mathx"17

# Convergence of Dynamics
on Inductive Systems of Banach Spaces

Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner

( Institut für Theoretische Physik, Leibniz Universität Hannover,
Appelstraße 2, 30167 Hannover, Germany
)

###### Abstract

Many features of physical systems, both qualitative and quantitative, become
sharply defined or tractable only in some limiting situation. Examples are
phase transitions in the thermodynamic limit, the emergence of classical
mechanics from quantum theory at large action, and continuum quantum field
theory arising from renormalization group fixed points. It would seem that few
methods can be useful in such diverse applications. However, we here present a
flexible modelling tool for the limit of theories, soft inductive limits,
constituting a generalization of inductive limits of Banach spaces. In this
context, general criteria for the convergence of dynamics will be formulated,
and these criteria will be shown to apply in the situations mentioned and
more.

###### Contents

  1. 1 Introduction
  2. 2 Convergent nets in inductive systems
  3. 3 Soft inductive limits
  4. 4 Nets of operations on inductive systems
  5. 5 Dynamics on inductive systems
  6. 6 Inductive systems of C*-algebras and completely positive dynamics
  7. 7 Examples and Applications
    1. 7.1 Quantum dynamics in the classical limit
    2. 7.2 Mean field limit
    3. 7.3 Spin systems, dynamics and the thermodynamic limit
    4. 7.4 Quantum scaling limits
    5. 7.5 Recovering symmetries: Thompson’s group actions à la Jones
  8. 8 Comparison with the literature
  9. A Convergence of implementors for dynamics in representations
  10. B Interchanging Lie-Trotter limits and inductive limits

##  1 Introduction

Many features of physical theories become really clear only in some limiting
situation. Quite often the limit is not just the limit of some parameters in a
fixed framework, but the structure of the theory changes in the limit. For
example, sharp phase transitions appear only in an infinite volume limit,
quantum theory becomes classical in the limit $\hbar\to 0$, or a quantum field
theory is defined after removing a length-scale cutoff $\varepsilon\to 0$. In
such cases one has to define carefully what the limit means, and it would seem
that the required techniques differ greatly from case to case. However, it
turns out that for the limit of dynamical evolutions there is a common
technical core, an abstract limit theorem, henceforth called the _evolution
theorem_ , that considerably simplifies and unifies the proofs of limit
theorems in quite diverse settings, from the classical and mean field limits,
to the thermodynamic limit and renormalization-group limits. This abstract
evolution theorem is the topic of the current paper, and will be illustrated
by salient examples.

The structural limits are essentially obtain via inductive-limit constructions
for a sequence of Banach spaces that describe either states or observables of
the approximating systems. Convergent sequences have an element in each of
these spaces. It will be convenient to further generalize the notion of
inductive limits and to tolerate norm-small deviations in a convergent
sequence. Hence the limits will be “in norm” in the sense that sequences whose
norm difference goes to zero have the same limit. The resulting _soft
inductive limits_ then generalize the completion construction.

The essential technical notion underlying soft inductive limits is a
generalization of the concept of Cauchy sequences to inductive limits. For an
inductive system $(E,j)$ of Banach spaces $E_{n}$ with connecting map $j_{nm}$
we coin this notion _$j$ -convergence_. It relies on the fact that the
connecting maps $j_{nm}$ allow for a notion of distance for elements in
different approximating spaces. We construct the limit space $E_{\infty}$ as
the quotient of the Banach space of convergent sequences with respect to null
sequences which is automatically complete. Thereby, we obtain an explicit
description for all its elements, in contrast with the standard inductive
limits where limit space arises as the completion of $\bigcup_{n}E_{n}$.

This is particularly beneficial in the context of the evolution theorem for
dynamics on the limit space, when the latter does not happen be uniformly
continuous as this requires a discussion of unbounded operators and their
domains. The main realization behind the evolution theorem is that central
notions of the semigroup theory on Banach spaces have a version for sequences
in an inductive limit. This splits the problem into showing that, on the one
hand, the generators have a dense set of convergent sequences in their domain,
allowing the definition of the generator in the limit space, and, on the other
hand, the resolvent of this limit operator has dense range. For this second
step one only needs to work in the limit space, which is often easier than
tracking properties of the approximating systems. Informally the evolution
theorem can be stated as follows:

######  Theorem A (The evolution theorem, cf. 27).

Given a (soft) inductive system $(E,j)$ along with approximating dynamics
$T_{n}(t)$ with generators $A_{n}$. Then we have the following equivalent
characterization of the dynamics on the limit space $E_{\infty}$:

  1. (1)

The approximating dynamics $T_{n}(t)$ preserves $j$-convergence, and the
resulting limit dynamics $T_{\infty}(t)$ are strongly continuous in $t$.

  2. (2)

The resolvents $R_{n}(\lambda)=(\lambda-A_{n})^{-1}$ preserve $j$-convergence,
and the resulting limit operators $R_{\infty}(\lambda)$ have dense range.

  3. (3)

There is a dense subspace of $j$-convergent sequences ${\mathcal{D}}$ such
that $(\lambda-A_{n}){\mathcal{D}}$ is also dense subspace of $j$-convergent
sequences.

  4. (4)

The limit generator $A_{\infty}$ is well-defined and generates a strongly
continuous dynamics $T_{\infty}(t)$.

We conclude that the limit dynamics $T_{\infty}(t)$ is a strongly continuous
one-parameter semigroups with generator is $A_{\infty}$. The domain of the
latter is obtained by acting with the limit resolvent $R_{\infty}(\lambda)$ on
$j$-convergent sequences.

The density mentioned in the third item is with respect to a natural seminorm
topology on $j$-convergent nets. Another important aspect is the observation
that soft inductive limits are categorically well behaved: If the inductive
system of Banach spaces carries additional structure which is respected by the
dynamics, the same will hold in the limit. As an example, consider an
inductive system where the Banach spaces are C*-algebras and the connecting
maps are asymptotically multiplicative and completely positive so that the
limit space is again a C*-algebra. If the dynamics is completely positive at
every scale and converges then the limiting dynamics will also be completely
positive.

Apart from their relevance in physics, inductive limits serve as major
constructive tool in mathematics. In particular, in the theory of operator
algebras various interesting objects can be constructed from simple building
blocks using inductive limits [Bla06]. For example, restricting to matrix
algebras as approximating objects leads to the class of AF algebras and
hyperfinite factors. To allow for a greater amount of flexibility, generalized
inductive limits have been proposed [BK97] emphasizing the importance of
asymptotic concepts to define the limit object, thereby allowing for the
construction of NF algebras. Only recently, it has been shown that the
asymptotic properties of generalized inductive systems can be further relaxed
allowing for all separable nuclear C*-algebras to be characterized by an
inductive limit construction [Cou23, CW23].

Our paper is organized as follows: In Section 2 we introduce and discuss the
notion of $j$-convergence, which is a Cauchy type criterion for sequences in
an inductive system, and use it to construct the limit space. This section
deals with inductive systems of Banach spaces, i.e., systems where the
transitivity relation $j_{nm}j_{ml}=j_{nl}$ holds exactly. We show in Section
3 that one can relax this criterion to an asymptotic version which still
allows for a construction of the limit space using $j$-convergent sequences.
This relaxed transitivity is the defining property of soft inductive systems.
Instead of introducing the concepts of soft inductive systems and that of
$j$-convergence at once we chose to split them into separate chapters for
pedagogical reasons. In Section 4 we analyze the convergence of sequences of
operations between inductive systems leading to operations between the limit
spaces. This is necessary as we are ultimately interested in the convergence
of dynamics on inductive system which is considered in Section 5. Because of
its central importance to our applications we explicitly consider dynamics and
$j$-convergence for (soft) inductive systems of C*-algebras with completely
positive connecting maps in Section 6. To illustrate our results, we discuss
in detail the four examples which motivated us to analyze convergence of
dynamics abstractly in Section 7 and illustrate how our abstract evolution
theorem helps in concrete situations. In Section 8 we conclude by comparing
our evolution theorem with a result on the convergence of dynamics in limits
of Banach spaces by Kurtz, our notion of $j$-convergence with continuous
fields of Banach spaces, and soft inductive limits of C*-algebras with
generalized inductive limits of C*-algebras. For convenience of the reader, we
discuss convergence of implemented dynamics in GNS representations in Appendix
A and interchangability of inductive limits and Lie-Trotter limits of
convergent dynamics in Appendix B.

##  2 Convergent nets in inductive systems

An inductive system $(E,j)$ of Banach spaces over a directed set $(N,\leq)$ is
a collection $\\{E_{n}\\}_{n\in N}$ of Banach spaces together with connecting
maps $\\{j_{nm}\\}_{n<m}$ which are linear contractions $j_{nm}:E_{m}\to
E_{n}$, $\lVert j_{nm}\rVert\leq 1$, whenever $n>m$, such that

$j_{nl}=j_{nm}\circ j_{ml},\quad n>m>l.$ (1)

For every inductive system there is a limit space $E_{\infty}$ and a net of
contractions $j_{\infty n}:E_{n}\to E_{\infty}$ such that $j_{\infty
m}=j_{\infty n}\circ j_{nm}$ whenever $n>m$. If the $j_{nm}$ are isometric, it
may be constructed by completing the union $\bigcup_{n}E_{n}$ with respect to
its natural norm. We will, however, discuss a different construction of the
limit space in terms of convergent nets. This construction has several
advantages and offers a direct description of every elements of $E_{\infty}$
(no completion is necessary).

A net in $(E,j)$ is a net $(x_{n})_{n\in N}$ of elements $x_{n}\in E_{n}$. We
will often denote nets in $(E,j)$ by
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$. We
denote the space of uniformly bounded nets by ${\mathbf{N}}(E,j)$ and equip it
with the norm

$\lVert x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}}\coloneqq\sup_{n}\ \lVert x_{n}\rVert.$ (2)

It is easy to see that
$({\mathbf{N}}(E,j),\lVert\,\cdot\,\rVert_{\mathbf{N}})$ is again a Banach
space. In fact, it is nothing but the Banach space product $\Pi_{n\in
N}E_{n}$. We further equip ${\mathbf{N}}(E,j)$ with the following seminorm

$\vvvert x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert\coloneqq\varlimsup_{n}\lVert x_{n}\rVert,$ (3)

where $\varlimsup_{n}$ denotes the limit superior along the directed set $N$.

We define $j_{nn}$ to be the identity map on $E_{n}$ for all $n\in N$ and we
set $j_{nm}=0$ whenever $n\ngeq m$. Given some $m\in N$ and some $x_{m}\in
E_{m}$, we define a uniformly bounded net
$j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}m}x_{m}$
in the obvious way, i.e., at index $n$ it is equal to $j_{nm}x_{m}$. We refer
to such nets as basic nets. We say that a net is $j$-convergent if it can be
approximated in seminorm by basic nets and we denote the space of
$j$-convergent nets by $\mathscr{C}(E,j)$, i.e.,
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$-convergent if

$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)\coloneqq\bar{\\{y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathbf{N}}(E,j)\nonscript\ |\allowbreak\nonscript\
\mathopen{}y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\text{ is basic }\\}}^{\vvvert\>\cdot\>\vvvert}.$ (4)

It is straightforward to check that $\mathscr{C}(E,j)$ is a vector space and
we equip it with the topology induced by the seminorm. Note that this turns
$j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}m}$ into
linear contraction from $E_{m}$ to $\mathscr{C}(E,j)$.

######  Lemma 1.

A net $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathbf{N}}(E,j)$ is $j$-convergent if and only if

$\mathop{\textup{lim}}_{n\gg m}\lVert x_{n}-j_{nm}x_{m}\rVert=0,$ (5)

where $\mathop{\textup{lim}}_{n\gg
m}=\mathop{\textup{lim}}_{m}\varlimsup_{n}$. If
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$-convergent then the limit $\mathop{\textup{lim}}_{n}\lVert x_{n}\rVert$
exists (hence is equal to $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert$).

It is clear that basic sequences
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}$ always satisfy (5) since the norm difference becomes zero for
all $n\geq m$. We can rewrite (5) in terms of the seminorm as

$\mathop{\textup{lim}}_{m}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}\vvvert=0.$ (6)

We will later see that there is another equivalent definition provided that
the connecting maps are (asymptotically) isometric.

###### Proof.

Eq. 6 already proves that nets satisfying (5) can be approximated by basic
nets. The converse follows from the triangle inequality: For $\varepsilon>0$
pick $y_{l}\in E_{l}$ such that $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert<\varepsilon$ where
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}l}y_{l}$.

$\displaystyle\mathop{\textup{lim}}_{m}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}\vvvert$
$\displaystyle\leq\mathop{\textup{lim}}_{m}\Big{\lparen}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert+\vvvert
y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}y_{m}\vvvert+\vvvert
j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}(x_{m}-y_{m})\vvvert\Big{\rparen}$
$\displaystyle\leq\varepsilon+0+\mathop{\textup{lim}}_{m}\vvvert
x_{m}-y_{m}\vvvert=\varepsilon.$

To see that the limit of the norms exists, note that $\varliminf_{n}\lVert
x_{n}\rVert\geq\varlimsup_{n}\lVert x_{n}\rVert$ follows from

$0=\mathop{\textup{lim}}_{n\gg m}~\lVert
x_{n}-j_{nm}x_{m}\rVert\geq\mathop{\textup{lim}}_{n\gg m}(\lVert
x_{n}\rVert-\lVert x_{m}\rVert)=\varlimsup_{n}\lVert
x_{n}\rVert-\varliminf_{m}\lVert x_{m}\rVert.\qed$

We say that a net
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is a
($j$-)null net, if $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=\mathop{\textup{lim}}_{n}\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert=0$ and we denote the subspace of null nets by
$\mathscr{C}_{0}(E,j)$. Since the zero net $0$ is basic and since $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-0\vvvert=\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=0$, all null nets are $j$-convergent. Two nets
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ and
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ have
vanishing seminorm distance if and only if
$(x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ is a null net.

Since the subspace $\mathscr{C}_{0}(E,j)$ is the preimage of $\\{0\\}$ under
the seminorm, it is a closed subspace of $\mathscr{C}(E,j)$ and we can
consider the quotient space

$E_{\infty}\coloneqq{{}^{\textstyle\mathscr{C}(E,j)}\big{/}_{\textstyle\mathscr{C}_{0}(E,j)}}.$
(7)

We will show that $E_{\infty}$ is the limit space of the inductive system
$(E,j)$ and we will refer to the equivalence class of a $j$-convergent net
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ as
its ($j$-)limit. The natural projection onto the quotient will be deoted by
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}:\mathscr{C}(E,j)\to
E_{\infty}$ and we write

$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\coloneqq
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in E_{\infty},\quad
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j).$ (8)

To keep notation concise we will often denote $j$-limit of a net
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
simply by $x_{\infty}$. The quotient structure induces a map $j_{\infty m}$
from $E_{m}$ to $E_{\infty}$ by assinging to $x_{m}$ the equivalence class of
the basic sequence
$j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}m}x_{m}$,
i.e.,

$j_{\infty
m}x_{m}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}j_{nm}x_{m}.$
(9)

Equivalently, $j_{\infty m}\coloneqq
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}\circ
j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}m}$. The
seminorm induces a norm on $E_{\infty}$, namely

$\lVert x_{\infty}\rVert=\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert,\quad
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j).$ (10)

The next result states that $E_{\infty}$ is a Banach space and satisfies a
property which

######  Proposition 2.

$E_{\infty}$ is a Banach space with the norm defined in (10). The pair
$(E_{\infty},j_{\infty{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}})$ is uniquely determined up to isometric isomorphism by the following
universal property:

Let $F$ be a Banach space and let $T_{n}:E_{n}\to F$ be a net of contractions
such that $T_{m}=T_{n}j_{nm}$ whenever $n>m$, then there is a contraction
$T_{\infty}:E_{\infty}\to F$ such that $T_{\infty}j_{\infty n}=T_{n}$ for all
$n$.

The universal property can even be strengthened, under the assumption on
$(F,T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
stated in the proposition,
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
guaranteed to map $j$-convergent nets to Cauchy nets in $F$ and the limit
operator satisfies
$T_{\infty}j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=\mathop{\textup{lim}}_{n}T_{n}x_{n}$.
We stress that completeness means that every element of the limit space arises
as the limit of some $j$-convergent sequence. In the standard construction
this is not the case and it is cumbersome to control elements outside the
union $\bigcup_{n}E_{n}$. This control is however much needed for the
discussion of unbounded operators on $E_{\infty}$.

###### Proof.

The proof of completeness of $E_{\infty}$ will be given under weaker
assumptions in Section 3. That
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ maps
$j$-convergent nets to Cauchy nets in $F$ follows from the estimate $\lVert
T_{n}x_{n}-j_{nm}T_{m}x_{m}\rVert=\lVert
T_{n}(x_{n}-j_{nm}x_{m})\rVert\leq\lVert x_{n}-j_{nm}x_{m}\rVert$. We obtain a
well-defined linear contraction $T_{\infty}:E_{\infty}\to F$ through
$T_{\infty}j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=\mathop{\textup{lim}}_{n}T_{n}x_{n}$.
That $j_{\infty n}T_{n}=T_{\infty}j_{\infty n}$ follows directly from
$T_{n}j_{nm}=j_{nm}T_{m}$. ∎

In the case of isometric connecting maps, we have the following equivalent
definition of $j$-convergence in terms of the maps $j_{\infty n}$:

######  Lemma 3.

Assume that the $j_{nm}$ are isometric, then a net
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$-convergent if and only if the net $j_{\infty n}x_{n}$ converges in
$E_{\infty}$.

In fact, it suffices if the $j_{nm}$ are asymptotically isometric in the sense
that

$\displaystyle\mathop{\textup{lim}}_{n\gg m}\Big{\lparen}\inf_{\lVert
x_{m}\rVert=1}\lVert j_{nm}x_{m}\rVert\Big{\rparen}=1\;.$

A proof is given in the next section.

Before moving on we discuss the notions we introduced above for the elementary
example of a constant inductive system. We will see that the notion of
$j$-convergence becomes equivalent to being a Cauchy net.

######  Example 4 (Constant inductive systems).

Let $E$ be a Banach space then the trivial inductive system
$(E,\operatorname{id})$ is obtained by setting $E_{n}=E$ and
$j_{nm}=\operatorname{id}_{E}$ and we also set $N=\mathbb{N}$ for simplicity.
The space ${\mathbf{N}}(E,\operatorname{id})$ consists of uniformly bounded
sequences in $E$. Basic sequences are just constant sequences, so a sequence
$(x_{n})$ is $j$-convergent if and only if it is approximated in seminorm by
constant sequences which is equivalent to being Cauchy in $E$, i.e., being
$j$-convergent is equivalent to being a convergent sequence because $E$ is
complete. In particular, we see that for a sequence $(x_{n})$ condition (5)
with $j_{nm}=\operatorname{id}$ is equivalent to being a Cauchy sequence,
which can also be seen directly from more elementary arguments. In standard
sequence space notation, the three space ${\mathbf{N}}(E,\operatorname{id})$,
$\mathscr{C}(E,\operatorname{id})$ and $\mathscr{C}_{0}(E,\operatorname{id})$
are are equal to $\ell^{\infty}(\mathbb{N};E)$, $c(\mathbb{N};E)$ and
$c_{0}(\mathbb{N};E)$, respectively.

Therefore, the construction of the limit space $E_{\infty}$ corresponds to the
standard construction of considering first the space of Cauchy sequences and
then taking the quotient with respect to null sequences which unsurprisingly
shows that $E_{\infty}\cong E$. In fact, this motivated the above construction
for the general case.

We collect some useful properties that we will repeatedly use later on. The
proof will be given in the next section in the more general setting of soft
inductive limits.

######  Proposition 5.

  1. (1)

Let $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be $j$-convergent and let
$(x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)})_{\alpha}$ be a net of $j$-convergent nets. The following
are equivalent,

    1. (i)

$\vvvert x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\vvvert\to 0$ as $\alpha\to\infty$,

    2. (ii)

$x_{\infty}^{(\alpha)}\to x_{\infty}$ in $E_{\infty}$ as $\alpha\to\infty$,

    3. (iii)

there are
$\tilde{x}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\in\mathscr{C}(E,j)$ such that
$\tilde{x}_{\infty}^{(\alpha)}=x_{\infty}^{(\alpha)}$ for all $\alpha$ and
such that

$\lVert x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\rVert_{\mathbf{N}}\to 0\quad\text{as $\alpha\to\infty$}.$

  2. (2)

For every $j$-convergent net
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$, the
net $(j_{\infty n}x_{n})_{n\in N}\subset E_{\infty}$ converges to
$x_{\infty}$.

  3. (3)

A subspace ${\mathcal{D}}\subset\mathscr{C}(E,j)$ is seminorm dense if and
only if ${\mathcal{D}}_{\infty}=\\{x_{\infty}\nonscript\
|\allowbreak\nonscript\
\mathopen{}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}\\}$ is dense in $E_{\infty}$.

  4. (4)

For any finite collection
$x^{(1)}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}},\ldots,x^{(k)}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ of $j$-convergent sequences and any $\varepsilon>0$, there are
$m\in N$ and $x_{m}^{(1)},\ldots,x_{m}^{(k)}\in E_{k}$ such that $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(i)}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}^{(i)}\vvvert<\varepsilon$.

We now turn to a brief discussion of the dual notion of weak* convergence for
nets of continuous linear functionals associated with $\mathscr{C}(E,j)$ (cf.
Section 7.4). To this end, we denote the continuous duals of the Banach spaces
$E_{n}$ by $E_{n}^{\prime}$ and we denote the dual pairing between $E_{n}$ and
$E_{n}^{\prime}$ by $\langle\,\cdot\,,\,\cdot\,\rangle$. A net of continuous
linear functionals
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ associated with $\mathscr{C}(E,j)$ is a collection
$\\{\varphi_{n}\\}_{n\in N}$ with $\varphi_{n}\in E_{n}^{\prime}$.

######  Definition 6.

A uniformly bounded net
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ of functionals $\varphi_{n}\in E_{n}^{\prime}$ is
$j^{*}$-convergent, if for all
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ the limit $\mathop{\textup{lim}}_{n}\langle
x_{n},\varphi_{n}\rangle$ exists. In this case
$x_{\infty}\mapsto\mathop{\textup{lim}}_{n}\langle x_{n},\varphi_{n}\rangle$
for some $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ with $j$-limit $x_{\infty}$ defines a bounded linear functional
$\varphi_{\infty}$ on $E_{\infty}$ with
$\lVert\varphi_{\infty}\rVert\leq\sup_{n}\lVert\varphi_{n}\rVert$, which we
also denote by
$j\mkern-0.5mu\hbox{${}^{*}$}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}\varphi_{n}$.

We have the following easy consequences of the definition.

######  Lemma 7.

  1. (1)

Let $\varphi_{\infty}\in E_{\infty}^{\prime}$ and set
$\varphi_{n}=\varphi_{\infty}\circ j_{\infty m}\in E_{n}^{\prime}$. Then
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j^{*}$-convergent and
$j\mkern-0.5mu\hbox{${}^{*}$}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}\varphi_{n}=\varphi_{\infty}$.

  2. (2)

Let $\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ be a uniformly bounded net of continuous linear functionals. Then
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j^{*}$-convergent if and only if for all $m\in N$, the nets
$(\varphi_{n}\circ j_{nm})$ are $w^{*}$-convergent in the limit $n\to\infty$.
In this case

$w^{*}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}\big{(}\varphi_{n}\circ
j_{nm}\big{)}=\big{(}j\mkern-0.5mu\hbox{${}^{*}$}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}\varphi_{n}\big{)}\circ
j_{\infty m}.$ (11)

Nets of functionals that arise from a functional $\varphi_{\infty}$ on
$E_{\infty}$ as in 1 are called projectively consistent [Tak55] and can
equivalently be characterized as those nets that satisfy $\varphi_{n}\circ
j_{nm}=\varphi_{m}$ for all $n>m$.

######  Remark 8.

Given a strict inductive system $(E,j)$, we may assume that each $j_{nm}$ is
injective because if we had $x_{m}\in\ker j_{nm}$ for some $n\geq m$, the
basic sequence $j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}(x_{m})$ would be null and, therefore, we would have
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}j_{nm}(x_{m})=0$.
This, in turn, would entail that we could restrict the inductive system to
$\tilde{E}_{m}=E_{m}/\cup_{n\geq m}\ker j_{nm}$.

##  3 Soft inductive limits

The construction of the limit space as well as the theory of $j$-convergence
works in a much more general setting if the assumption of inductive systems is
weakened to what we call soft inductive systems. This weaker notion is
characterized by relaxing the equality $j_{nl}=j_{nm}j_{ml}$ to an asymptotic
equality. Readers interested only in standard inductive systems can skip this
section and may ignore the word “soft” in subsequent sections. All results
stated for inductive systems in this paper hold in this generalized setting if
not explicitly said otherwise, and all proofs are already written so that they
are valid in this setting.

######  Definition 9.

A soft inductive system of Banach spaces over a directed set $(N,\leq)$ is a
tuple $(E,j)$, where $E=\\{E_{n}\\}_{n\in N}$ is a family of Banach spaces and
where $j=\\{j_{nm}\\}_{n\geq m}$ is a collection of linear contractions such
that

$\mathop{\textup{lim}}_{n\gg
m}\lVert(j_{nl}-j_{nm}j_{ml})x_{l}\rVert=0\quad\forall l\in N,\ x_{l}\in
E_{l}.$ (12)

Standard inductive systems of Banach spaces are soft inductive systems with
the additional property that $j_{nl}=j_{nm}j_{ml}$. These will be called
strict to emphasize that they are a special case. Under these assumptions the
usual construction of the limit space as a completion of the union fails and
the universal property of the limit space becomes meaningless. In fact, the
latter can be replaced by an asymptotic version much in the same way as (12)
is an asymptotic version of (1). The way of constructing the limit space via
equivalence classes of $j$-convergent nets will however still be possible by
the same arguments. In fact, we will not have to change a single word in the
proofs presented in Section 2 as the directly apply to this generalized
setting.

As in the case of strict inductive system, we denote by ${\mathbf{N}}(E,j)$
the Banach space of uniformly bounded nets with the sup-norm
$\lVert\,\cdot\,\rVert_{\mathbf{N}}$. Basic nets are nets of the form
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}$ for some $m$, $x_{m}\in E_{m}$, where we set
$j_{mm}=\operatorname{id}_{E_{m}}$ and $j_{nm}=0$ if $n\ngeq m$. We now come
to the importance of the assumption (12), which can be rewritten as
$\mathop{\textup{lim}}_{n\gg m}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}\vvvert=0$ for all basic nets with the seminorm defined as in
(3). This guarantees that the proof of 1 still works and we get:

######  Lemma 10.

Let $(E,j)$ be a soft inductive system. Then a net
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathbf{N}}(E,j)$ can be approximated in seminorm by basic nets
if and only if

$\mathop{\textup{lim}}_{n\gg m}\lVert
x_{n}-j_{nm}x_{m}\rVert=\mathop{\textup{lim}}_{m}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}x_{m}\vvvert=0.$ (13)

As in the strict case, such nets will be called $j$-convergent and the space
of $j$-convergent nets is denoted $\mathscr{C}(E,j)$. An important class of
$j$-convergent nets are the null nets which are the nets such that
$\mathop{\textup{lim}}_{n}\lVert x_{n}\rVert=0$ and we denote the subspace of
null nets again by $\mathscr{C}_{0}(E,j)$. The limit space $E_{\infty}$ is
defined as the quotient of $\mathscr{C}(E,j)$ by $\mathscr{C}_{0}(E,j)$ as
before and equip it with the induced norm (see Eq. 10). The projection onto
the quotient is denoted by
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}$ and we write
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ and the maps $j_{\infty m}:E_{m}\to E_{\infty}$ are defined by
setting $j_{\infty
m}x_{m}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}j_{nm}x_{m}$.

######  Lemma 11.

Both $\mathscr{C}(E,j)$ and $\mathscr{C}_{0}(E,j)$ are closed with respect to
the $\lVert\,\cdot\,\rVert_{\mathbf{N}}$-norm topology and the Banach space
quotient is isomometrically isomorphic to $E_{\infty}$, i.e.,

${{}^{\textstyle(\mathscr{C}(E,j),\lVert\,\cdot\,\rVert_{\mathbf{N}})}\big{/}_{\textstyle(\mathscr{C}_{0}(E,j),\lVert\,\cdot\,\rVert_{\mathbf{N}})}}\cong
E_{\infty}.$ (14)

In particular, $E_{\infty}$ is a Banach space and the norm is given by

$\lVert x_{\infty}\rVert=\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=\inf_{y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}_{0}(E,j)}\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}+y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}},\quad
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j).$ (15)

###### Proof.

Suppose $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}$ is a $\lVert\,\cdot\,\rVert_{\mathbf{N}}$-Cauchy
sequence in $\mathscr{C}(E,j)$ and let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
its limit in ${\mathbf{N}}(E,j)$. Then
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ follows from

$\displaystyle\mathop{\textup{lim}}_{n\gg m}\lVert x_{n}-j_{nm}x_{m}\rVert$
$\displaystyle\leq~\mathop{\textup{lim}}_{n\gg m}\Big{(}\lVert
x_{n}-x_{n}^{(\alpha)}\rVert_{\mathbf{N}}+\lVert
x_{n}^{(\alpha)}-j_{nm}x_{m}^{(\alpha)}\rVert+\lVert
j_{nm}(x_{m}-x_{m}^{(\alpha)})\rVert\Big{)}$ $\displaystyle\leq 2\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\rVert\xrightarrow{\alpha\to\infty}0.$

Now suppose that
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}$ are all null nets. Then $\mathop{\textup{lim}}_{n}\lVert
x_{n}\rVert\leq\mathop{\textup{lim}}_{n}\big{(}\lVert
x_{n}-x_{n}^{(\alpha)}\rVert+\lVert x_{n}^{(\alpha)}\rVert\big{)}\leq\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\rVert\to 0$ and thus
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}_{0}(E,j)$.

To show the norm equality, let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ and define a net
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(m)}\in\mathscr{C}_{0}(E,j)$ by setting $y_{n}^{(m)}=-x_{n}$ if
$n<m$ and $y_{n}^{(m)}=0$ else. We have

$\lVert x_{\infty}\rVert=\varlimsup_{n}\ \lVert
x_{n}\rVert=\mathop{\textup{lim}}_{m}\ \sup_{n>m}\ \lVert
x_{n}\rVert=\mathop{\textup{lim}}_{m}\ \lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(m)}\rVert_{\mathbf{N}}\geq\inf_{y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}_{0}(E,j)}\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}+y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert$

and thus

$\lVert
x_{\infty}\rVert\geq\inf_{y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}_{0}(E,j)}\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}+y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}}\geq\inf_{y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}_{0}(E,j)}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}+y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=\lVert x_{\infty}\rVert.$

∎

######  Lemma 12.

5 holds for soft inductive systems.

###### Proof.

1: The equivalence of (i) and (ii) is obvious since $\lVert
x_{\infty}-x_{\infty}^{(\alpha)}\rVert=\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\vvvert$ and it is clear that (iii) implies (i). To show
(i) implies (iii), we define the net
$\tilde{x}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}$ as follows

$\tilde{x}_{n}^{(\alpha)}\coloneqq{\begin{cases}x_{n}-\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\vvvert\,\frac{x_{n}-x_{n}^{(\alpha)}}{\lVert
x_{n}-x_{n}^{(\alpha)}\rVert},&\text{ if~}x_{n}^{(\alpha)}\neq x_{n}\\\
x_{n},&\text{ if }x_{n}^{(\alpha)}=x_{n}\end{cases}}\ .$

This definition clearly guarantees that
$\tilde{x}_{\infty}^{(\alpha)}=x_{\infty}^{(\alpha)}$ and $\lVert
x_{n}-x_{n}^{(\alpha)}\rVert\leq\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\vvvert$ which goes to zero as $\alpha\to\infty$ but is
independent of $n$.

2: This follows immediately from (5) because $\mathop{\textup{lim}}_{m}\lVert
x_{\infty}-j_{\infty m}x_{m}\rVert=\mathop{\textup{lim}}_{n\gg m}\lVert
x_{n}-j_{nm}x_{m}\rVert=0$.

3: Observe that
${\mathcal{D}}_{\infty}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}{\mathcal{D}}$.
Since the projection of a dense subspace onto a quotient is dense this shows
that density of ${\mathcal{D}}$ implies density of ${\mathcal{D}}_{\infty}$.
For the converse let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ be given then by density of
${\mathcal{D}}_{\infty}$ there are
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}$ so that $x_{\infty}^{(\alpha)}\to x_{\infty}$. But this
implies that $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\alpha)}\vvvert=\lVert x_{\infty}-x_{\infty}^{(\alpha)}\rVert\to
0$ and thus that density of ${\mathcal{D}}_{\infty}$ implies density of
${\mathcal{D}}$.

4: Pick basic sequences
$j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m_{i}}x_{m_{i}}$ such that $\vvvert
x^{(i)}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m_{i}}x_{m_{i}}\vvvert<\varepsilon/2$ for all $i=1,\ldots,k$. Now pick
$m$ large enough so that $\vvvert
j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m_{i}}x_{m_{i}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}j_{mm_{i}}x_{m_{i}}\vvvert<\varepsilon/2$ for all $i=1,\ldots,k$. We
set $x_{m}^{(i)}j_{mm_{i}}x_{m_{i}}$ and the claim follows from the triangle
inequality. ∎

The limit space of a soft inductive system satisfies the following the
universal property.

######  Proposition 13.

Let $(E,j)$ be a soft inductive system of Banach spaces and let $F$ be another
Banach space. Let
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be a
net of linear contractions $T_{n}:E_{n}\to F$ which maps $j$-convergent nets
to Cauchy nets in $F$, then there is an operator $T_{\infty}:E_{\infty}\to F$
such that
$T_{\infty}(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n})=\mathop{\textup{lim}}_{n}T_{n}(x_{n})$.

This is a special case of a general result on convergence of operations
between inductive systems (see 22). The importance of this property is that it
uniquely determines the limit space $E_{\infty}$ and the maps
$j_{\infty{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}}$:

######  Proposition 14.

Let $(E,j)$ be a soft inductive system of Banach spaces. Let
$\tilde{E}_{\infty}$ be a Banach space and let ${\tilde{\jmath}}_{\infty
n}:E_{n}\to\tilde{E}_{\infty}$ be linear contractions, such that
$\mathop{\textup{lim}}_{n}\lVert({\tilde{\jmath}}_{\infty
n}j_{nm}-{\tilde{\jmath}}_{\infty m})x_{m}\rVert=0$ for all $x_{m}\in E_{m}$.
Then ${\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}:\mathscr{C}(E,j)\to\tilde{E}_{\infty},\,x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\mapsto{\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=\mathop{\textup{lim}}_{n}{\tilde{\jmath}}_{\infty
n}x_{n}$ is a linear contraction. If 14 holds for $\tilde{E}_{\infty}$ and
${\tilde{\jmath}}\mkern 0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}$, then
there is an isometric isomorphism $\psi:E_{\infty}\to\tilde{E}_{\infty}$ such
that $\psi\circ j_{\infty n}={\tilde{\jmath}}_{\infty n}$.

The isomorphism $\psi$ is simply given by
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\mapsto{\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}$.

###### Proof.

We claim that for any
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$, the limit ${\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\coloneqq\mathop{\textup{lim}}_{n}{\tilde{\jmath}}_{\infty
n}x_{n}$ exists in $F$. This follows from

$\mathop{\textup{lim}}_{n\gg m}\lVert{\tilde{\jmath}}_{\infty
n}x_{n}-{\tilde{\jmath}}_{\infty m}x_{m}\rVert=\mathop{\textup{lim}}_{n\gg
m}\lVert{\tilde{\jmath}}_{\infty
n}(x_{n}-j_{nm}x_{m})\rVert+0\leq\mathop{\textup{lim}}_{n\gg m}\lVert
x_{n}-j_{nm}x_{m}\rVert=0.$

It is also clear, that for any two
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}},y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ with $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=0$ we get ${\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}={\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}y_{n}$ (just check that
${\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}z_{n}=0$ for all
$z\in\mathscr{C}_{0}(E,j)$).

Consider the nets $j_{\infty n}:E_{n}\to E_{\infty}$ and
${\tilde{\jmath}}_{\infty n}:E_{n}\to E_{\infty}$ which take $j$-convergent
nets to Cauchy nets and apply the universal properties of $\tilde{E}_{\infty}$
and $E_{\infty}$, respectively. Therefore, there are contractions
$j_{\infty\infty}:\tilde{E}_{\infty}\to E_{\infty}$ and
$\psi\coloneqq{\tilde{\jmath}}_{\infty\infty}:E_{\infty}\to\tilde{E}_{\infty}$,
such that

$j_{\infty\infty}({\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n})=\mathop{\textup{lim}}_{n}\,j_{\infty
n}x_{n}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\quad\text{and}\quad{\tilde{\jmath}}_{\infty\infty}(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n})={\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}.$

Therefore $\psi$ is an isometric isomorphism between $E_{\infty}$ and
$\tilde{E}_{\infty}$. It remains to be shown that ${\tilde{\jmath}}_{\infty
n}$ is equal to $j_{\infty n}$ up to this isomorphism. This follows from

$\psi(j_{\infty
m}x_{m})=\psi\big{\lparen}j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}j_{nm}x_{m}\big{\rparen}={\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}j_{nm}x_{m}=\mathop{\textup{lim}}_{n}{\tilde{\jmath}}_{\infty
n}j_{nm}x_{m}={\tilde{\jmath}}_{\infty m}x_{m}.\qed$

######  Remark 15.

Only the asymptotic properties of the connecting maps $j_{nm}$ and the spaces
$E_{n}$ matter for the structure of the limit space (see also 18). One can
even relax the assumption that all $E_{n}$ are Banach spaces and just assume a
soft inductive system of normed spaces. In this case, the limit space will
automatically be complete, i.e., a Banach space, and it agrees with the limit
space of the system of Banach spaces that is obtained by completion of the
normed spaces and continuous extension of the connecting maps.

######  Lemma 16.

Assume that the connecting contractions $j_{nm}$ are asymptotically isometric
in the sense that

$\mathop{\textup{lim}}_{n\gg
m}\lambda_{nm}=1,\quad\lambda_{nm}=\inf_{\begin{subarray}{c}x_{m}\in E_{m}\\\
\lVert x_{m}\rVert=1\end{subarray}}\lVert j_{nm}x_{m}\rVert.$ (16)

Then a net $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j$-convergent if and only if
$\mathop{\textup{lim}}_{n}j_{\infty n}x_{n}$ exists (and one has
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=\mathop{\textup{lim}}_{n}j_{\infty
n}x_{n}$).

###### Proof.

The “only if” part of the assertion holds for all soft inductive systems. For
the converse assume that the limit $\mathop{\textup{lim}}_{n}j_{\infty
n}x_{n}$ exists. Now (16) implies that $\mathop{\textup{lim}}_{m}\lVert
x_{m}\rVert=\mathop{\textup{lim}}_{n\gg m}\lVert j_{nm}x_{m}\rVert$. We find
that

$\displaystyle\mathop{\textup{lim}}_{n\gg m}\lVert x_{n}-j_{nm}x_{m}\rVert$
$\displaystyle\leq\mathop{\textup{lim}}_{n\gg m}\lVert j_{\infty
n}(x_{n}-j_{nm}x_{m})\rVert$ $\displaystyle\leq\mathop{\textup{lim}}_{n\gg
m}\lVert j_{\infty n}x_{n}-j_{\infty m}x_{m}\rVert+\mathop{\textup{lim}}_{n\gg
m}\lVert j_{\infty m}x_{m}-j_{\infty n}j_{nm}x_{m}\rVert=0.$

The first term is zero since we assumed $j_{\infty n}x_{n}$ to be Cauchy and
the second one vanishes (already in the limit $n\to\infty$) because basic
sequences satisfy (13). ∎

######  Remark 17.

If the directed set $N$ is countable and if all Banach spaces $E_{n}$ are
separable, then so is $E_{\infty}$. This can be seen by noting that a
collection of a countable total subset $\\{x_{i}^{(n)}\nonscript\
|\allowbreak\nonscript\ \mathopen{}i\in\mathbb{N}\\}$ in each $E_{n}$ gives us
a countable subset
$\\{j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}n}a_{i}^{(i)}\nonscript\ |\allowbreak\nonscript\
\mathopen{}i\in\mathbb{N},\ n\in N\\}\subset\mathscr{C}(E,j)$ with seminorm-
dense span.

We are interested in the degree to which the notion of $j$-convergence and the
limit space depend on the explicit choice of connecting maps $j_{nm}$.

######  Proposition 18.

Let $\\{E_{n}\\}_{n\in N}$ be a family of Banach spaces indexed by a directed
set $N$ and let $j_{nm},{\tilde{\jmath}}_{nm}:E_{n}\to E_{m}$ be two families
of connecting maps so that $(E,j)$ and $(E,{\tilde{\jmath}}\,)$ both are soft
inductive systems. The following are equivalent

  1. (1)

$j$-convergence is equivalent to $\tilde{\jmath}$-convergence, i.e.
$\mathscr{C}(E,j)=\mathscr{C}(E,{\tilde{\jmath}}\,)$.

  2. (2)

For all $l$ and $x_{l}\in E_{l}$ one has

$\mathop{\textup{lim}}_{n\gg
m}\lVert(j_{nm}{\tilde{\jmath}}_{ml}-j_{nl})x_{l}\rVert=\mathop{\textup{lim}}_{n\gg
m}\lVert({\tilde{\jmath}}_{nm}j_{ml}-{\tilde{\jmath}}_{nl})x_{l}\rVert=0.$
(17)

In this case, the limit spaces of both inductive systems are isometrically
isomorphic via the identification

$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\longleftrightarrow{\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}.$ (18)

If one constructs the limit space as in (7) then they are even equal (not just
isomorphic).

###### Proof.

1 is equivalent to $j_{nm}$ and ${\tilde{\jmath}}_{nm}$ defining the same
notion of convergence. This in turn equivalent to $j$-convergence of all
$\tilde{\jmath}$-basic nets and $\tilde{\jmath}$-convergence of all $j$-basic
nets which is precisely the condition 2.

The second part is clear: Both the $j$-convergent and null nets are the same
for both systems, so that the quotients
$\mathscr{C}(E,j)/\mathscr{C}_{0}(E,j)$ and
$\mathscr{C}(E,{\tilde{\jmath}}\,)/\mathscr{C}_{0}(E,{\tilde{\jmath}}\,)$
agree. ∎

Equipped with this, we briefly discuss the notion of a split inductive system.
This is an extra structure that singles out a $j$-convergent net for every
point of the limit space. This structure is present in several of the examples
that we discuss in Section 7 and does not trivialize in the case of strict
systems.

######  Definition 19.

A split inductive system $(E,j,s)$ is soft inductive system $(E,j)$ together
with a linear contraction
$s_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}:E_{\infty}\to\mathscr{C}(E,j)$ so that
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}s_{n}(y)=y$ for all
$y\in E_{\infty}$, i.e.,
$s_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is a
right inverse of
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}:\mathscr{C}(E,j)\to
E_{\infty}$.

We have the following application of 18:

######  Corollary 20.

Let $(E,j,s)$ be a split inductive system and define new connecting maps
${\tilde{\jmath}}_{nm}=j_{\infty n}\circ s_{n}$. Then $(E,{\tilde{\jmath}}\,)$
is a soft inductive system and the notions of $j$-convergence and
$\tilde{\jmath}$-convergence are equivalent (hence they define the same limit
space). Furthermore, the following stronger version of (12) holds for the
${\tilde{\jmath}}$-maps

$\mathop{\textup{lim}}_{m}\sup_{n}\,\lVert({\tilde{\jmath}}_{nl}-{\tilde{\jmath}}_{nm}~{\tilde{\jmath}}_{ml})x_{l}\rVert=0\quad\forall
l\in N,\,x_{l}\in E_{l}.$ (19)

A natural way to obtain a split inductive system is the following111This is
the abstract version of the soft inductive system that we will use for the
classical limit in Section 7.1. Here the limit space is a space of functions
on the classical phase space and the maps $i$ and $p$ are suitable
quantization and dequantization maps.: Suppose that we are given a net of
Banach spaces $E_{n}$ a space $E_{\infty}$ and nets of contractions
$i_{n}:E_{n}\to E_{\infty}$ and $p_{n}:E_{\infty}\to E_{n}$ so that
$i_{n}\circ p_{n}~\to\operatorname{id}_{E_{\infty}}$ strongly. Then we obtain
a soft inductive system by setting $j_{nm}=p_{n}\circ i_{m}$. In fact,
$(E,j,p)$ is a split inductive system and by 20, all split inductive systems
are essentially of this form.

##  4 Nets of operations on inductive systems

The universal property of the limit space of a strict inductive system can be
regarded as a convergence result for certain operations. But if we regard it
as such, the assumption $T_{n}j_{nm}=j_{nm}T_{m}$ for all $n>m$ is
unnecessarily restrictive. We will now define convergence for nets of
operators between two inductive systems $(E,j)$ and
$(\tilde{E},{\tilde{\jmath}}\,)$ but we will almost exclusively work with the
cases where either both are the same system or one is constant (as in 4).
Whenever we consider two inductive systems we will assume that they are
defined w.r.t. the same directed set. All definitions, statements and proofs
given in this section also apply to the broader class of soft inductive
systems introduced in the previous section.

######  Definition 21.

Let $(E,j)$ and $(\tilde{E},{\tilde{\jmath}}\,)$ be inductive systems and let
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be a
uniformly bounded net of linear operators $T_{n}:E_{n}\to\tilde{E}_{n}$. We
say that $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j$$\tilde{\jmath}$-convergent if it maps $j$-convergent nets to
$\tilde{\jmath}$-convergent nets, i.e., if for every
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ one has
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(\tilde{E},{\tilde{\jmath}}\,)$.

In the case that $(\tilde{E},{\tilde{\jmath}}\,)=(E,j)$ this
$j$$j$-convergence means holds if and only if
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
preserves $j$-convergence. It follows that there always is a well-defined
limit for such operations which is an operator between the limit spaces.

######  Proposition 22.

Let $(E,j)$ and $(\tilde{E},{\tilde{\jmath}}\,)$ be inductive systems and let
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
an $j$$\tilde{\jmath}$-convergent net of operators. Then there is linear
operator $T_{\infty}:E_{\infty}\to\tilde{E}_{\infty}$, such that

$T_{\infty}\big{(}j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\big{)}={\tilde{\jmath}}\mkern
0.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}T_{n}x_{n},\quad
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j).$ (20)

Its norm is bounded by $\lVert
T_{\infty}\rVert_{\mathcal{L}(E_{\infty},\tilde{E}_{\infty})}\leq\varlimsup_{n}\lVert
T_{n}\rVert_{\mathcal{L}(E_{n},\tilde{E}_{n})}$.

###### Proof.

Well-definedness follows from uniform boundedness of the net
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$: If
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ and
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ are
$j$-convergent with the same limit, then $\lVert
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}T_{n}x_{n}-j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}T_{n}y_{n}\rVert\leq\mathop{\textup{lim}}_{n}\lVert
T_{n}\rVert\lVert x_{n}-y_{n}\rVert=0$. The norm bound is also immediate:
$\lVert
T_{\infty}j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\rVert=\mathop{\textup{lim}}_{n}\lVert
T_{n}x_{n}\rVert\leq\varlimsup_{n}\lVert T_{n}\rVert\lVert
x_{n}\rVert=\varlimsup_{n}\lVert T_{n}\rVert\lVert
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\rVert$. ∎

13, which states that
$(E_{\infty},j_{\infty{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}})$ satisfies the universal property of the limit space, follows as the
special case where $(\tilde{E},{\tilde{\jmath}}\,)=(F,\operatorname{id}_{F})$
is a constant inductive system. Another application is the notion of
$j^{*}$-convergence discussed in 6 which is $j$$\tilde{\jmath}$-convergence if
$(\tilde{E},{\tilde{\jmath}}\,)=(\mathbb{C},\operatorname{id})$. One can view
$j$$\tilde{\jmath}$-convergence as a generalization of strong convergence:

######  Example 23.

Let $(E,\operatorname{id}_{E})$ and $(F,\operatorname{id}_{F})$ be constant
inductive systems. Then a uniformly bounded sequence of operators $T_{n}:E\to
F$ is $\operatorname{id}_{E}\operatorname{id}_{F}$-convergent if and only if
it is strongly convergent and the limit $T_{\infty}$ is the strong limit.

Another special case of $j$$\tilde{\jmath}$-convergence is the convergence of
operations $T_{n}:E_{n}\to E_{f(n)}$ which change the index within a fixed
inductive systems. This allows for greater flexibility in describing
operations on the limit space (see for example Section 7.5). If $f:N\to N$ is
a monotone cofinal mapping, i.e., $n\leq m\implies f(n)\leq f(m)$ and
$\mathop{\textup{lim}}_{n}f(n)=\infty$, then we obtain a soft inductive system
by setting $\tilde{E}_{n}=E_{f(n)}$ and ${\tilde{\jmath}}_{nm}=j_{f(n)f(m)}$.
The limit space of this inductive system $(\tilde{E},{\tilde{\jmath}}\,)$ is
canonically isomorphic to $E_{\infty}$. Thus a $j$$\tilde{\jmath}$-convergent
net of operations defines an operation $T_{\infty}$ on $E_{\infty}$.

When it comes to proving $j$$\tilde{\jmath}$-convergence we have the following
criterion:

######  Lemma 24.

Let $(E,j)$ and $(\tilde{E},{\tilde{\jmath}}\,)$ be inductive systems and let
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be a
uniformly bounded net of of linear operators $T_{n}:E_{n}\to\tilde{E}_{n}$.

  1. (1)

If $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
maps $j$-basic sequences to $\tilde{\jmath}$-convergent sequences, then it is
$j$$\tilde{\jmath}$-convergent.

  2. (2)

$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$$\tilde{\jmath}$-convergent if and only if

$\mathop{\textup{lim}}_{n\gg
m}\lVert({\tilde{\jmath}}_{nm}T_{m}-T_{n}j_{nm})x_{m}\rVert=0\quad\forall
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j).$ (21)

  3. (3)

If $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
is $j$$\tilde{\jmath}$-convergent and
$S_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is a
${\tilde{\jmath}}\hat{\jmath}$-convergent net of uniformly bounded operators
from $(\tilde{E},{\tilde{\jmath}}\,)$ to an inductive system
$(\hat{E},\hat{\jmath})$, then the composition
$S_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j\hat{\jmath}$-convergent and
$(ST)_{\infty}=S_{\infty}T_{\infty}$.

###### Proof.

3 is clear. 1: We assume that
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ maps
basic sequences to $j$-convergent ones and set $M=\sup_{n}\allowbreak\lVert
T_{n}\rVert_{\mathcal{L}(E_{n},\tilde{E}_{n})}$. Let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
$j$-convergent. For $\varepsilon>0$, pick a basic net
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ such
that $\vvvert x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert<\varepsilon$, then

$\mathop{\textup{lim}}_{m}\,\vvvert
T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}T_{m}x_{m}\vvvert\leq\,M\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert+\mathop{\textup{lim}}_{m}\vvvert
y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}T_{m}x_{m}\vvvert+M\mathop{\textup{lim}}_{m}\lVert
x_{m}-y_{m}\rVert<2M\varepsilon.$

2 Assume $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ to be $j$$\tilde{\jmath}$-convergent, then

$\mathop{\textup{lim}}_{n\gg
m}\lVert({\tilde{\jmath}}_{nm}T_{m}-T_{n}j_{nm})x_{m}\rVert\leq\mathop{\textup{lim}}_{n\gg
m}\Big{\lparen}\lVert T_{n}x_{n}-{\tilde{\jmath}}_{nm}T_{m}x_{m}\rVert+\lVert
T_{n}\rVert\lVert x_{n}-j_{nm}x_{m}\rVert\Big{\rparen}=0.$

For the converse, we have

$\mathop{\textup{lim}}_{n\gg
m}\,\lVert(T_{n}x_{n}-{\tilde{\jmath}}_{nm}T_{m}x_{m}\rVert\leq\mathop{\textup{lim}}_{n\gg
m}\,\lVert T_{n}\rVert\lVert x_{n}-j_{nm}x_{m}\rVert+0=0.$

∎

The universal property of the soft inductive limit space follows from 22 as
the special case with a constant inductive system.

Let us assume that we have a, say, isometric action of a group $G$ on each
Banach space $E_{n}$. If the action preserves $j$-convergence (i.e. if
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\mapsto g\cdot
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$$j$-convergent), then the limit operations form an action of $G$ on
$E_{\infty}$. Actually the same holds, if $G$ is just a semigroup. If the
action of the group is strongly continuous with respect to some topology on
$G$, then we would want the same to hold for the limiting action. In the one-
parameter case, we will reduce the problem to studying convergence properties
of the infinitesimal generators. This requires generalizing $j$$j$-convergence
to nets of unbounded operators:

######  Definition 25.

Let $(E,j)$ and $(\tilde{E},{\tilde{\jmath}}\,)$ be inductive systems and let
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be a
net of unbounded operators $A_{n}:D(A_{n})\to\tilde{E}_{n}$. We define the net
domain of $A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ as

$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})\coloneqq\\{x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)\nonscript\ |\allowbreak\nonscript\
\mathopen{}x_{n}\in D(A_{n}),\
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(\tilde{E},{\tilde{\jmath}}\,)\,\\}.$ (22)

We say that $A_{\infty}$ is well-defined if
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is the same for all
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
with the same $j$-limit. In this case we define the limit operator on
$D(A_{\infty})=\\{j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\nonscript\
|\allowbreak\nonscript\
\mathopen{}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})\\}$ by

$A_{\infty}:D(A_{\infty})\to\tilde{E}_{\infty},\
A_{\infty}(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n})\coloneqq
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}x_{n}.$ (23)

Observe that a net of contractions
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ the
well-defined of the limit is always guaranteed and that
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$$\tilde{\jmath}$-convergent if and only if
$D(T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})=\mathscr{C}(E,j)$.

######  Lemma 26.

Let $(E,j)$ be an inductive system and let
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be a
net of operators $A_{n}:D(A_{n})\to E_{n}$. Then:

  1. (1)

$A_{\infty}$ is well-defined if and only if for all
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
with $\mathop{\textup{lim}}_{n}\lVert x_{n}\rVert=0$ we have
$\mathop{\textup{lim}}_{n}\lVert A_{n}x_{n}\rVert=0$,

  2. (2)

if all $A_{n}$ are closed operators, then
$\Gamma_{\infty}=\\{j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}\oplus
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}x_{n}\nonscript\
|\allowbreak\nonscript\
\mathopen{}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})\\}$ is a closed subspace of $E_{\infty}\oplus E_{\infty}$.

  3. (3)

Assume that there a $w^{*}$-dense subset of the dual space $E_{\infty}^{*}$
which arises as limits of $j^{*}$-convergent sequences
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ with the properties that $\varphi_{n}\in D(A_{n}^{*})$ and that
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{*}\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j^{*}$-convergent. Then $A_{\infty}$ is well-defined.

Now assume that $A_{\infty}$ is well-defined. Then

  1. 4.

$A_{\infty}$ is densely defined if and only if
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
is seminorm dense in $\mathscr{C}(E,j)$.

  2. 5.

if all $A_{n}$ are closable, then $A_{\infty}$ is a closed operator,

  3. 6.

if for each $n$, ${\mathcal{D}}_{n}$ is a core for $A_{n}$, then every
$x_{\infty}\in D(A_{\infty})$ is the $j$-limit of an
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
such that $x_{n}\in{\mathcal{D}}_{n}$ for all $n$.

###### Proof.

Denote the graph of $A_{n}$ by $\Gamma_{n}\subset E_{n}\oplus E_{n}$, and by
$\lVert\,\cdot\,\rVert_{A_{n}}$ the restriction of the norm on $E_{n}\oplus
E_{n}$ to $\Gamma_{n}$, i.e., $\lVert x_{n}\rVert_{A_{n}}=\lVert
x_{n}\rVert+\lVert A_{n}x_{n}\rVert$.

1: This is immediate because the difference of two nets in
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
with the same limit is an element of
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
such that $j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=0$.

2: We will show that $\Gamma_{\infty}$ is complete by proving that every
absolutely summable series converges in $\Gamma_{\infty}$ [RS12, Thm. III.3]
which will follow from the same property for all $\Gamma_{n}$. Let
$x_{\infty}^{(k)}\oplus y_{\infty}^{(k)}\in~\Gamma_{\infty}$ be an absolutely
summable sequence, i.e., $\sum_{k}\lVert x_{\infty}^{(k)}\oplus
y_{\infty}^{(k)}\rVert_{E_{\infty}\oplus E_{\infty}}<\infty$. We can pick
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(k)}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$ so
that
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}^{(k)}=x_{\infty}^{(k)}$,
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}x_{n}^{(k)}=y_{\infty}^{(k)}$
and $\lVert x_{n}^{(k)}\oplus A_{n}x_{n}^{(k)}\rVert_{E_{n}\oplus
E_{n}}=\lVert x_{\infty}^{(k)}\oplus y_{\infty}^{(k)}\rVert_{E_{\infty}\oplus
E_{\infty}}$. Then the series $\sum_{k}(x_{n}^{(k)}\oplus A_{n}x_{n}^{(k)})$
is absolutely and hence converges to some $x_{n}^{(\infty)}\oplus
A_{n}x_{n}^{(\infty)}\in\Gamma_{n}$. We claim that
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{(\infty)}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
and that $\sum_{k}(x_{\infty}^{(k)}\oplus
y_{\infty}^{(k)})=x_{\infty}^{(\infty)}\oplus
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}(A_{n}x_{n}^{(\infty)})$.
This follows from the fact that these series approximate their limits
uniformly in the index $n$.

3: Let $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ converges to zero, then

$\displaystyle\lVert
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}x_{n}\rVert$
$\displaystyle=\sup\mathop{\textup{lim}}_{n}\lvert\langle
A_{n}x_{n},\varphi_{n}\rangle\rvert$
$\displaystyle=\sup\mathop{\textup{lim}}_{n}\lvert\langle
x_{n},A_{n}^{*}(\varphi_{n})\rangle\rvert=\sup\lvert\langle
x_{\infty},j\mkern-0.5mu\hbox{${}^{*}$}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}(\varphi_{n})\rangle\rvert=0$

where the supremum is over all nets
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ with $\lVert
j\mkern-0.5mu\hbox{${}^{*}$}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}\varphi_{n}\rVert=1$
which satisfy the specified assumptions.

4: This follows item 3 of 5.

5: Follows from item 2.

6: Let $\varepsilon_{n}\searrow 0$ as $n\to 0$ and let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$.
Pick for each $n$ a $y_{n}\in{\mathcal{D}}_{n}$ such that $\lVert
x_{n}-y_{n}\rVert_{A_{n}}<\varepsilon_{n}$. Then $\mathop{\textup{lim}}_{n\gg
m}\lVert y_{n}-j_{nm}y_{m}\rVert\leq\mathop{\textup{lim}}_{n\gg
m}(\varepsilon_{n}+\varepsilon_{m}+\lVert x_{n}-j_{nm}x_{m}\rVert)=0$, i.e.,
$y\in\mathscr{C}(E,j)$ and $\lVert
x_{\infty}-y_{\infty}\rVert<\mathop{\textup{lim}}_{n}\varepsilon_{n}=0$. By a
similar argument one gets
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$. ∎

##  5 Dynamics on inductive systems

We say that $\\{T(t)\\}_{t\geq 0}$ is a dynamical semigroup on a Banach space,
if it is a strongly continuous one-parameter semigroup such that each $T(t)$
is a contraction. The generator of a dynamical semigroup is the closed
dissipative operator

$Ax=\mathop{\textup{lim}}_{t\searrow 0}\,\frac{T(t)x-x}{t},\quad
D(A)=\left\\{x\in E\nonscript\ \middle|\allowbreak\nonscript\
\mathopen{}t\mapsto T(t)x\text{ is in~}C^{1}(\mathbb{R}_{+},E)\,\right\\}.$
(24)

The resolvents of the generator are the operators
$R(\lambda)=(\lambda-A)^{-1}$, $\lambda\in\mathbb{C}$ with
$\operatorname{Re}\lambda>0$, they are an important tool in the theory of
dynamical semigroups and enjoy many nice properties, e.g., they are analytic
in $\lambda$ and one has $D(A)=\operatorname{Ran}R(\lambda)$. For an
introduction to dynamical semigroups we refer the reader to [EN01].

Consider now a dynamical semigroup $T_{n}(t)$ on each Banach space $E_{n}$ of
an inductive system $(E,j)$ and let $A_{n}$ be the net of generators. We will
need the notion of the net domain for nets of unbounded operators introduced
in 25. The main result of this section is the following theorem which relates
$j$-convergence preservation of the semigroups to a type of convergence of the
infinitesimal generators and $j$-convergence preservation of the resolvents.
It can be regarded as a generalization of the Trotter-Kato approximation
theorems, to which our result reduces in the case of a constant inductive
system.

######  Theorem 27.

Let $(E,j)$ be an inductive system and let
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
be a net of dynamical semigroups with
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ the
corresponding net of generators. Let $\lambda\in\mathbb{C}$ be such that
$\operatorname{Re}\lambda>0$. The following are equivalent:

  1. (1)

$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
is $j$$j$-convergent and the limit operators $T_{\infty}(t)$ are strongly
continuous in $t$.

  2. (2)

The net $R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)$ of resolvents $R_{n}(\lambda)=(\lambda-A_{n})^{-1}$ is
$j$$j$-convergent and the limit operation $R_{\infty}(\lambda)$ has dense
range.

  3. (3)

There is a seminorm dense subspace ${\mathcal{D}}\subset
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
such that $(\lambda-
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}){\mathcal{D}}$ is also seminorm dense.

  4. (4)

$A_{\infty}$ is well-defined, hence closed, and generates a dynamical
semigroup.

If these hold, the limit operations $T_{\infty}(t)$ form a strongly continuous
one-parameter semigroups and their generator is $A_{\infty}$. Furthermore, the
net domain is given by
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})=R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)\mathscr{C}(E,j)$ and the limits
${\mathcal{R}}_{\infty}(\lambda)$ of the resolvents are the resolvents of
$A_{\infty}$. In particular, these claims hold for some $\lambda$ if and only
if they hold for all $\lambda$.

If one only assumes that all
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
are $j$$j$-convergent, then it still follows that $T_{\infty}(t)$ is a one-
parameter semigroup. This is because
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(s)=T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t+s)$ holds as an equation of operators acting $\mathscr{C}(E,j)$.

Let us discuss some consequences of this theorem. In the case where one
already has a candidate for the semigroup on the limit space and is interested
in showing that the dynamics converges, we have the following criterion:

######  Corollary 28.

Let $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be a net of dynamical semigroups and let
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
the net of generators. Let $S(t)$ be a dynamical semigroup on $E_{\infty}$
with generator $B$ and let ${\mathcal{D}}_{\infty}$ be a core for $B$. Suppose
that there is a ${\mathcal{D}}\subset
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
with
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}{\mathcal{D}}={\mathcal{D}}_{\infty}$,
such that
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}x_{n}=B(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n})$
for all $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}$. Then
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
is $j$$j$-convergent, $A_{\infty}=B$ and $T_{\infty}(t)=S_{\infty}(t)$.

###### Proof.

We check item 3 of 27. ${\mathcal{D}}$ is seminorm dense because
${\mathcal{D}}_{\infty}$ is dense in $E_{\infty}$ (see 5). Similarly
$[(\lambda-A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}){\mathcal{D}}]_{\infty}=(\lambda-B){\mathcal{D}}_{\infty}$ is dense
because ${\mathcal{D}}_{\infty}$ is a core for $B$. ∎

Before we come to the we briefly discuss the analogous theorem for uniformly
continuous semigroups. I.e., for semigroup so that $T(t)$ depends continuously
on $t$ in the operator norm topology of $\mathcal{L}(E)$. For such semigroup,
the generator is always bounded and the semigroup is equal to the exponential
series $T(t)=e^{tA}$. For example, the convergence of this series and
$j$-convergence preservation of
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
implies $j$-convergence preservation of the semigroup under the right
assumptions.

######  Corollary 29.

Let $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be a net of uniformly continuous dynamical semigroups and let
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
the net of generators. The following are equivalent

  1. (1)

$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
is $j$$j$-convergent for all $t\geq 0$ and $T_{\infty}(t)$ is a uniformly
continuous semigroup,

  2. (2)

$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$$j$-convergent.

In this case it holds that $T_{\infty}(t)=e^{tA_{\infty}}$.

###### Proof of 27.

For the proof we introduce two additional statements

  1. (1′)

For all $t\geq 0$,
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
is $j$$j$-convergent and
$\\{x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)\nonscript\ |\allowbreak\nonscript\
\mathopen{}x_{n}\in D(A_{n}),\,\lVert
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}}<\infty\\}$ is seminorm dense in
$\mathscr{C}(E,j)$.

  2. (2′)

The resolvents $R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\mu)$ are $j$-convergent for all $\mu$ with $\operatorname{Re}\mu>0$
and the range of $R_{\infty}(\mu)$ is dense for all $\mu$.

We will show the following implications

${(1)}$${(2^{\prime})}$${(4)}$${(1^{\prime})}$${(2)}$${(3)}$

1 $\Rightarrow$ (2′): We use the integral formula for the resolvent
$R_{n}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}T_{n}(t)\,dt$ [EN01, Ch. II]
and get

$\vvvert R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}R_{m}(\lambda)x_{m}\vvvert\leq\int_{0}^{\infty}e^{-\operatorname{Re}\lambda
t}\vvvert T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}m}T_{m}(t)x_{m}\vvvert\,sdt$

for $j$-convergent
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
because of dominated convergence (for exchanging the limit in the definition
of the seminorm with the integral). Applying dominated convergence again to
the limit in $m$ and using that
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
$j$$j$-convergent shows that
$R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)$ also is. This argument works for all $\lambda$.

Similar to the above one can check that the integral formula and the resolvent
formula remain valid for the limits, i.e.,
$R_{\infty}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}T_{\infty}(t)\,dt$ and
$R_{\infty}(\lambda)-R_{\infty}(\mu)=(\mu-\lambda)R_{\infty}(\lambda)R_{\infty}(\mu)$.
The resolvent formula shows that the range of $R_{\infty}(\lambda)$ is
independent of $\lambda$. We can now approximate any $x_{\infty}\in
E_{\infty}$ by elements of the form $\lambda R_{\infty}(\lambda)x_{\infty}$
with $\lambda>0$:

$\lVert x_{\infty}-R_{\infty}(\lambda)\rVert\leq\int_{0}^{\infty}\lambda
e^{-\lambda t}\lVert x_{\infty}-T_{\infty}(t)x_{\infty}\rVert\,dt.$

The density of $\operatorname{Ran}R_{\infty}(\lambda)$ is equivalent to
density of the range of
$R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)$ as an operator on $\mathscr{C}(E,j)$. From the resolvent
equation $R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)-R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\mu)=(\mu-\lambda)R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\mu)$ it follows that this range is independent of $\lambda$. We can
now approximate any
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ by $\lambda
R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ with large $\lambda>0$:

$\displaystyle\mathop{\textup{lim}}_{\lambda\to\infty}\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)\vvvert$
$\displaystyle\leq\mathop{\textup{lim}}_{\lambda\to\infty}\mathop{\textup{lim}}_{n}\int_{0}^{\infty}\lambda
e^{-\lambda t}\lVert x_{n}-T_{n}(t)x_{n}\rVert\,dt$
$\displaystyle=\mathop{\textup{lim}}_{\lambda\to\infty}\int_{0}^{\infty}e^{-t}\lVert
x_{\infty}-T_{\infty}(t/\lambda)x_{\infty}\rVert\,dt=0$

where we used dominated convergence (twice) and strong continuity of
$T_{\infty}(t)$. In particular, it also follows that $R_{\infty}$ satisfies
the resolvent equation.

(2′) $\Rightarrow$ (1′): We use the following formula [Kat13, Ch. X§1.2] valid
for dynamical semigroups

$\left\lVert
T_{n}(t)x_{n}-\left\lparen(t/k)R_{n}(t/k)\right\rparen^{k}x_{n}\right\rVert\leq\frac{t^{2}}{2k}\lVert
A_{n}^{2}x_{n}\rVert,\quad x_{n}\in D(A_{n}^{2}),\,k\in\mathbb{N}.$

By assumption
$\left\lparen(t/k)R_{n}(t/k)\right\rparen^{k}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j$-convergent for all $j$-convergent
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ and
the right-hand side is uniformly bounded if
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$. Therefore
$T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}(t)$
preserves $j$-convergence of
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ in
${\mathcal{D}}=R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$. It remains to be shown that this subspace is seminorm dense. This
will then automatically show density of the subspace
$\\{x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)\nonscript\ |\allowbreak\nonscript\
\mathopen{}x_{n}\in D(A_{n}),\,\lVert
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}}<\infty\\}$. It is straightforward to see that
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})=R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)\mathscr{C}(E,j)$ and hence that
${\mathcal{D}}=R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)^{2}\mathscr{C}(E,j)$. Therefore the seminorm density of
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
is guaranteed by assumption of the density of the range of
$R_{\infty}(\lambda)$. Now let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ and $\varepsilon>0$. Pick a
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
such that $\vvvert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert<\frac{\varepsilon}{2}$. Then we can pick a $\lambda>0$ such
that $\vvvert y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-\lambda
R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert<\frac{\varepsilon}{2}$ with
$z_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=\lambda
R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}$, we have $\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-z_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert\leq\lVert
x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert+\lVert
y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-z_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert<\varepsilon$. This shows that ${\mathcal{D}}$ is dense.

(1′) $\Rightarrow$ 1: By assumption, it suffices to show strong continuity on
the dense subspace ${\mathcal{D}}_{\infty}$ of $j$-limits of nets in
${\mathcal{D}}=\\{x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)\nonscript\ |\allowbreak\nonscript\
\mathopen{}x_{n}\in D(A_{n}),\,\lVert
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}}<\infty\\}$. Let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}$. We use the formula
$T_{n}(t)x_{n}=x_{n}+\int_{0}^{t}T_{n}(s)A_{n}x_{n}\,ds$ to get the estimate

$\lVert T_{n}(t)x_{n}-x_{n}\rVert\leq\int_{0}^{t}\lVert
T_{n}(s)A_{n}x_{n}\rVert\,ds\leq t\lVert
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\rVert_{\mathbf{N}}\quad\forall n.$

Since the right-hand side goes to zero as $t\to 0$, we obtain $\vvvert
T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert=\lVert T_{\infty}(t)x_{\infty}-x_{\infty}\rVert\to 0$ and
hence strong continuity of $T_{\infty}(t)$.

(2′) $\Rightarrow$ 4: It is straightforward that the limits
$R_{\infty}(\lambda)$ satisfy the resolvent equation
$R_{\infty}(\lambda)-R_{\infty}(\mu)=(\mu-\lambda)R_{\infty}(\lambda)R_{\infty}(\mu)$.
This implies that the range of $R_{\infty}(\lambda)$ is independent of
$\lambda$ (and dense by assumption). Such families of operators are well-
studied under and are usually called “pseudoresolvents” (see [EN01, Ch. III,
4.6]). In fact, it follows that there is a closed operator
$B:E_{\infty}\supset D(B)\to E_{\infty}$ with
$D(B)=\operatorname{Ran}R_{\infty}(\lambda)$ and
$R_{\infty}(\lambda)=(\lambda-B)^{-1}$. We can use this to prove well-
definedness of $A_{\infty}$ (in the sense of 25): For
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
with $j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=0$, we
have

$\displaystyle
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}x_{n}$
$\displaystyle=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}(\lambda-
A_{n})x_{n}$
$\displaystyle=(\lambda-B)R_{\infty}(\lambda)j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}(\lambda-
A_{n})x_{n}$
$\displaystyle=(\lambda-B)j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}R_{n}(\lambda)(\lambda-
A_{n})x_{n}$
$\displaystyle=(\lambda-B)j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}=0.$

It is readily checked that we have
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})=R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)\mathscr{C}(E,j)$ and this implies that
$D(A_{\infty})=\operatorname{Ran}R_{\infty}(\lambda)=D(B)$. This can now be
used to show that $A=B$, by considering

$\displaystyle
BR_{\infty}(\lambda)j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}$
$\displaystyle=(1+\lambda
R_{\infty}(\lambda))j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}$
$\displaystyle=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}(x_{n}+\lambda
R_{n}(\lambda)x_{n})$
$\displaystyle=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}A_{n}R_{n}(\lambda)x_{n}$
$\displaystyle=A_{\infty}R_{\infty}(\lambda)j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}x_{n}$

We can now use the Lumer-Phillips theorem [EN01, Ch. III, Thm. 3.15] to prove
that $A_{\infty}=B_{\infty}$ generates a dynamical semigroup. Since we already
now that $\operatorname{Ran}(\lambda-
A_{\infty})^{-1}=\operatorname{Ran}R_{\infty}(\lambda)$ is dense, we only have
to show that $A_{\infty}$ is dissipative, i.e., that $\lVert(\lambda-
A_{\infty})x_{\infty}\rVert\geq\lambda\lVert x_{\infty}\rVert$. This follows
from dissipativity of all $A_{n}$ as for any
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$ we
find $\lVert(\lambda-A_{n})x_{n}\rVert\geq\lambda\lVert x_{n}\rVert$ and
taking the limit $n\to\infty$ shows the desired inequality.

4 $\Rightarrow$ 3: Put
${\mathcal{D}}=D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ which is seminorm dense because
${\mathcal{D}}_{\infty}=D(A_{\infty})$ is dense by assumption. Then
${\mathcal{D}}_{\infty}=\operatorname{Ran}R_{\infty}(\lambda)$ where
$R_{\infty}(\lambda)$ is the resolvent of $A_{\infty}$.

3 $\Rightarrow$ 2: Let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ be a net in the dense subspace $(\lambda-
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}){\mathcal{D}}$ and let
$y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}$ be so that
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=(\lambda-
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$. Then
$R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=y_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$. Since the subspace we chose
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ from
is seminorm dense, this shows that
$R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)$ is $j$$j$-convergent. Furthermore
$R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)\mathscr{C}(E,j)$ is dense because it contains the dense
subspace $R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda-
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}){\mathcal{D}}={\mathcal{D}}$.

(2) $\Rightarrow$ (2′): Let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
$j$-convergent and let $\mu$ be so that
$\lvert\lambda-\mu\rvert<\operatorname{Re}\lambda$. Then the series expansion

$R_{n}(\mu)x_{n}=R_{n}(\lambda)(1+(\mu-\lambda)R_{n}(\lambda))^{-1}x_{n}=R_{n}(\lambda)\sum_{k=0}^{\infty}[(\mu-\lambda)R_{n}(\lambda)]^{k}x_{n}$

converges uniformly in $n$. Therefore
$R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\mu)x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is approximated by $j$-convergent nets in the
$\lVert\,\cdot\,\rVert_{\mathbf{N}}$-norm and hence is itself $j$-convergent.
By iterating the argument we find that $j$-convergence holds for all $\mu$ in
the right half-plane of $\mathbb{C}$. ∎

In applications, there might symmetries that are not present on the spaces
$E_{n}$ but emerge in the limit. An example is the translation symmetry of
$L^{2}(\mathbb{R})$ viewed as the inductive limit of the spaces $L^{2}(I)$
where $I\subset\mathbb{R}$ are finite-length intervals (ordered by inclusion
with the obvious embeddings between the $L^{2}$-spaces). Nevertheless it can
make sense to approximate the generator of such a symmetry by operators which
do not generate anything (yet).

Recall that an operator $A$ is called _dissipative_ , if for all $\lambda>0$,

$\lVert(\lambda-A)y\rVert\geq\lambda\lVert y\rVert\quad\forall y\in D(A).$
(25)

All generators of dynamical semigroups are dissipative. In fact, a dissipative
operator $A$ generates a dynamical semigroup if and only if
$\operatorname{Ran}(\lambda-A)^{-1}$ is dense (this is the Lumer-Phillips
theorem [EN01, III, Thm. 3.15]).

######  Corollary 30.

Let $A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be a net of dissipative closed operators. Let ${\mathcal{D}}\subset
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$ be
seminorm dense in $\mathscr{C}(E,j)$ and assume that $(\lambda-
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}){\mathcal{D}}$ is also seminorm dense. Then $A_{\infty}$ is well-
defined, dissipative and generates a dynamical semigroup. Furthermore, the
resolvents $R_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)$ are $j$$j$-convergent for all $\lambda$ with
$\operatorname{Re}\lambda>0$ and $R_{\infty}(\lambda)=(\lambda-
A_{\infty})^{-1}$.

###### Proof.

We can proceed as in the proof of 27. The arguments used for the implications
3 $\Rightarrow$ 2 $\Rightarrow$ 4 still work under our assumptions. ∎

It is not hard to see that 27 is stable under perturbations of the generators
by $j$$j$-convergent nets of operators. In fact, we can even allow for certain
unbounded perturbations. To state the conditions, we recall the notion of
relative-boundedness. A linear operator $B:D(B)\to F$ on a Banach space $F$ is
_$A$ -bounded_ with respect to an operator $A:D(A)\to F$, if $D(A)\subset
D(B)$ and if there are constants $a,b>0$ such that

$\lVert By\rVert\leq a\lVert Ay\rVert+b\lVert x\rVert\quad\forall y\in D(A).$
(26)

The infimum $a_{0}$ over constants $a>0$ such that there is an $b>0$ for which
(26) holds, is called the $A$-bound of $B$. The classical perturbation result
that we will make use of is the following: Let $T(t)$ be a dynamical semigroup
with generator $A$ and let $B$ be a dissipative $A$-bounded operator with
$A$-bound $a_{0}<1$, then $A+B$ generates a dynamical semigroup.

######  Proposition 31.

Let $T_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)$ be a net of dynamical semigroups which is convergent in the
sense of 27 and let
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be
the net of generators. Let
$B_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ be a
net of dissipative operators, such that

  * •

$D(B_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})\supset
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$.
In particular, $D(B_{n})\supset D(A_{n})$ for all $n$,

  * •

there are $a<1$ and $b>0$, such that

$\lVert B_{n}x_{n}\rVert\leq a\lVert A_{n}x_{n}\rVert+b\lVert
x_{n}\rVert\quad\forall n,\,x_{n}\in E_{n}.$ (27)

Let $S_{n}(t)$ be the dynamical semigroup generated by $A_{n}+B_{n}$. Then the
net $S_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)$ of semigroups converges in the sense of 27 and the generator of
$S_{\infty}(t)$ is $A_{\infty}+B_{\infty}$.

###### Proof.

We will check condition 4 of 27. We define $C_{n}=A_{n}+B_{n}$ on
$D(C_{n})=D(B_{n})$. The assumption
$D(B_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})\supset
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
implies $D(C_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})=D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$, $D(A_{\infty})=D(C_{\infty})$ and $D(B_{\infty})\supset
D(A_{\infty})$. We check well-definedness of $C_{\infty}$: Let
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}\in
D(C_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
such that $x_{\infty}=0$, then $\mathop{\textup{lim}}_{n}\lVert
C_{n}x_{n}\rVert\leq\mathop{\textup{lim}}_{n}(\lVert A_{n}x_{n}\rVert+\lVert
B_{n}x_{n}\rVert)\leq(1+a)\lVert A_{\infty}x_{\infty}\rVert+b\lVert
x_{\infty}\rVert=0$. Finally, the relative-boundedness inequality (27) carries
over to the limit space by simply taking limits with
$x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
being a $j$-convergent net in
$D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$.
In particular, we know that $C_{\infty}=A_{\infty}+B_{\infty}$. Now the
standard perturbation theorem for dynamical semigroups [EN01, III, Thm. 2.7]
implies that $C_{\infty}=A_{\infty}+B_{\infty}$ generates a dynamical
semigroup. ∎

We can also use the technique of analytic vectors on $E_{\infty}$ to ensure
existence of the limit dynamics. This follows from the easy Lemma:

######  Lemma 32.

Let $A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be a net of closed dissipative operators and assume that there is a seminorm-
dense space ${\mathcal{D}}\subset
D(A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$ of
nets which are analytic in seminorm in the sense that there is a $t>0$ for
each $x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}$ so that

$\sum_{k\in\mathbb{N}}\,\frac{t^{k}\,\vvvert
A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{k}x_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\vvvert}{k!}<\infty.$ (28)

If $A_{\infty}$ is well-defined, then $A_{\infty}$ is a closed dissipative
operator on $E_{\infty}$ and
${\mathcal{D}}_{\infty}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}({\mathcal{D}})\subset
D(A_{\infty})$ is a dense set of analytical vectors.

In many situations this implies that $A_{\infty}$ generates a semigroup of
contractions (see, e.g., [BR87, Prop. 3.1.18–3.1.22], [Nel59] or [RS75]).

##  6 Inductive systems of C*-algebras and completely positive dynamics

Our techniques can be combined with additional structure. Roughly speaking,
any property that is asymptotically respected by the connecting maps $j_{nm}$
will passes onto the limit. In this section we illustrate this by analyzing
(soft) inductive systems of C*-algebras. For the connecting maps we allow for
completely positive contractions instead of requiring *-homomorphisms. For the
limit space to be a C*-algebra, we will need to assume asymptotic
multiplicativity (similar to [BK97]). This will, for example, allow for a
commutative limit space of non-commutative algebras (see Section 7.2) which
would otherwise not be possible. We will see that many expected results
automatically hold for such a setup. For example, the $j^{*}$-limit of a
$j^{*}$-convergent net of states
$\varphi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is a state on $\mathcal{A}_{\infty}$ and the limit operation of a
$j$$\tilde{\jmath}$-convergent net of completely positive contractions is a
completely positive contraction. In particular, the evolution theorem holds in
the category of C*-algebras and completely positive contractions.

######  Definition 33.

A soft inductive system of C*-algebras is a soft inductive system
$(\mathcal{A},j)$ of Banach spaces, such that all $\mathcal{A}_{n}$ are
C*-algebras, the connecting maps $j_{nm}$ are completely positive contractions
and satisfy the asymptotically homomorphism property:

$\mathop{\textup{lim}}_{n\gg
m}\,\big{\lVert}j_{nm}((j_{ml}a_{l})(j_{ml}b_{l}))-(j_{nl}a_{l})(j_{nl}b_{l})\big{\rVert}=0.$
(29)

If we assume that the C*-algebras are unital then we should also assume that
this structure is preserved by the connecting maps, i.e., we assume that
$j_{nm}{\mathbf{1}}_{\mathcal{A}_{m}}={\mathbf{1}}_{\mathcal{A}_{n}}$. In this
case the net of units is automatically $j$-convergent. It follows that the
limit space of a soft inductive system of (unital) C*-algebras is again a
(unital) C*-algebra:

######  Proposition 34.

Let $(\mathcal{A},j)$ be a soft inductive system of (unital) C*-algebras. Then
the adjoint $a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{*}\coloneqq(a_{n}^{*})$ of a $j$-convergent net
$a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ is
again $j$-convergent and products of $j$-convergent nets
$a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ and
$b_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ are
also $j$-convergent. The limit space becomes a (unital) C*-algebra with the
operations

$\left\lparen
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}\right\rparen^{*}\coloneqq
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}^{*}\quad\text{and}\quad\left\lparen
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}\right\rparen\left\lparen
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}\right\rparen\coloneqq
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}b_{n}.$ (30)

In the unital case, the net of units
${\mathbf{1}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is always $j$-convergent and the unit of $\mathcal{A}_{\infty}$ is
${\mathbf{1}}_{\mathcal{A}_{\infty}}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}{\mathbf{1}}_{\mathcal{A}_{n}}$.

###### Proof.

Suppose $a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is $j$-convergent, then
$a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}^{*}$
is also $j$-convergent because

$\lVert
a_{n}^{*}-j_{nm}a_{m}^{*}\rVert=\lVert(a_{n}-j_{nm}a_{m})^{*}\rVert=\lVert
a_{n}-j_{nm}a_{m}\rVert.$

It suffices to check $j$-convergence of products on basic nets
$a_{n}=j_{nl}a_{l}$ and $a_{n}=j_{nl}b_{l}$. We can assume both basic nets to
start at the same index because of 5. Set
$c_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}b_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}=(a_{n}b_{n})$, then

$\displaystyle\mathop{\textup{lim}}_{n\gg m}\lVert c_{n}-j_{nm}c_{m}\rVert$
$\displaystyle=\mathop{\textup{lim}}_{n\gg
m}\lVert(j_{nl}a_{l})(j_{nl}b_{l})-j_{mm}((j_{ml}a_{l})(j_{ml}b_{l}))\rVert=0.\qed$

In fact, Eq. 29 is equivalent to $j$-convergence of products of $j$-convergent
nets. For the remainder of this section $(\mathcal{A},j)$ denotes a unital
soft inductive system of C*-algebras. For a (soft) inductive system of
C*-algebras ${\mathbf{N}}(\mathcal{A},j)=\prod\mathcal{A}_{n}$ is also a
C*-algebra with $\mathscr{C}(\mathcal{A},j)$ being a C*-subalgebra (now
equipped with $\lVert\,\cdot\,\rVert_{\mathbf{N}}$) and by the above
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}:\mathscr{C}(\mathcal{A},j)\to\mathcal{A}_{\infty}$
is a *-homomorphism. In fact, we know that
$\ker(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}})$ is the closed
two-sided ideal of null nets, so that $\mathcal{A}_{\infty}$ is isomorphic to
the C*-quotient $\mathscr{C}(\mathcal{A},j)/\mathscr{C}_{0}(\mathcal{A},j)$:

The C*-algebras $\mathcal{A}_{n}$ (including the limit space) form a
_continuous field of C*-algebras_ [Dix82, Ch. 10] over the topological space
$N\cup\\{\infty\\}$ (equipped with the order topology) where the continuous
sections are precisely defined to be $j$-convergent nets including their
limit. This is discussed in more detail in Section 8.

######  Corollary 35.

  1. (1)

Let $a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be $j$-convergent. If all $a_{n}$ are unitary/normal/self-adjoint/ projections
then so is the $j$-limit
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}$.

  2. (2)

Let $a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be $j$-convergent, then
$\operatorname{Sp}(a_{\infty})\subset\bigcap_{k}\bigcup_{k>n}\operatorname{Sp}(a_{k})$.

  3. (3)

Let $a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
be a $j$-convergent net of (normal) elements. Let $\Omega\subset\mathbb{C}$ be
open and let $f:\Omega\to\mathbb{C}$ be an analytic (continuous) function. If
$\operatorname{Sp}a_{n}\subset\Omega$ for all $n$, then the analytic
(continuous) functional calculi
$f(a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}})$
are also $j$-convergent and
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}f(a_{n})=f(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n})$.

  4. (4)

An element of the limit space $a_{\infty}\in\mathcal{A}_{\infty}$ is positive
if and only if there is a $j$-convergent net
$a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(\mathcal{A},j)$ so that
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}=a_{\infty}$.

###### Proof.

1: All of these follow from the two properties that the adjoint and the
product preserve $j$-convergence and correspond to the adjoint and product on
$\mathcal{A}_{\infty}$.

2: Consider the C* inclusion
$\mathscr{C}(\mathcal{A},j)\subset{\mathbf{N}}(\mathcal{A},j)$. To compute
$\operatorname{Sp}(a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ we may regard
$a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ as
an element of $\prod\mathcal{A}_{n}={\mathbf{N}}(\mathcal{A},j)$ and we have
$z\in\operatorname{Sp}(a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ if and only if
$(z{\mathbf{1}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}-a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ is non-invertible if and only if $(z{\mathbf{1}}-A_{n})$ non-
invertible for some $n$, so that
$\operatorname{Sp}(a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})\subset\bigcup_{n}\operatorname{Sp}(a_{n})$. Since
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}:\mathscr{C}(\mathcal{A},j)\to\mathcal{A}_{\infty}$
is a *-homomorphism we have
$\operatorname{Sp}(j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n})\subset\operatorname{Sp}(a_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$.

3: One can start by showing the claim for polynomials and then extend to
continuous/analytic functions by approximation.

4: $a_{\infty}$ is positive if and only if
$a_{\infty}=(b_{\infty})^{*}b_{\infty}$ for some $b_{\infty}$. Now pick
$b_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}(E,j)$ with
$b_{\infty}=j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}b_{n}$.
Then $b_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}^{*}b_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ is also $j$-convergent and its limit is equal to $a_{\infty}$ which
proves the claim. ∎

A dynamical semigroup on a C*-algebra is a strongly continuous222This is also
called “point-norm continuous” elsewhere. one-parameter group of completely
positive contractions. We include the following characterization of generators
of dynamical semigroups on C*-algebras which is a consequence of the Arendt-
Chernoff-Kato theorem [ACK82]:

######  Lemma 36.

Let ${\mathcal{T}}(t)$ be a strongly continuous one-parameter semigroup on a
C*-algebra $\mathcal{A}$ and let ${\mathcal{L}}$ be its infinitesimal
generator. Then ${\mathcal{T}}(t)$ is completely positive if and only if the
generator is conditionally completely positive in the sense that for all
$0\leq x\in
D({\mathcal{L}}\otimes\operatorname{id}_{n})\subset{\mathbb{M}}_{n}(\mathcal{A})$
and all states $\omega$ on $\mathfrak{S}({\mathbb{M}}_{n}(\mathcal{A}))$:

$\omega(x)=0\implies\omega(({\mathcal{L}}\otimes\operatorname{id}_{n})(x))\geq
0.$ (31)

Another equivalent condition is the complete positivity of the resolvent
${\mathcal{R}}(\lambda)=(\lambda-{\mathcal{L}})^{-1}$ for all $\lambda>0$.

Here ${\mathbb{M}}_{n}(\mathcal{A})$ is the C*-algebra of $n\times n$ matrices
with entries in $\mathcal{A}$, which is isomorphic with
${\mathbb{M}}_{n}(\mathbb{C})\otimes\mathcal{A}$. The domain of
${{\mathcal{L}}\otimes\operatorname{id}_{n}}$ consists of all matrices whose
entries are in $D({\mathcal{L}})$, i.e.,
$D({\mathcal{L}}\otimes\operatorname{id}_{n})={\mathbb{M}}_{n}(D({\mathcal{L}}))$.

###### Proof.

It is sufficient to prove that ${\mathcal{T}}(t)$ is positive if and only if
Eq. 31 holds for $n=1$ which is the content of the Arendt-Chernoff-Kato
theorem. ∎

The Lemma and the Lumer-Phillips Theorem [EN01, Ch. III, Thm. 3.15] imply that
generators of dynamical semigroups are precisely the conditionally completely
positive closed dissipative operators ${\mathcal{L}}$ so that
$(\lambda-{\mathcal{L}})^{-1}$ has full range. We finish this section by
repeating the convergence theorem for nets of completely positive semigroups
on soft inductive systems of C*-algebras.

######  Theorem 37.

Let $(\mathcal{A},j)$ be a (soft) inductive system of C*-algebras and let
${\mathcal{T}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ be a net of dynamical semigroups with
${\mathcal{L}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ denoting the corresponding net of generators. Let
$\lambda\in\mathbb{C}$ be such that $\operatorname{Re}\lambda>0$. The
following are equivalent:

  1. (1)

${\mathcal{T}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)$ is $j$$j$-convergent and the limit operators
${\mathcal{T}}_{\infty}(t)$ are strongly continuous in $t$, hence form a
dynamical semigroup.

  2. (2)

The net
${\mathcal{R}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)$ of resolvents
${\mathcal{R}}_{n}(\lambda)=(\lambda-{\mathcal{L}}_{n})^{-1}$ is
$j$$j$-convergent and the limit operation ${\mathcal{R}}_{\infty}(\lambda)$
has dense range.

  3. (3)

There is a seminorm dense subspace ${\mathcal{D}}\subset
D({\mathcal{L}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})$ such that
$(\lambda-{\mathcal{L}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}){\mathcal{D}}$ is also seminorm dense.

  4. (4)

${\mathcal{L}}_{\infty}$ is well-defined, hence closed, and generates a
dynamical semigroup on $\mathcal{A}_{\infty}$.

If these hold, then it follows that
$D({\mathcal{L}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}})={\mathcal{R}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(\lambda)\mathscr{C}(E,j)$, that the limits of the resolvents are the
resolvents of ${\mathcal{L}}_{\infty}$ and that the semigroup generated
${\mathcal{L}}_{\infty}$ are the limits of
${\mathcal{T}}_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}(t)$. In particular, these claims hold for some $\lambda$ if and only
if they hold for all $\lambda$ with positive real part.

###### Proof.

One only has to check that (conditional) complete positivity is preserved in
the limit. That these are equivalent is a consequence of 36. That complete
positivity passes to the limit follows from combining the fact that $j$-limits
of positive nets are positive and that the matrix amplifications
${\mathbb{M}}_{\nu}(\mathcal{A}_{n})$ form a soft inductive system with the
connecting maps $j_{nm}\otimes\operatorname{id}_{\nu}$ whose limit space is
naturally isomorphic with ${\mathbb{M}}_{\nu}(\mathcal{A}_{\infty})$. The
former implies that the $T_{\infty}(t)$ are positive semigroups and the latter
implies that the same applies to all matrix amplifications, so that
$T_{\infty}(t)$ is completely positive. ∎

Finally, we discuss tensor products of soft inductive limits of C*-algebras.
If $\mathcal{A}$ and $\mathcal{B}$ are unital C*-algebras we denote by
$\mathcal{A}\odot\mathcal{B}$, $\mathcal{A}\otimes_{min}\mathcal{B}$ and
$\mathcal{A}\otimes_{max}\mathcal{B}$ the algebraic, minimal and maximal (C*-)
tensor product respectively. Let $(\mathcal{A},j)$ and $(\mathcal{B},k)$ be
soft inductive limits of C*-algebras over the same directed set $N$ and set

$(\mathcal{A}\otimes_{*}\mathcal{B})_{n}=\mathcal{A}_{n}\otimes_{*}\mathcal{B}_{n}\quad\text{and}\quad(j\otimes
k)_{nm}=j_{nm}\otimes k_{nm},$

where $*$ is either _min_ or _max_. Note that $(j\otimes k)_{nm}$ are unital
completely positive maps in both cases.

######  Proposition 38.

Let $\otimes_{*}$ denote either the minimal or maximal C*-tensor product.

  1. (1)

$(\mathcal{A}\otimes_{*}\mathcal{B},j\otimes k)$ is a soft inductive system of
C*-algebras. If
$a_{i,{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}}\in\mathscr{C}(\mathcal{A},j)$ and
$b_{i,{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}}$,
$i=1,\ldots k$, then
$\sum_{i=1}^{k}a_{i,{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}}\otimes b_{i,{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}}\in\mathscr{C}(\mathcal{A}\otimes_{*}\mathcal{B},j\otimes k)$. This
induces an embedding of the algebraic tensor product
$\mathcal{A}_{\infty}\odot\mathcal{B}_{\infty}$ into the limit space
$(\mathcal{A}\otimes_{*}\mathcal{B})_{\infty}$. This embedding has dense
range.

  2. (2)

The maps $\phi_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}$ with $\phi_{n}:=j_{\infty n}\otimes k_{\infty
n}:\mathcal{A}_{n}\otimes_{*}\mathcal{B}_{n}\to\mathcal{A}_{\infty}\otimes_{*}\mathcal{B}_{\infty}$
take $(j\otimes k)$-convergent nets to Cauchy nets. The limit operation
$\phi_{\infty}:(\mathcal{A}\otimes_{*}\mathcal{B})_{\infty}\to\mathcal{A}_{\infty}\otimes_{*}\mathcal{B}_{\infty}$
is a surjective *-homomorphism.

  3. (3)

$(\mathcal{A}\otimes_{max}\mathcal{B})_{\infty}$ is the maximal C*-tensor
product of $\mathcal{A}_{\infty}$ and $\mathcal{B}_{\infty}$.

  4. (4)

$(\mathcal{A}\otimes_{min}\mathcal{B})_{\infty}$ is a C*-tensor product of
$\mathcal{A}_{\infty}$ and $\mathcal{B}_{\infty}$. $\phi_{\infty}$ is the
canonical homomorphism onto the minimal C*-tensor product.

In general, $(\mathcal{A}\otimes_{min}\mathcal{B})_{\infty}$ is not equal to
the minimal C*-tensor product. Counterexamples exist already for strict
inductive systems $(\mathcal{A},j)$ of C*-algebras where the $j_{nm}$ are
*-homomorphisms [Bla06, II.9.6.5].

###### Proof.

1: It suffices to check $(j\otimes k)$-convergence and asymptotic
multiplicaticity on basic sequences of the form $(j\otimes
k)_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}l}x_{l}$
with $x_{l}\in\mathcal{A}_{l}\odot\mathcal{B}_{l}$. For these the properties
follows from their counterparts on $(\mathcal{A},j)$ and $(\mathcal{B},k)$ and
the triangle inequality. Furthermore such basic sequences are seminorm dense
which implies that the embedding of the algebraic tensor product is dense.

2: It suffices to check basic nets with $(j\otimes
k)_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}l}x_{l}$
with $x_{l}=\sum a_{i}\otimes b_{i}\in\mathcal{A}_{n}\odot\mathcal{B}_{n}$
where the claim follows again from the triangle inequality. That
$\phi_{\infty}$ is a *-homomorphism is clear from the asymptotic
multiplicativity of $j_{\infty n}$ and $k_{\infty n}$. That it is surjective
follows from the fact that the algebraic tensor product
$\mathcal{A}_{\infty}\odot\mathcal{B}_{\infty}$ is dense in
$\mathcal{A}_{\infty}\otimes_{*}\mathcal{B}_{\infty}$ and is part of the range
of $\phi$.

3 and 4: We start by showing that
$(\mathcal{A}\otimes_{*}\mathcal{B})_{\infty}$ is a C*-tensor product, i.e.,
that $\lVert\sum_{i}a_{i,\infty}\otimes
b_{i,\infty}\rVert_{\beta}=\lVert(j\\!\otimes\\!k)\mkern-0.5mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{i,n}\otimes
b_{i,n}\rVert$ is a C*-norm on the algebraic tensor product [Tak01, Sec.
IV.4]. The triangle inequality, submultiplicativity and the C*-property
$\lVert x^{*}x\rVert_{\beta}=\lVert x\rVert_{\beta}^{2}$ are inherited from
the norm on the limit space. Furthermore,
$\lVert(j\\!\otimes\\!k)\mkern-0.5mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{n}a_{n}\otimes
b_{n}\rVert=\mathop{\textup{lim}}_{n}\lVert a_{n}\otimes
b_{b}\rVert\leq\mathop{\textup{lim}}_{n}\lVert a_{n}\rVert\lVert
b_{n}\rVert\leq\lVert a_{\infty}\rVert\lVert b_{\infty}\rVert$. Non-degeneracy
of $\lVert\,\cdot\,\rVert_{\beta}$ holds by 2 which proves
$\lVert\,\cdot\,\rVert_{\beta}\leq\lVert\,\cdot\,\rVert_{*}$. If $*=$ _max_ ,
then this implies $\lVert\,\cdot\,\rVert_{\beta}=\lVert\,\cdot\,\rVert_{max}$
[Tak01, Sec. IV.4]. ∎

##  7 Examples and Applications

###  7.1 Quantum dynamics in the classical limit

An approach to the classical limit via soft inductive limits of C*-algebras
was published in [Wer95]. We discuss here a corresponding version in the
Schrödinger picture with an emphasis on convergence of quantum dynamics to
classical dynamics in the classical limit. We are planning to publish these
and more results on the classical limit with full proofs in the future and
will focus on conveying the bigger picture here.

The underlying directed set for the classical limit is the interval $(0,1]$
directed towards zero. For this reason we write “$0$” instead of “$\infty$” to
denote limit objects. An element $\hbar\in(0,1]$ is thought of as an _action
scale_ of the quantum system with $d$ canonical degrees of freedom with
Hilbert space $L^{2}(\mathbb{R}^{d},dx)$. The role of the connecting maps
$j_{\hbar\hbar^{\prime}}$ that we will introduce is to change the action scale
from $\hbar$ to $\hbar^{\prime}$. We will refer to a family of objects (e.g.
states) indexed by $\hbar\in(0,1]$ as an ($\hbar$-)sequence because the
directed set is totally (and strictly) ordered so that “net” seems to be an
unnecessarily complicated term.

The Banach space in which the states of the quantum system at scale $\hbar$
live is the trace class
${\mathfrak{T}}_{\hbar}\coloneqq{\mathfrak{T}}(\mathcal{H})$. To define the
connecting maps
$j_{\hbar\hbar^{\prime}}:{\mathfrak{T}}_{\hbar^{\prime}}\to{\mathfrak{T}}_{\hbar}$,
we start with the well-known coherent-state quantization and its
dequantization counterpart which we denote by $j_{\hbar 0}$ and $j_{0\hbar}$,
respectively. By $|z\rangle_{\hbar}$, $z\in\mathbb{R}^{2d}$, we denote the
coherent-state vectors obtained from displacing the ground state
$|0\rangle_{\hbar}$ of the harmonic oscillator333The Harmonic oscillator has
the $\hbar$-scaling $H_{\hbar}=\frac{1}{2}(x^{2}-\hbar^{2}\Delta_{x})$ and the
displacement operators are $W_{z}^{\hbar}\psi(x)=\exp\\{(i/\hbar)(p\cdot
x+i\hbar\nabla_{q})\\}\psi(x)$, $z=(q,p)$. If $|0\rangle_{\hbar}$ is the
ground state of $H_{\hbar}$, then the coherent states are
$|z\rangle_{\hbar}=W_{z}^{\hbar}|0\rangle_{\hbar}$.. With these we define

$\displaystyle j_{\hbar 0}(\rho_{0})$
$\displaystyle\coloneqq\int_{\mathbb{R}^{2d}}\rho_{0}(z)\,|z\rangle\langle
z|_{\hbar}\ dz,$ $\displaystyle\rho_{0}$ $\displaystyle\in
L^{1}(\mathbb{R}^{2d})$ (32) $\displaystyle j_{0\hbar}(\rho_{\hbar})(z)$
$\displaystyle\coloneqq\tfrac{1}{(2\pi\hbar)^{d}}\ {\langle
z|\rho_{\hbar}|z\rangle_{\hbar}},$ $\displaystyle\rho_{\hbar}$
$\displaystyle\in{\mathfrak{T}}_{\hbar}.$ (33)

These maps take probability distributions (or rather their densities) on phase
space to quantum states and vice versa444This is a consequence of the
overcompleteness property $\int|z\rangle\langle
z|_{\hbar}\,dz=(2\pi\hbar)^{d}{\mathbf{1}}$.. They are also known as upper and
lower symbol or Wick dequantization and anti-Wick quantization, respectively
(see [Fol16]). The connecting maps are simply defined by

$j_{\hbar\hbar^{\prime}}\coloneqq j_{\hbar 0}\circ
j_{0\hbar^{\prime}}:{\mathfrak{T}}_{\hbar^{\prime}}\to{\mathfrak{T}}_{\hbar}.$
(34)

By construction the connecting maps are completely positive and trace-
preserving (so called _quantum channels_). In particular, they are linear
contractions.

######  Lemma 39.

$({\mathfrak{T}},j)$ is a soft inductive limit of Banach spaces, i.e.,

$\mathop{\textup{lim}}_{\hbar\ll\hbar^{\prime}}\lVert(j_{\hbar\hbar^{\prime\prime}}-j_{\hbar\hbar^{\prime}}j_{\hbar^{\prime}\hbar^{\prime\prime}})\rho_{\hbar^{\prime\prime}}\rVert_{1}=0\quad\forall{\hbar^{\prime\prime}}>0,\
\rho_{\hbar^{\prime\prime}}\in{\mathfrak{T}}_{\hbar^{\prime\prime}}.$ (35)

The limit space ${\mathfrak{T}}_{0}$ is isometrically isomorphic with
$L^{1}(\mathbb{R}^{2d})$ with the isomorphism being defined by

${\mathfrak{T}}_{0}\cong L^{1}(\mathbb{R}^{2d})\quad\text{via}\quad
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}\rho_{\hbar}\mapsto\mathop{\textup{lim}}_{\hbar}j_{0\hbar}\rho_{\hbar}.$
(36)

Indeed, if an $\hbar$-sequence
$\rho_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$
is $j$-convergent, then its dequantizations $j_{0\hbar}\rho_{\hbar}$ converge
in the topology of $L^{1}(\mathbb{R}^{2d})$. The abstract maps $j_{0\hbar}$
(defined as in Eq. 9) and the one defined in Eq. 33 coincide (modulo the above
isomorphism), i.e. one has
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}j_{\hbar\hbar^{\prime}}\rho_{\hbar^{\prime}}=j_{0\hbar^{\prime}}\rho_{\hbar^{\prime}}$.

###### Proof.

As we have
$(j_{\hbar\hbar^{\prime\prime}}-j_{\hbar\hbar^{\prime}}j_{\hbar^{\prime}\hbar^{\prime\prime}})=j_{\hbar
0}(\operatorname{id}-j_{0\hbar}j_{\hbar 0})j_{0\hbar^{\prime\prime}}$, it
follows that the norm in Eq. 35 is bounded by
$\lVert(\operatorname{id}-j_{0\hbar^{\prime}}j_{\hbar^{\prime}0})\sigma_{0}\rVert_{1}$
with $\sigma_{0}=j_{0\hbar^{\prime\prime}}\rho_{\hbar^{\prime\prime}}$. It is
well-known that $j_{0\hbar}j_{\hbar 0}$ is the heat transform at time $\hbar$,
i.e. it convolves with a Gaussian having variance $\hbar$, and the claim
follows from strong-continuity of the heat semigroup on
$L^{1}(\mathbb{R}^{2n})$.

One sees that $j_{0\hbar}\rho_{\hbar}$ is a Cauchy sequence in $\hbar$ for any
$\rho_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}({\mathfrak{T}},j)$ by considering

$\displaystyle\lVert
j_{0\hbar}\rho_{\hbar}-j_{0\hbar^{\prime}}\rho_{\hbar^{\prime}}\rVert_{1}$
$\displaystyle\leq\lVert
j_{0\hbar}(\rho_{\hbar}-j_{\hbar\hbar^{\prime}}\rho_{\hbar^{\prime}})\rVert_{1}+\lVert(j_{0\hbar}j_{\hbar
0}-\operatorname{id})j_{0\hbar^{\prime}}\rho_{\hbar^{\prime}}\rVert_{1}\xrightarrow{\hbar\ll\hbar^{\prime}}0.\qed$

A special feature of this soft inductive system, not assumed in the abstract
setup of soft inductive systems, are the quantization maps $j_{\hbar 0}$ and
their properties. For any $\rho_{0}\in L^{1}(\mathbb{R}^{2d})$ one has that
$j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}0}\rho_{0}$ is $j$-convergent and the limit is
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}j_{\hbar
0}\rho_{0}=\rho_{0}$. This has the consequence that for any dense subspace
${\mathcal{D}}_{0}\in L^{1}(\mathbb{R}^{2d})$, the space
${\mathcal{D}}=\\{j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}0}\rho_{0}\nonscript\ |\allowbreak\nonscript\
\mathopen{}\rho_{0}\in{\mathcal{D}}_{0}\\}$ is seminorm dense in
$\mathscr{C}(E,j)$. Furthermore, this makes for the following simplification:
$j^{*}$-convergence of a sequence
$A_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip 1.0pt}{}}$ of
observables $A_{\hbar}\in{\mathcal{L}}(\mathcal{H})$ is equivalent to
existence of the limit
$w^{*}\mkern-3.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}j_{0\hbar}A_{\hbar}$
in $L^{\infty}(\mathbb{R}^{2d})=(L^{1}(\mathbb{R}^{2d}))^{\prime}$. Another
interesting property of this specific soft inductive system is that even
though the spaces ${\mathfrak{T}}_{\hbar}$ are the same for all $\hbar$ the
limit space is different.

We now turn to the discussion of dynamics in the classical limit. This is done
separately for Hamiltonian dynamics case and irreversible dynamics generated
by a Lindblad operator. Both of these cases will be proved by applying 28,
which reduces the problem to an infinitesimal one provided that one already
has good control over the expected limit dynamics. We start with the
Hamiltonian case, where the main result is:

######  Theorem 40.

Let $V:\mathbb{R}^{d}\to\mathbb{R}$ be a $C^{2}$-potential with bounded
second-order derivatives and consider the Schrödinger operator
$H_{\hbar}=-\frac{\hbar^{2}}{2}\Delta+V(x)$ and the classical Hamiltonian
function $H_{0}(q,p)=\frac{1}{2}p^{2}+V(q)$. Then the quantum dynamics
$\rho_{\hbar}\mapsto
e^{-\frac{i}{\hbar}tH_{\hbar}}\rho_{\hbar}e^{\frac{i}{\hbar}tH_{\hbar}}$ is
convergent in the sense of 27. The limiting operation is the classical time
evolution generated by $H_{0}$, i.e., the classical limit of time-evolved
quantum states follows the classical flow generated by $H_{0}$. In particular,

$\frac{d}{dt}\bigg{|}_{t=0}\
j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}\Big{\lparen}e^{-\frac{i}{\hbar}H_{\hbar}}\rho_{\hbar}e^{\frac{i}{\hbar}tH_{\hbar}}\Big{\rparen}=\Big{\\{}H_{0},j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}\rho_{\hbar}\Big{\\}}\quad\forall\rho_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in\mathscr{C}({\mathfrak{T}},j)$ (37)

provided that
$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}\rho_{\hbar}$
is suitably differentiable, where $\\{f,g\\}$ denotes the Poisson bracket of
functions $f$ and $g$.

The assumptions on the potential guarantee that $H_{\hbar}$ is essentially
self-adjoint on the domain of Schwartz-functions for every $\hbar$ [Bau+23]
and for the dynamics of $H_{0}$ to exist for all times and all initial values
(by the Picard-Lindelöf theorem).

###### Sketch of proof.

It can readily be checked that $C_{c}^{2}(\mathbb{R}^{2d})$ is a core for the
classical dynamics generated by $H_{0}$ on $L^{1}(\mathbb{R}^{2d})$. Applying
28 to
${\mathcal{D}}=\\{j_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}0}\rho_{0}\nonscript\ |\allowbreak\nonscript\ \mathopen{}\rho_{0}\in
C_{c}^{2}(\mathbb{R}^{2d})\\}$ reduces the problem to showing that commutators
converge to Poisson brackets on ${\mathcal{D}}$, i.e. showing that

$j\mkern-1.0mu\hbox{-}\mkern-3.0mu\mathop{\textup{lim}}_{\hbar}\Big{\lparen}-\frac{i}{\hbar}\big{[}H_{\hbar},\rho_{\hbar}\big{]}\Big{\rparen}=\\{H_{0},\rho_{0}\\}\quad\forall\rho_{{{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\hskip
1.0pt}{}}\in{\mathcal{D}}.$ (38)

One can reduce the prove to the case where $H=j_{\hbar 0}H_{0}$ where the
claim can proved explicitly using the explicit form of the coherent state
quantization. ∎

The proof in fact works for a much larger class of Hamiltonians containing all
coherent-state quantized Hamiltonians $H_{\hbar}=(2\pi\hbar)^{-d}\int
H_{0}(z)|z\rangle\langle z|_{\hbar}dz$ of $C^{2}$-functions with bounded
second-order derivatives and all Weyl quantizations of $C^{2d+3}$-functions
with uniformly bounded derivatives of second and higher orders (both of these
conditions guarantee essential self-adjointness [Bau+23, FL23]). A version of
this theorem for bounded Hamiltonians was already included in [Wer95]. The
extension to unbounded Hamiltonians is taken from [Lui21] and again builds on
27.

We now turn to discussing irreversible dynamics. On the quantum side,
irreversible dynamics correspond to strongly continuous semigroups of trace-
preserving completely positive maps [SHW17]. Consider the Lindbladian of a
Gaussian (or quasi-free) dynamical semigroup [DW22, BW]

$\displaystyle{\mathcal{L}}_{\hbar}(\rho_{\hbar})$
$\displaystyle=-\frac{i}{2\hbar}\sum_{jk}\Big{\lparen}A_{jk}\big{[}R_{j}R_{k},\rho_{\hbar}\big{]}+iM_{jk}\big{\lparen}R_{j}[R_{k},\rho_{\hbar}]+[\rho_{\hbar},R_{j}]R_{k}\big{\rparen}\Big{\rparen}$
(39)

where $R=(x_{1},\ldots,x_{d},-i\hbar\partial_{1},\ldots,-i\hbar\partial_{d})$
is the vector of canonical operators and where $A$ is a symmetric real matrix
and $M$ is a positive semi-definite complex matrix. We remark that Eq. 39 can
be transferred to the form
${\mathcal{L}}_{\hbar}(\rho_{\hbar})=-(i/\hbar)[H,\rho]+(1/2\hbar)\sum_{j}(L_{j}[\rho_{\hbar},L_{j}^{*}]+[L_{j},\rho_{\hbar}]L_{j}^{*})$
with jump operators $L_{j}=\sum_{k}(\sqrt{M})_{jk}R_{k}$ and Hamiltonian
$H=\frac{1}{2}R\cdot AR$. Denote the symplectic matrix by $\sigma$.

######  Theorem 41.

For any complex positive semi-definite matrix $M$ and any real symmetric
matrix $A$, the Gaussian dynamical semigroup generated by Eq. 39 is convergent
in the sense of 27. The limit semigroup on $L^{1}(\mathbb{R}^{2d})$ is
generated by the first-order differential operator
${\mathcal{L}}_{0}=z^{\top}\cdot(A-\operatorname{Im}M)\sigma\nabla$,
$z\in\mathbb{R}^{2d}$.

The notation $\operatorname{Im}M$ means $\frac{i}{2}(M^{*}-M)$. Note that any
real $2d\times 2d$ matrix can be written as $K=(A-\operatorname{Im}M)\sigma$
for a symmetric real matrix $A$ and a complex positive semi-definite matrix