal{K}_{s,t}$,
$I=\\{s=\iota_{0}<\iota_{1}<\iota_{2}<\,\dots<\iota_{m}<\iota_{m+1}=t\\},$ we
consider the product measure space $(\Omega_{I},\mathcal{F}_{I},\mu_{I})$ over
$I$, where
$\Omega_{I}=\Omega_{\iota_{0},\iota_{1}}\times\Omega_{\iota_{1},\iota_{2}}\times\dots\times\Omega_{\iota_{m},\iota_{m+1}}$,
$\mathcal{F}_{I}=\mathcal{F}_{\iota_{0},\iota_{1}}\otimes\mathcal{F}_{\iota_{1},\iota_{2}}\otimes\dots\otimes\mathcal{F}_{\iota_{m},\iota_{m+1}},$
and
$\mu_{I}=\mu_{\iota_{0},\iota_{1}}\times\mu_{\iota_{1},\iota_{2}}\times\dots\times\mu_{\iota_{m},\iota_{m+1}}.$

For any refinement $J\in\mathcal{K}_{s,t}$ of $I$, we also consider the
mapping $T_{I,J}:\Omega_{J}\to\Omega_{I}$ defined as follows:

(i) If $I$ is the trivial partition, $I=\\{s,t\\}$, and
$J=\\{s=j_{0}<j_{1}<j_{2}<\,\dots<j_{n}<j_{n+1}=t\\},$ then
$T_{I,J}:\Omega_{J}\to\Omega_{s,t}$ is defined iteratively as

$\displaystyle T_{I,J}=\begin{cases}T_{j_{0},j_{1},j_{2}}&\,\mbox{if}\,n=1\\\
T_{j_{0},j_{n},j_{n+1}}(T_{\\{j_{0},j_{n}\\},J\setminus\\{j_{n+1}\\}}\times\operatorname{id}_{\Omega_{j_{n},j_{n+1}}})&\,\mbox{if}\,n\geq
2\end{cases}$

where $J\setminus\\{j_{n+1}\\}\in\mathcal{K}_{s,j_{n}}$ is the partition
obtained by removing the endpoint $t=j_{n+1}$ from $J$.

(ii) If
$I=\\{s=\iota_{0}<\iota_{1}<\iota_{2}<\,\dots<\iota_{m}<\iota_{m+1}=t\\},$
$I\subseteq J$, are arbitrary partitions, then by writing $J$ it as in (5), we
define

$\displaystyle T_{I,J}=T_{\\{\iota_{0},\iota_{1}\\},I_{0}}\times
T_{\\{\iota_{1},\iota_{2}\\},I_{1}}\times\dots
T_{\\{\iota_{m},\iota_{m+1}\\},I_{m}},$

where
$T_{\\{\iota_{k},\iota_{k+1}\\},I_{k}}:\Omega_{I_{k}}\to\Omega_{\iota_{k},\iota_{k+1}}$
are as in (i).

(iii) If $I=J$, we set $T_{I,I}=\operatorname{id}_{\Omega_{I}}$.

The construction carried out above ensures that
$T_{I,J}:(\Omega_{J},\mathcal{F}_{J},\mu_{J})\to(\Omega_{I},\mathcal{F}_{I},\mu_{I})$
is a morphism of probability spaces, for all $I,\,J\in\mathcal{K}_{s,t}$,
$I\subseteq J$. The newly created system is projective, as shown below.

######  Proposition 3.1.

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a convolution system. Then the system
$(\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I})\\},\\{T_{I,J}\\})$ is a projective
system of probability spaces over the directed set
$(\mathcal{K}_{s,t},\subseteq)$, for all $s<t$ in $\mathbb{S}$.

###### Proof.

Let $s<t$ be two elements of $\mathbb{S}$. We show that
$T_{I,K}=T_{I,J}T_{J,K}$, for all partitions $I,\,J,\,K\in\mathcal{K}_{s,t}$,
$I\subseteq J\subseteq K$. If $I=\\{s,t\\}$ is the trivial partition, then the
identity can be deduced easily. For a general partition
$I=\\{s=\iota_{0}<\iota_{1}<\cdots<\iota_{m}<\iota_{m+1}=t\\}$, we use (5) to
decompose $J$ in terms of $I$ as $J=I_{0}\cup I_{1}\cup\cdots\cup I_{m}$,
where $I_{i}\in\mathcal{K}_{\iota_{i},\iota_{i+1}}$, for all $0\leq i\leq m$.
Similarly, we can decompose $K$ in terms of $J$, thus $K=J_{0}\cup
J_{1}\cup\cdots\cup J_{\ell}$, where
$J_{0}\in\mathcal{K}_{\iota_{0_{0}},\iota_{0_{1}}}$,
$J_{1}\in\mathcal{K}_{\iota_{0_{1}},\iota_{0_{2}}}$, $\cdots$,
$J_{\ell}\in\mathcal{K}_{\iota_{m_{n_{I_{m}}}},\iota_{m+1}}$. However, we can
also decompose $K$ in terms of $I$. Explicitly, we get $K=I^{\prime}_{0}\cup
I^{\prime}_{1}\cup\cdots\cup I^{\prime}_{m}$, where
$I^{\prime}_{0}=\bigcup\limits_{i=0}^{n_{I_{0}}}J_{i}$ and
$I^{\prime}_{i}=\bigcup\limits_{j=n_{I_{i-1}+1}}^{n_{I_{i}}}J_{j}$ for all
$1\leq i\leq m$. Via this new decomposition, we have that
$T_{I_{0},I^{\prime}_{0}}=T_{\\{\iota_{0_{0}},\iota_{0_{1}}\\},J_{0}}\times
T_{\\{\iota_{0_{1}},\iota_{0_{2}}\\},J_{1}}\times\cdots\times
T_{\\{\iota_{0_{n_{I_{0}}}},\iota_{1}\\},J_{n_{I_{0}}}}$, and
$T_{I_{k},I^{\prime}_{k}}=T_{\\{\iota_{k_{0}},\iota_{k_{1}}\\},J_{n_{I_{k-1}}+1}}\times
T_{\\{\iota_{k_{1}},\iota_{k_{2}}\\},J_{n_{I_{k-1}}+2}}\times\cdots\times
T_{\\{\iota_{k_{n_{I_{k}}}},\iota_{k+1}\\},J_{n_{I_{k}}}}$, for all $1\leq
k\leq m$, so $T_{J,K}=\bigtimes_{k=0}^{m}T_{I_{k},I_{k}^{\prime}}$.
Consequently,

$T_{I,K}=\bigtimes_{k=0}^{m}T_{\\{\iota_{k},\iota_{k+1}\\},I_{k}^{\prime}}=\bigtimes_{k=0}^{m}T_{\\{\iota_{k},\iota_{k+1}\\},I_{k}}T_{I_{k},I_{k}^{\prime}}=T_{I,J}T_{J,K}$

as required. $\hfill\square$

######  Definition 3.1.

A convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
is said to be convergent if for any two elements $s<t$ of $\mathbb{S}$, the
projective system $(\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I})\\},\\{T_{I,J}\\})$
over $(\mathcal{K}_{s,t},\subseteq)$ is convergent.

We denote by
$(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})$ the
projective limit,

$(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})=\varprojlim_{\mathcal{K}_{s,t}}\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),T_{I,J}\\},$

of the convergent projective system
$\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),T_{I,J}\\}$ over
$(\mathcal{K}_{s,t},\subseteq)$, and by
$T^{\flat}_{I}:(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})\to(\Omega_{I},\mathcal{F}_{I},\mu_{I})$
the canonical epimorphism (coordinate projection), for all
$I\in\mathcal{K}_{s,t}$; they satisfy the compatibility relation
$T_{I}^{\flat}=T_{I,J}T_{J}^{\flat}$, for all $I,\,J\in\mathcal{K}_{s,t}$,
$I\subseteq J$.

The following result allows the construction of a CPPS from a convergent
convolution system.

######  Theorem 3.2.

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a convergent convolution system. For any $r<s<t$ in $\mathbb{S}$, there
exists an isomorphism of probability spaces

$T^{\flat}_{r,s,t}:(\Omega^{\flat}_{r,s},\mathcal{F}_{r,s}^{\flat},\mu_{r,s})\times(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})\to(\Omega^{\flat}_{r,t},\mathcal{F}_{r,t}^{\flat},\mu_{r,t}^{\flat})$

that makes the system
$\mathscr{S}^{\flat}=\\{\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat},T_{r,s,t}^{\flat}\\}_{\mathbb{S}}$
into a CPPS.

###### Proof.

Let $r,\,t\in\mathbb{S}$ with $r<t$. For any $s\in\mathbb{S}$ satisfying
$r<s<t$, let $\mathcal{K}_{r,s,t}$ be the set of all partitions
$I\in\mathcal{K}_{r,t}$ such that $s\in I$. Since $\mathcal{K}_{r,s,t}$ is a
cofinal subset of $(\mathcal{K}_{r,t},\subseteq)$, we have that
$(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})$ is the
projective limit of the convergent projective system
$\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),T_{I,J}\\}$ over
$(\mathcal{K}_{r,s,t},\subseteq)$. We freely identify the poset
$(\mathcal{K}_{r,s,t},\subseteq)$ with
$\mathcal{K}_{r,s}\times\mathcal{K}_{s,t}$, considered with the product order,
via the order isomorphism $\mathcal{K}_{r,s,t}\ni I=I_{s}\cup{}_{s}I\mapsto
I_{s}\times{}_{s}I\in\mathcal{K}_{r,s}\times\mathcal{K}_{s,t}$. Given that
$(\Omega_{I},\mathcal{F}_{I},\mu_{I})=(\Omega_{I_{s}},\mathcal{F}_{I_{s}},\mu_{I_{s}})\times(\Omega_{{}_{s}I},\mathcal{F}_{{}_{s}I},\mu_{{}_{s}I})$,
for all $I\in\mathcal{K}_{r,s,t}$, and $T_{I,J}=T_{I_{s},J_{s}}\times
T_{{}_{s}I,{}_{s}J}$, for all $I\subseteq J$ in $\mathcal{K}_{r,t}$, we infer
that the product probability space
$(\Omega^{\flat}_{r,s},\mathcal{F}_{r,s}^{\flat},\mu_{r,s}^{\flat})\times(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})$
is the least upper bound of the convergent projective system
$\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),T_{I,J}\\}$ over
$(\mathcal{K}_{s,t},\subseteq)$. Consequently, there exists an isomorphism of
probability spaces
$T^{\flat}_{r,s,t}:(\Omega^{\flat}_{r,s},\mathcal{F}_{r,s}^{\flat},\mu_{r,s}^{\flat})\times(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})\to(\Omega^{\flat}_{r,t},\mathcal{F}_{r,t}^{\flat},\mu_{r,t}^{\flat})$
such that

(7) $\displaystyle T^{\flat}_{I}T^{\flat}_{r,s,t}=T^{\flat}_{I_{s}}\times
T^{\flat}_{{}_{s}I}$

for all $I=I_{s}\cup{}_{s}I\in\mathcal{K}_{r,s,t}$.

It remains to show that the family $\\{T_{r,s,t}^{\flat}\\}$ thus constructed
is associative. Let then $r<s<t<u$ be some elements of $\mathbb{S}$.
Identifying, as above, the poset $(\mathcal{K}_{r,s,t,u},\subseteq)$ of all
partitions $I\in\mathcal{K}_{r,u}$ such that $s,\,t\in I$ with the product
poset $\mathcal{K}_{r,s}\times\mathcal{K}_{s,t}\times\mathcal{K}_{t,u}$ via
$I=I_{s}\cup{}_{s}I_{t}\cup{}_{t}I\mapsto
I_{s}\times{}_{s}I_{t}\times{}_{t}I$, we have

$\displaystyle
T^{\flat}_{I}T^{\flat}_{r,s,u}(\operatorname{id_{\Omega_{r,s}^{\flat}}}\times
T^{\flat}_{s,t,u})$ $\displaystyle=$ $\displaystyle(T^{\flat}_{I_{s}}\times
T^{\flat}_{{}_{s}I_{t}\cup{}_{t}I})(\operatorname{id_{\Omega_{r,s}^{\flat}}}\times
T^{\flat}_{s,t,u})$ $\displaystyle=$ $\displaystyle T^{\flat}_{I_{s}}\times
T^{\flat}_{{}_{s}I_{t}}\times T^{\flat}_{I_{t}}.$

Similarly,
$T_{I}^{\flat}T^{\flat}_{r,t,u}(T^{\flat}_{r,s,t}\times\operatorname{id}_{\Omega_{t,u}^{\flat}})=T^{\flat}_{I_{s}}\times
T^{\flat}_{{}_{s}I_{t}}\times T^{\flat}_{I_{t}}$, and the conclusion follows.
$\hfill\square$

######  Definition 3.2.

The system
$\mathscr{S}^{\flat}=\\{\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat},T_{r,s,t}^{\flat}\\}_{\mathbb{S}}$
will be referred as the projective CPPS associate with $\mathscr{S}$.

If
$T^{\flat}_{s,t}:(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t})\to(\Omega_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t})$
is the canonical epimorphism corresponding to the trivial partition
$I=\\{s,t\\}$, then the resulting family
$\tau_{\mathscr{S}}=\\{T^{\flat}_{s,t}\\}_{s<t}$ is an epimorphism of
convolution systems from $\mathscr{S}^{\flat}$ to $\mathscr{S}$. The following
result follows immediately from Theorem 3.2 and from the properties of the
projective limit.

######  Corollary 3.2.1.

If $\theta:\mathscr{S}_{1}\to\mathscr{S}_{2}$ is an isomorphism of convergent
convolution systems, then there exists an isomorphism of CPPSs
$\theta^{\flat}:\mathscr{S}_{1}^{\flat}\to\mathscr{S}_{2}^{\flat}$ such that
$\theta\tau_{\mathscr{S}_{1}}=\tau_{\mathscr{S}_{2}}\theta^{\flat}$.

We end this section by noticing that the product system of Hilbert spaces
$L^{2}(\mathscr{S}^{\flat})$ of the projective CPPS $\mathscr{S}^{\flat}$
associated with a convergent convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
can be described in terms of the subproduct system of Hilbert spaces
$L^{2}(\mathscr{S})$ trough an inductive limit construction. Indeed, for all
$s<t$ in $\mathbb{S}$, let

$H_{s,t}=\mathop{\oalign{lim\cr$\longrightarrow$\cr}}_{I\in\mathcal{K}_{s,t}}L^{2}(\mu_{I})$

be the inductive limit of the inductive system of Hilbert spaces
$(L^{2}(\mu_{I}),U_{T_{I,J}})$ over $(\mathcal{K}_{s,t},\subseteq)$ with
connecting isometries $V_{I}:L^{2}(\mu_{I})\to H_{s,t}$,
$I\in\mathcal{K}_{s,t}$ satisfying the compatibility relation
$V_{J}U_{T_{I,J}}=V_{I}$ for all $I\subseteq J$ in $\mathcal{K}_{s,t}$.
Arguing as in Theorem 3.2, one can find a unique unitary operator
$U_{r,s,t}:H_{r,t}\rightarrow H_{r,s}\otimes H_{s,t}$ that satisfies the
identity $U_{r,s,t}V_{I}=V_{I_{s}}\otimes V_{{}_{s}I}$, for all
$I=I_{s}\cup{}_{s}I\in\mathcal{K}_{r,s,t}$. The system

$\mathscr{H}=\\{H_{s,t},U_{r,s,t}\\}_{\mathbb{S}}$

is a product system of Hilbert spaces. We then have:

######  Theorem 3.3.

If
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
is a convergent convolution system, then the product systems of Hilbert spaces
$\mathscr{H}=\\{H_{s,t},U_{r,s,t}\\}_{\mathbb{S}}$ and
$L^{2}(\mathscr{S}^{\flat})$ are isomorphic.

###### Proof.

Because $U_{T_{J}^{\flat}}U_{T_{I,J}}=U_{T_{I}^{\flat}}$, for all $I\subseteq
J$, $I,\mathcal{,}J\in\mathcal{K}_{s,t}$, and the set
$\bigcup_{I\in\mathcal{K}_{s,t}}U_{T_{I}^{\flat}}(L^{2}(\mu_{I}))$ is
everywhere dense in $L^{2}(\mu_{s,t}^{\flat})$, it follows from the universal
property of the inductive limit that there exists a unique unitary operator
$\theta_{s,t}:H_{s,t}\to L^{2}(\mu_{s,t}^{\flat})$ such that
$\theta_{s,t}V_{I}=U_{T^{\flat}_{I}}$, for all $I\in\mathcal{K}_{s,t}$.
Moreover, since

$\displaystyle(\theta_{r,s}\otimes\theta_{s,t})U_{r,s,t}V_{I}$
$\displaystyle=$
$\displaystyle\theta_{r,s}V_{I_{s}}\otimes\theta_{s,t}V_{{}_{s}I}=U_{T^{\flat}_{I_{s}}}\otimes
U_{T^{\flat}_{{}_{s}I}}$ $\displaystyle=$ $\displaystyle
U_{T^{\flat}_{r,s,t}}U_{T^{\flat}_{I}}=U_{T^{\flat}_{r,s,t}}\theta_{r,t}V_{I},$

for all $I=I_{s}\cup{}_{s}I\in\mathcal{K}_{r,s,t}$, and $r<s<t$ in
$\mathbb{S}$, we deduce that
$(\theta_{r,s}\otimes\theta_{s,t})U_{r,s,t}=U_{T^{\flat}_{r,s,t}}\theta_{r,t}$,
i.e., $\\{\theta_{s,t}\\}$ is an isomorphism of product
systems.$\hfill\square$

##  4\. $K$-convergent convolution systems and associated flow systems

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a convolution system. Consider two partitions $I,\,J\in\mathcal{K}$,
$I\subseteq J$, and write $J={I}_{L}\cup\widetilde{I}\cup{I}_{R}$ as in (6).
We set

$\displaystyle X_{I,J}=\begin{cases}T_{I,J}&\text{if
}I,J\in\mathcal{K}_{s,t},\,\text{for some}\,s<t\,\text{in}\;\mathbb{S}\\\
(T_{I,\widetilde{I}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\pi_{\widetilde{I},J})&\text{otherwise}\end{cases}$

where $\pi_{\widetilde{I},J}:\Omega_{J}\to\Omega_{\widetilde{I}}$ is the
coordinate projection. Then
$X_{I,J}:(\Omega_{J},\mathcal{F}_{J},\mu_{J})\to(\Omega_{I},\mathcal{F}_{I},\mu_{I})$
is a morphism of probability spaces. Moreover:

######  Proposition 4.1.

The system $\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),X_{I,J}\\}$ is a
projective system of probability spaces over the directed set
$(\mathcal{K},\subseteq)$.

###### Proof.

The compatibility relation $X_{I,J}X_{J,K}=X_{I,K}$, for all
$I,\,J,\,K\in\mathcal{K}$ satisfying $I\subseteq J\subseteq K$, is verified
using the same approach as in Proposition 3.1. For this, suppose
$I\in\mathcal{K}_{q,r}$, $J\in\mathcal{K}_{s,t}$, and $K\in\mathcal{K}_{u,v}$.
where $u<s<q<r<t<v$. Following (6), we decompose $J$ with respect to $I$ as
$J={I}_{L}\cup\widetilde{I}\cup{I}_{R}$, $K$ with respect to $J$ as
$K={J}_{L}\cup\widetilde{J}\cup{J}_{R}$, and $K$ with respect to $I$ as
$K={I}_{L}^{\prime}\cup\widetilde{I^{\prime}}\cup{I}_{R}^{\prime}.$ Note that
the partitions $I,\widetilde{I},\widetilde{I^{\prime}}\in\mathcal{K}_{q,r}$
satisfy $I\subseteq\widetilde{I}\subseteq\widetilde{I^{\prime}}$, and the
partitions $J,\widetilde{J}\in\mathcal{K}_{s,t}$ satisfy
$J\subseteq\widetilde{J}$. It then follows that
$\widetilde{I^{\prime}}\subseteq\widetilde{J},$ and decomposition (6) of
$\widetilde{J}$ with respect to $\widetilde{I^{\prime}}$ is given by
$\widetilde{J}=(\widetilde{J}\cap[u,s])\cup\widetilde{I^{\prime}}\cup(\widetilde{J}\cap[t,v])$.
Using this, we have

$\displaystyle\pi_{\widetilde{I},J}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T_{J,\widetilde{J}})$
$\displaystyle=$
$\displaystyle\pi_{\widetilde{I},J}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T_{J\cap[u,s],\widetilde{J}\cap[u,s]}\times
T_{J\cap[s,t],\widetilde{J}\cap[s,t]}\times
T_{J\cap[t,v],\widetilde{J}\cap[t,v]})$ $\displaystyle=$
$\displaystyle\pi_{\widetilde{I},J}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T_{I_{L},\widetilde{J}\cap[u,s]}\times
T_{\widetilde{I},\widetilde{I^{\prime}}}\times
T_{I_{R},\widetilde{J}\cap[t,v]})$ $\displaystyle=$
$\displaystyle(T_{\widetilde{I},\widetilde{I^{\prime}}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}\pi_{\widetilde{I^{\prime}},\widetilde{J}}.$

Therefore

$\displaystyle X_{I,J}X_{J,K}$ $\displaystyle=$
$\displaystyle(T_{I,\widetilde{I}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\pi_{\widetilde{I},J})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T_{J,\widetilde{J}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\pi_{\widetilde{J},K})=(T_{I,\widetilde{I}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T_{\widetilde{I},\widetilde{I^{\prime}}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\pi_{\widetilde{I^{\prime}},\widetilde{J}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\pi_{\widetilde{J},K})$
$\displaystyle=$ $\displaystyle X_{I,K},$

as needed. $\hfill\square$

######  Definition 4.1.

A convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
is said to be $K$-convergent if the projective system
$\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),X_{I,J}\\}$ over
$(\mathcal{K},\subseteq)$ is convergent.

If
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
is $K$-convergent, then we denote by
$(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})$ the projective limit of this
projective system,

$(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})=\varprojlim_{\mathcal{K}}\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),X_{I,J}\\},$

and by
$X_{I}:(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})\to(\Omega_{I},\mathcal{F}_{I},\mu_{I})$
the canonical epimorphism; it satisfies the compatibility relation
$X_{I}=X_{I,J}X_{J}$, for all $I,\,J\in\mathcal{K}$, $I\subseteq J$. For a
trivial partition $I=\\{s<t\\}$, we simply write $X_{s,t}$ instead of
$X_{\\{s,t\\}}$. Using the properties of the projective limit, one can readily
see that

$((\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat}),\\{X_{s,t}\\})_{\mathscr{S}}$

is a flow system over $\mathscr{S}$. In addition, we have the following.

######  Proposition 4.2.

Any $K$-convergent convolution system is convergent.

###### Proof.

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a $K$-convergent convolution system, and $s<t$ be two fixed elements in
$\mathbb{S}$. It is clear that the Set-projective system
$\\{\Omega_{I},T_{I,J}\\}$ over $(\mathcal{K}_{s,t},\subseteq)$ is simply
maximal. Furthermore, if $\pi_{s,t}:\Omega^{\flat}\to\Omega_{s,t}^{\flat}$ is
the coordinate projection onto $\Omega^{\flat}$, then
$T^{\flat}_{I}\pi_{s,t}=X_{I}$, for all $I\in\mathcal{K}_{s,t}$, so
$(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})$ is an upper bound of
$\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),T_{I,J}\\}$. Therefore the the
projective system $\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),T_{I,J}\\}$ over
$(\mathcal{K}_{s,t},\subseteq)$ must be convergent. $\hfill\square$

Next, we focus on finding a necessary and sufficient condition for a
convergent convolution system to be $K$-convergent. For this purpose, we start
with a convergent convolution system
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$,
and let
$\mathscr{S}^{\flat}=\\{\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat},T_{r,s,t}^{\flat}\\}_{\mathbb{S}}$
be the projective CPPS associate with $\mathscr{S}$. For any elements
$u<s<t<v$ in $\mathbb{S}$, let
$T^{\flat}_{(s,t),(u,v)}:\Omega^{\flat}_{u,v}\to\Omega^{\flat}_{s,t}$ be the
mapping

$T^{\flat}_{(s,t),(u,v)}=\pi_{(s,t)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,t,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{u,s,v})^{-1},$

where
$\pi_{(s,t)}:\Omega^{\flat}_{u,s}\times\Omega^{\flat}_{s,t}\times\Omega^{\flat}_{t,v}\to\Omega^{\flat}_{s,t}$
is the coordinate projection onto $\Omega^{\flat}_{s,t}$. If $u=s$, and/or
$t=v$, then $T^{\flat}_{(s,t),(u,v)}$ is defined accordingly. It is clear that
$T^{\flat}_{(s,t),(u,v)}:(\Omega^{\flat}_{u,v},\mathcal{F}_{u,v}^{\flat},\mu_{u,v}^{\flat})\to(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})$
is an epimorphism of probability spaces. Furthermore, we have:

######  Lemma 4.3.

For any elements $u<s<t<v$ in $\mathbb{S}$ and partitions
$I\in\mathcal{K}_{s,t}$ and $J\in\mathcal{K}_{u,v}$ we have
$T^{\flat}_{I}T^{\flat}_{(s,t),(u,v)}=X_{I,J}T^{\flat}_{J}$.

###### Proof.

By decomposing $J=I_{L}\cup\widetilde{I}\cup I_{R}$ as in (6) , we have

$\displaystyle T^{\flat}_{I}T^{\flat}_{(s,t),(u,v)}$ $\displaystyle=$
$\displaystyle
T^{\flat}_{I}\,\pi_{(s,t)}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,t,v})^{-1}(T^{\flat}_{u,s,v})^{-1}$ $\displaystyle=$
$\displaystyle
T_{I,\widetilde{I}}T^{\flat}_{\widetilde{I}}\pi_{(s,t)}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,t,v})^{-1}(T^{\flat}_{u,s,v})^{-1}$ $\displaystyle=$
$\displaystyle
T_{I,\widetilde{I}}\pi_{\widetilde{I},J}(T^{\flat}_{I_{R}}\times
T^{\flat}_{\widetilde{I}}\times
T^{\flat}_{I_{L}})(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,t,v})^{-1}(T^{\flat}_{u,s,v})^{-1}$ $\displaystyle=$
$\displaystyle X_{I,J}(T^{\flat}_{I_{R}}\times T^{\flat}_{\widetilde{I}}\times
T^{\flat}_{I_{L}})(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,t,v})^{-1}(T^{\flat}_{u,s,v})^{-1}$
$\displaystyle\stackrel{{\scriptstyle(\ref{sf})}}{{=}}$ $\displaystyle
X_{I,J}T^{\flat}_{J},$

as needed. $\hfill\square$

######  Proposition 4.4.

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a convergent convolution system. The system
$(\\{(\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})\\},T_{(s,t),(u,v)})$
is a projective system of probability spaces over the partially ordered set of
ordered pairs $\mathcal{C}=\\{(s,t)\,|\,s,\,t\in\mathbb{S},\,s\leq t\\}$ with
partial order $(s,t)\subseteq(u,v)$ if $u\leq s\leq t\leq v$.

###### Proof.

The compatibility relation
$T^{\flat}_{(q,r),(s,t)]}T^{\flat}_{(s,t),(u,v)}=T^{\flat}_{(q,r),(u,v)}$, for
all $(q,r)\subseteq(s,t)\subseteq(u,v)$, can be checked by using the laws of
associativity. The process is, however, lengthy and convoluted. We will
highlight the key intermediate steps and leave the detailed completion to the
reader.

$\displaystyle T^{\flat}_{(q,r),(s,t)}T^{\flat}_{(s,t),(u,v)}$
$\displaystyle=$
$\displaystyle\pi_{(q,r)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{s,q,r}\times\operatorname{id}_{\Omega^{\flat}_{r,t}})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{s,r,t})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}$
$\displaystyle\pi_{(s,t)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,t,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{u,s,v})^{-1}$
$\displaystyle=$
$\displaystyle\pi_{(q,r)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{s,q,r}\times\operatorname{id}_{\Omega^{\flat}_{r,t}})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{s,r,t})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}$
$\displaystyle
T^{\flat}_{s,r,t}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}\pi_{(s,r)\times(r,t)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times\operatorname{id}_{\Omega^{\flat}_{s,r}}\times
T_{r,t,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}$
$\displaystyle(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,r,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{u,s,v})^{-1}$
$\displaystyle=$
$\displaystyle\pi_{(q,r)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times\operatorname{id}_{\Omega^{\flat}_{s,q}}\times\operatorname{id}_{\Omega^{\flat}_{q,r}}\times
T_{r,t,v}^{\flat})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}$
$\displaystyle(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,q,r}\times\operatorname{id}_{\Omega^{\flat}_{r,v}})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,r,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{u,s,v})^{-1}$
$\displaystyle=$
$\displaystyle\pi_{(q,r)}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{u,s,q}\times\operatorname{id}_{\Omega^{\flat}_{q,r}}\times\operatorname{id}_{\Omega^{\flat}_{r,v}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,q,r}\times\operatorname{id}_{\Omega^{\flat}_{r,v}})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}$
$\displaystyle(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,r,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(T^{\flat}_{u,s,v})^{-1}$

where
$\pi_{(s,r)\times(r,t)}:\Omega^{\flat}_{u,s}\times\Omega^{\flat}_{s,r}\times\Omega^{\flat}_{r,t}\times\Omega^{\flat}_{t,v}\to\Omega^{\flat}_{s,r}\times\Omega^{\flat}_{r,t}$
is the coordinate projection. Noticing that
$(\operatorname{id}_{\Omega^{\flat}_{u,q}}\times\circ
T^{\flat}_{q,r,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}T^{-\flat}_{u,q,v}=(T^{\flat}_{u,s,q}\times\operatorname{id}_{\Omega^{\flat}_{q,r}}\times\operatorname{id}_{\Omega^{\flat}_{r,v}})\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,q,r}\times\operatorname{id}_{\Omega^{\flat}_{r,v}})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}(\operatorname{id}_{\Omega^{\flat}_{u,s}}\times
T^{\flat}_{s,r,v})^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}T^{-\flat}_{u,s,v}$,
it follows that
$T^{\flat}_{(q,r),[s,t]}T^{\flat}_{(s,t),(u,v)}=T^{\flat}_{(q,r),(u,v)}$, as
required. $\hfill\square$

We conclude with the following general description of the $K$-convergence of a
convergent convolution system.

######  Theorem 4.5.

Let
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
be a convergent convolution system such that the Set-projective system
$\\{\Omega_{I},X_{I,J}\\}$ over $\mathcal{K}$ is simply maximal. The following
conditions are equivalent:

  1. (1)

The convolution system $\mathscr{S}$ is $K$-convergent;

  2. (2)

The projective system
$\\{(\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat}),T_{(s,t),(u,v)}\\}$
is convergent.

###### Proof.

(1) $\Rightarrow$ (2). Suppose that $\mathscr{S}$ is $K$-convergent. Because
$T_{I,J}X_{J}=X_{I}$, for any two elements $s<t$ in $\mathbb{S}$ and any two
partitions $I,\,J\in\mathcal{K}_{s,t}$, $I\subseteq J$, it follows that there
exits an epimorphism
$X_{s,t}^{\flat}:(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})\to(\Omega^{\flat}_{s,t},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})$
such that

$\displaystyle T^{\flat}_{I}X_{s,t}^{\flat}=X_{I},$

for all $I\in\mathcal{K}_{s,t}$ and all $s<t$ in $\mathbb{S}$. Using Lemma
4.3, one can readily see that
$T^{\flat}_{(s,t),(u,v)}X^{\flat}_{u,v}=X^{\flat}_{s,t}$, for all $u<s<t<v$ in
$\mathbb{S}$. Indeed,

$T^{\flat}_{I}T^{\flat}_{(s,t),(u,v)}X^{\flat}_{u,v}=X_{I,J}T^{\flat}_{J}X^{\flat}_{u,v}=X_{I,J}X_{J}=X_{I},$

for all $I\in\mathcal{K}_{s,t}$, $J\in\mathcal{K}_{u,v}$, $I\subseteq J$, and
the conclusion follows. Consequently
$(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})$ is an upper limit for the
projective system
$\\{(\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat}),T_{(s,t),(u,v)}\\}$,
so this must be convergent.

(2) $\Rightarrow$ (1). Suppose that the projective system
$\\{(\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat}),T_{(s,t),(u,v)}\\}$
is convergent, and let

(11)
$\displaystyle({}^{\flat}\Omega,{}^{\flat}\mathcal{F},{}^{\flat}P)=\mathop{\oalign{lim\cr$\longleftarrow$\cr}}_{\mathcal{C}}\\{(\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat}),T_{(s,t),(u,v)}\\}$

be its projective limit with canonical epimorphisms
$T^{\prime}_{s,t}:({}^{\flat}\Omega,{}^{\flat}\mathcal{F},{}^{\flat}P)\to(\Omega_{s,t}^{\flat},\mathcal{F}_{s,t}^{\flat},\mu_{s,t}^{\flat})$,
for all $s<t$ in $\mathbb{S}$. For any $I\in\mathcal{K}$, say
$I\in\mathcal{K}_{s,t}$, for some $s<t$ in $\mathbb{S}$, consider the
composite epimorphism
$T_{I}^{\flat}T^{\prime}_{s,t}:({}^{\flat}\Omega,{}^{\flat}\mathcal{F},{}^{\flat}P)\to(\Omega_{I},\mathcal{F}_{I},\mu_{I})$.
We deduce from Lemma 4.3 that
$T_{I}^{\flat}T^{\prime}_{s,t}=X_{I,J}T^{\flat}_{J}T^{\prime}_{u,v}$ for all
$I\subseteq J$, $I\in\mathcal{K}_{s,t}$, $J\in\mathcal{K}_{u,v}$, $u\leq
s<t\leq v$. Thus $({}^{\flat}\Omega,{}^{\flat}\mathcal{F},{}^{\flat}P)$ is an
upper bound for $\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),X_{I,J}\\}$ over
$(\mathcal{K},\subseteq)$, which leads to the convergence of the projective
system $\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),X_{I,J}\\}$. $\hfill\square$

It is worth mentioning that if
$\mathscr{S}=\\{\Omega_{s,t},\mathcal{F}_{s,t},\mu_{s,t},T_{r,s,t}\\}_{\mathbb{S}}$
is a $K$-convergent convolution system, then the probability space
$({}^{\flat}\Omega,{}^{\flat}\mathcal{F},{}^{\flat}P)$ defined in (11) is the
least upper bound for the projective system
$\\{(\Omega_{I},\mathcal{F}_{I},\mu_{I}),X_{I,J}\\}$ over
$(\mathcal{K},\subseteq)$. We postpone the proof of this fact, as well as the
discussion of some relevant examples, for another occasion.

## References


# Polarity switch of PMMA powder transported through a PMMA duct

Wenchao Xu1 Holger Grosshans1,2 1Physikalisch-Technische Bundesanstalt (PTB),
Braunschweig, Germany  2Otto von Guericke University of Magdeburg, Institute
of Apparatus- and Environmental Technology, Magdeburg, Germany
<EMAIL_ADDRESS>
###### Abstract

During pneumatic conveying, powder electrifies rapidly due to the high flow
velocities. In our experiments, the particles even charge if the conveying
duct is made of the same material, which might be caused by
triboelectrification between two asymmetric contact surfaces. Surprisingly, we
found the airflow rate to determine the polarity of the overall powder charge.
This study investigates the charging of microscale PMMA particles in turbulent
flows passing through a square PMMA duct. The particles are spherical and
monodisperse. A Faraday at the duct outlet measured the total charge of the
particles. At low flow velocities, the particles charged negatively after
passing through the duct. However, the powder’s overall charge switched to a
positive polarity when increasing the flow velocity.

##  1 Introduction

Triboelectric charging, also known as contact electrification, is the exchange
of electric charge between two objects when they touch against each other. The
pneumatic conveyance of powders is commonly involved in industrial
productions. Being transported pneumatically through ducts or pipelines,
powders or particles undergo undesired charging via triboelectrification due
to particle–particle and particle–surface interactions. The charge can
accumulate on non–conductive materials, leading to a ”cone discharge” and
igniting dust clouds [1].

The transfer of charge in triboelectrification processes typically follows a
consistent direction and is determined by uniform global parameters. For
example, in the case of different metals, the direction of charge transfer is
determined by factors such as the work function [2]. For insulators, Lewis
basicity is one of many factors that can explain the observed charge transfer
behavior [3]. However, it is important to note that there are cases where the
charge or charging direction reverses.

In a study by Tanoue et al. [4], the polarity change of particles impacting a
rotating metal target was investigated. The researchers observed that the
tribo-charge polarity depended on the impact angle, resulting in charge
reversal under certain conditions. Furthermore, charge reversal has also been
observed during sliding contacts [5], [6]. These findings highlight that in
specific situations, such as impact angles or sliding contacts, the direction
of charge transfer can be reversed or exhibit different charging behavior than
what is typically observed.

Grosjean and Waitukaitis [7] have proposed an explanation for triboelectric
charging using the mosaic model, which incorporates donor/acceptor
asymmetries. By considering the asymmetrical geometry of the surfaces, they
successfully reproduced the observed sign flips in contact sliding
experiments.

##  2 Experiment setup and measurement

Figure 1 illustrates the schematics of our enhanced experimental facility, an
upgrade from our previous setup [8]. The key components include a square-
shaped test duct, an air blower, a powder feeder, and a Faraday cage.

To prevent powder leakage during feeding, the air blower is positioned at the
outlet of the test duct to create a negative pressure within the duct.
Equipped with a frequency converter, the blower allows for precise control of
rotation speed and airflow velocity. For measuring the bulk flow velocity, we
employ a Pitot-tube anemometer. The anemometer is installed in a duct with
identical geometry, connected to the air blower’s outlet. This setup enables
measurement of the streamwise velocity without disturbing the particle-laden
airflow.

The test section is a 2-meter-long square-shaped polymethyl methacrylate
(PMMA) duct with inner dimensions of 46 mm $\times$ 46 mm, as depicted in fig.
1. During each test, we apply a syringe to inject PMMA particles into the test
rig through an opening located 200 mm from the inlet of the duct. In the
experiments described in this paper, we used monodisperse spherical PMMA
particles with two different diameters, namely 100 $\upmu$m and 150 $\upmu$m.

InletTest ductPowder feedingFaraday cageDuctAir blowerChargeamplifier &
electrometer200 mm1800 mm

Figure 1: Schematics of the pneumatic conveying system.

The outlet of the test duct leads to the measurement section, which consists
of a PMMA duct enclosed within a cylindrical Faraday cage. Inside the
measurement section, a 50 $\mathrm{\upmu m}$ pore-size filter bag is employed
to separate particles from the airflow passing through.

To measure the charge of the collected particles, a charge amplifier is
connected to the Faraday cage. It is important to note that the electrical
circuit experiences consistent charge leakage. However, during the experiment,
the time scale for charging measurement was set significantly shorter than the
charge leakage time scale of the circuit, so the error due to circuit leakage
was negligible. Furthermore, since triboelectrification is sensitive to
ambient conditions, we incorporated a high-accuracy temperature and humidity
sensor to monitor and record the temperature and humidity during the
experiments.

During the experiment, particles of various diameters were introduced into the
test rig, and their resulting charge-to-mass ratios were measured. We obtained
the charge-to-mass ratio of particles after their passage through the duct at
various air velocities by conducting multiple measurements with different
particle sizes and conveying flow velocities.

It is possible that a small number of particles may deposit onto the inner
surface of the duct or leak from the duct, thereby evading collection by the
filter. To account for this, we measured the weight of particles before and
after their introduction into the test rig. The weight difference is carefully
controlled to remain below 1$\%$.

Figure 2: Experiment measurement of charge-to-mass ratio as a function of air
velocity at $T=20.1^{\circ}C$ and $RH=58.1\%$.

Figure 2 presents the charge-to-mass ratio of particles after traversing the
PMMA duct at various air velocities. We observed a polarity switch in the
charging behavior for both particle sizes (100 $\upmu$m and 150 $\upmu$m) at
an air velocity range of approximately 8-10 m/s. Specifically, when the air
velocity falls below 8 m/s, both particles acquire a negative charge within
the PMMA duct. Conversely, they exhibit a positive charge when the air
velocity surpasses 9 m/s.

##  3 Simulations

In a recent study by Grosjean and Waitukaitis [7], the switch in charging
polarity has been elucidated using ”Mosaic models”. These models assume that
surfaces are composed of a random patchwork of microscopic donor/acceptor
sites, leading to persistent global differences even among surfaces made from
the same material. Building upon this framework, they expanded the mosaic
model by incorporating variations in the densities of donor/acceptor sites
across the surfaces.

Following their approach, we generated two surfaces, denoted as $A$ and $B$,
with dimensions $L\times L$, where $L=50$. These surfaces are assumed to
consist of donor sites with an initial probability of $p_{A}$ and $p_{B}$,
respectively, while the probabilities for acceptor sites are given by
$1-p_{A}$ and $1-p_{B}$. To mimic the physics of surface formation, the
process also takes into account the favorability of neighboring donor/acceptor
sites and includes a time dependency. The transition probability of a given
site depends on its neighboring sites and is described by the equations

$P_{A}(n_{d})=P_{0}\exp(-Kn_{d})~{}\mathrm{and}~{}P_{D}(n_{d})=P_{0}\exp(-K(4-n_{d})),$
(1)

where $P_{A}$ is the probability of a site transiting into an acceptor and
$P_{D}$ is the opposite. Here, $n_{d}\in[0,4]$ represents the number of donors
in the neighboring grids, and $K$ is a constant that signifies the impact of
each neighbor on the local energy barrier [9]. The surfaces generated using
this approach are illustrated in fig. 3. Figure 3(a) depicts the surface of a
particle with $p_{A}=0.40$, while fig. 3(b) represents the inner surface of
the PMMA duct with $p_{B}=0.35$.

In the simulation, surface $A$ comes into contact with surface $B$ repeatedly
to mimic the collisions of particles on the inner duct walls. After each
contact, surface $B$ is regenerated randomly with the same parameters. During
the contact, donors on surface $A$ transfer a charge of $\mathrm{e}$ to facing
acceptors on surface $B$ with the probability $\alpha$ of 0.6, meanwhile
acceptors on surface $A$ can also acquire charge from the facing donors on
surface $B$ with the same probability.

Figure 4 illustrates the accumulated charge of grid $A$ as a function of the
number of contacts. Notably, the charging polarity undergoes a reversal during
the contacting process. Initially, grid $A$ loses electrons in the first few
contacts, but subsequently gains electrons in the following contacts. This
pattern of electron exchange demonstrates the dynamic charging behavior
observed during the contact between surfaces $A$ and $B$.

(a)

(b)

Figure 3: A square grid $A$ with $L\times L$ sites contacts on a rectangular
grid $B$ with $L\times L$ sites. Red cells represent donor sites and blue
cells represent acceptor sites. $L=50,~{}p_{A}=0.40,~{}p_{B}=0.35$. Figure 4:
Accumulated charge of the square grid $A$ as a function of contact numbers
after repetitively contacting with grid $B$.

Turning to the case of particle-laden flows, since we lacked a direct method
to measure the frequency of particle-wall contacts, we conducted a series of
Large-Eddy simulations (LES) with our open-source CFD tool pafiX [10]. The
numerical work simulated 100 $\upmu$m particles entrained in turbulent air
flow passing through a square-shaped duct with dimensions of 45
$\mathrm{mm}\times 45~{}\mathrm{mm}\times 2000~{}\mathrm{mm}$, which
corresponds to the meshgrid of $80\times 80\times 200$. The boundary
conditions in spanwise directions are no-slip walls, whereas in streamwise
directions are periodic boundaries. The fluid is solved in the Eulerian and
the particles are solved in the Lagrangian framework using a four-way coupled
approach. More detailed descriptions of the mathematical model and numerical
methods implemented in pafiX were given by Grosshans et al.[11].

Figure 5 displays the average number of particle-wall collisions at various
air velocities. Our simulations indicate that air velocity significantly
impacts the frequency of particle-wall contacts, highlighting the strong
influence of airflow velocity on these interactions, which eventually affects
the polarity of the triboelectric charge of the PMMA particles.

Figure 5: Average wall collision numbers per particle in particle-laden flows
as a function of air velocities. The results are from LES simulations with the
particle diameter of 100 $\upmu$m.

##  4 Conclusions

In our experimental investigation of triboelectric charging of PMMA particles
in turbulent duct flows, we observed that the polarity of particle charge
switches as the airflow velocity changes. Specifically, particles were charged
negatively at low velocities and positively at high air velocities.

One explanation for the observed phenomenon is the presence of donor/acceptor
asymmetries between the surfaces of particles and the inner walls of the duct.
These asymmetries could arise from impurities, contamination, or different
surface treatments. To investigate this hypothesis, we employed the mosaic
model and simulated a grid with a higher probability of donors repetitively
contacting surfaces with a lower donor probability. We successfully reproduced
a scenario where the charge polarity reverses during repeated contact. In this
scenario, the polarity of charge is dependent on the number of contacts.
Moreover, we conducted a series of Computational Fluid Dynamics (CFD)
simulations to study turbulent particle-laden flows. The simulation results
strongly indicated that the airflow velocity significantly influences the
number of particle-wall contacts, further influencing the particles’ charging
polarity.

This project has received funding from the European Research Council (ERC)
under the European Union Horizon 2020 research and innovation program (grant
agreement No. 947606 PowFEct).

## References

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