# The strongly Leibniz property and the Gromov–Hausdorff propinquity

Konrad Aguilar  Department of Mathematics and Statistics, Pomona College, 610
N. College Ave., Claremont, CA 91711<EMAIL_ADDRESS>https://aguilar.sites.pomona.edu ,  Stephan Ramon Garcia  Department of
Mathematics and Statistics, Pomona College, 610 N. College Ave., Claremont, CA
91711<EMAIL_ADDRESS>https://pages.pomona.edu/~sg064747 ,  Elena
Kim  Department of Mathematics, Massachusetts Institute of Technology, 77
Massachusetts Ave, Cambridge, MA 02139<EMAIL_ADDRESS>and  Frédéric
Latrémolière  Department of Mathematics, University of Denver, Denver, CO
80208<EMAIL_ADDRESS>https://math.du.edu/~frederic/

###### Abstract.

We construct a new version of the dual Gromov–Hausdorff propinquity that is
sensitive to the strongly Leibniz property. In particular, this new distance
is complete on the class of strongly Leibniz quantum compact metric spaces.
Then, given an inductive limit of C*-algebras for which each C*-algebra of the
inductive limit is equipped with a strongly Leibniz L-seminorm, we provide
sufficient conditions for placing a strongly Leibniz L-seminorm on an
inductive limit such that the inductive sequence converges to the inductive
limit in this new Gromov–Hausdorff propinquity. As an application, we place
new strongly Leibniz L-seminorms on AF-algebras using Frobenius–Rieffel norms,
for which we have convergence of the Effros–Shen algebras in the
Gromov–Hausdorff propinquity with respect to their irrational parameter.

###### Key words and phrases:

Gromov–Hausdorff propinquity, quantum metric spaces, Effros–Shen algebras

###### 2000 Mathematics Subject Classification:

Primary: 46L89, 46L30, 58B34.

The first author gratefully acknowledges the financial support from the
Independent Research Fund Denmark through the project ‘Classical and Quantum
Distances’ (grant no. 9040-00107B)

The second author is partially supported by NSF grants DMS-1800123 and
DMS-2054002

The third author is supported by the National Science Foundation Graduate
Research Fellowship under Grant No. 1745302

##  1\. Introduction and Background

What is the analogue of a Lipschitz seminorm in noncommutative geometry?
Interest in this question stems from the observation that the Lipschitz
seminorm defined by the metric of a compact metric space encodes the
underlying metric at the level of its C*-algebra of ${\mathds{C}}$-valued
continuous functions. Thus, an appropriate notion of a noncommutative
Lipschitz seminorm is the core ingredient in a noncommutative metric space. A
good definition of a noncommutative Lipschitz seminorm should be flexible
enough to include many interesting examples and strong enough to enable the
development of an interesting theory. For our purposes, an interesting theory
is a theory of convergence which extends the Gromov–Hausdorff distance [9]
between compact metric spaces to noncommutative geometry.

The origin of the theory of quantum compact metric spaces is found in Connes’
work on spectral triples, which provide noncommutative generalizations of
Riemannian manifolds [6, 5]. Rieffel then addressed the question of defining
noncommutative analogues of metric spaces without reference to a differential
structure [25, 26], designed to be flexible while still enabling him to define
the quantum Gromov–Hausdorff distance between such spaces [27]. Quantum
compact metric spaces were given by ordered pairs
$({\mathfrak{A}},{\mathsf{L}})$ of an order unit space ${\mathfrak{A}}$ and a
seminorm ${\mathsf{L}}$ that induces a metric on the state space
${\mathscr{S}}({\mathfrak{A}})$ of ${\mathfrak{A}}$ that metrizes the weak*
topology, called the Monge–Kantoroich metric, denoted
${\mathsf{mk}_{{\mathsf{L}}}}$. The metric ${\mathsf{mk}_{{\mathsf{L}}}}$
generalizes the Monge–Kantorovich distance [11], built from the usual
Lipschitz seminorm, and Connes distance [5], built from spectral triples.

The study of Rieffel’s quantum Gromov–Hausdorff distance leads to two natural
questions. One concerns the coincidence property. If ${\mathfrak{A}}$ and
${\mathfrak{B}}$ are two unital C*-algebras such that
$(\mathfrak{sa}({\mathfrak{A}}),{\mathsf{L}}_{\mathfrak{A}})$ and
$(\mathfrak{sa}({\mathfrak{B}}),{\mathsf{L}}_{\mathfrak{B}})$, where
$\mathfrak{sa}({\mathfrak{A}})$ is the order unit space of self-adjoint
elements of ${\mathfrak{A}}$, are Rieffel’s quantum compact metric spaces
which are at distance zero with respect to Rieffel’s distance, then it is not
clear that ${\mathfrak{A}}$ and ${\mathfrak{B}}$ are *-isomorphic. Attempts at
strengthening Rieffel’s construction to get this desirable property typically
involve restricting the notion of a quantum compact metric spaces, for
example, to a class of operator systems [12] or to C*-algebras [17]. The other
question is whether one can extend Rieffel’s distance to encompass more
structures than the quantum metrics. For example, when working with quantum
compact metric spaces of the form
$(\mathfrak{sa}({\mathfrak{A}}),{\mathsf{L}})$ with ${\mathfrak{A}}$ a unital
C*-algebra, we are interested in convergence for modules, group actions, and
other such higher structures over the underlying C*-algebras. The two matters
are related: they require an analogue of the Gromov–Hausdorff distance which
is well adapted to C*-algebras. For instance, Rieffel’s work on convergence of
modules [28, 29, 32, 31, 34] highlighted that a relationship between
L-seminorms and the multiplicative structure of the underlying C*-algebra of a
quantum compact metric space was desirable. Rieffel introduced the notion of a
_strongly Leibniz_ L-seminorm to this end. However, the quantum
Gromov–Hausdorff distance is not well adapted to working with such seminorms
(see the discussion of the proximity in [31]).

A possible answer to the search for a noncommutative analogue of the
Gromov–Hausdorff distance adapted to C*-algebras and Leibniz seminorms was
proposed by the fourth author, with the introduction of the propinquity [17,
15, 18, 14] on a class of quantum compact metric spaces constructed out of
C*-algebras and L-seminorms satisfying some form of Leibniz inequality.
Moreover, the dual Gromov–Hausdorff propinquity [15] is complete on the class
of Leibniz quantum compact metric spaces. However, the strongly Leibniz
property has not been addressed. Thus, in Section 2, we introduce a form of
the dual Gromov–Hausdorff propinquity on the class of strongly Leibniz quantum
compact metric spaces and show that it is complete on this class. In Section
3, we apply this to construct strongly Leibniz L-seminorms on certain
inductive limits of C*-algebras for which the inductive sequence converges to
the inductive limit in the Gromov–Hausdorff propinquity. To do this, we assume
there exist strongly Leibniz L-seminorms on the C*-algebras of the inductive
sequence that satisfy natural assumptions, as done in [1]. Moreover, we find
strongly Leibniz L-seminorms on all unital AF-algebras equipped with faithful
tracial state using Frobenius–Rieffel norms [24, 2] following a suggestion
from Rieffel. These new seminorms still preserve the convergence results of
[3], including the convergence of Effros–Shen algebras with respect to their
irrational parameters.

We begin with the following notion of quantum compact metric spaces.

######  Notation 1.1.

We denote the norm of a normed vector space $E$ by
$\left\|{\cdot}\right\|_{E}$.

######  Definition 1.2.

For $(A,B)\in[1,\infty)\times[0,\infty)$, an $(A,B)$-quantum compact metric
space $({\mathfrak{A}},{\mathsf{L}})$ is a unital C*-algebra ${\mathfrak{A}}$
and a seminorm ${\mathsf{L}}$ defined on a dense Jordan-Lie subalgebra
${\operatorname*{dom}({{\mathsf{L}}})}$ of $\mathfrak{sa}({\mathfrak{A}})$
that satisfies the following.

  1. (1)

$\\{a\in{\operatorname*{dom}({{\mathsf{L}}})}:{\mathsf{L}}(a)=0\\}={\mathds{R}}1_{\mathfrak{A}}$.

  2. (2)

The Monge-Kantorovich distance ${\mathsf{mk}_{{\mathsf{L}}}}$ defined on the
state space ${\mathscr{S}}({\mathfrak{A}})$ of ${\mathfrak{A}}$ by:

$\forall\varphi,\psi\in{\mathscr{S}}({\mathfrak{A}})\quad{\mathsf{mk}_{{\mathsf{L}}}}(\varphi,\psi)=\sup\left\\{\left|\varphi(a)-\psi(a)\right|:a\in{\operatorname*{dom}({{\mathsf{L}}})},{\mathsf{L}}(a)\leqslant
1\right\\}$

metrizes the weak* topology of ${\mathscr{S}}({\mathfrak{A}})$.

  3. (3)

For all $a,b\in{\operatorname*{dom}({{\mathsf{L}}})}$,

$\max\left\\{{\mathsf{L}}\left(\frac{ab+ba}{2}\right),{\mathsf{L}}\left(\frac{ab-
ba}{2i}\right)\right\\}\leqslant
A\left({\mathsf{L}}(a)\left\|{b}\right\|_{{\mathfrak{A}}}+\left\|{a}\right\|_{{\mathfrak{A}}}{\mathsf{L}}(b)\right)+B{\mathsf{L}}(a){\mathsf{L}}(b)\text{.}$

  4. (4)

$\\{a\in{\operatorname*{dom}({{\mathsf{L}}})}:{\mathsf{L}}(a)\leqslant 1\\}$
is closed in ${\mathfrak{A}}$.

We call ${\mathsf{L}}$ an $(A,B)$-L-seminorm or an L-seminorm when context is
clear.

To ease our notation, we adopt the following convention.

######  Convention 1.3.

We fix a class of $(A,B)$-quasi-Leibniz quantum compact metric spaces for some
$A\geqslant 1$ and $B\geqslant 0$. All quantum compact metric spaces belong to
this class without further mention.

The class of quantum compact metric spaces can be turned into the objects of a
category [13]; for our purpose, quantum isometries, defined below, will
provide us with an adequate notion of morphisms.

######  Definition 1.4.

A _quantum isometry_
$\pi:({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})\rightarrow({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$
between two quantum compact metric spaces
$({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ is a surjective *-morphism
$\pi:{\mathfrak{A}}\rightarrow{\mathfrak{B}}$ such that
$\pi({\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{A}}})})\subseteq{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{B}}})}$
and

$\forall
b\in{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{B}}})}\quad{\mathsf{L}}_{\mathfrak{B}}(b)=\inf\left\\{{\mathsf{L}}_{\mathfrak{A}}(a):a\in{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{A}}})},\pi(a)=b\right\\}\text{.}$

A _full quantum isometry_
$\pi:({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})\rightarrow({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$
is a quantum isometry such that $\pi$ is a *-isomorphism and $\pi^{-1}$ is a
quantum isometry as well.

The propinquity is a metric, up to full quantum isometry, between quantum
compact metric spaces, defined as follows.

######  Definition 1.5.

Let $({\mathfrak{A}}_{1},{\mathsf{L}}_{1})$ and
$({\mathfrak{A}}_{2},{\mathsf{L}}_{2})$ be quantum compact metric spaces. A
_tunnel_ $\tau=({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}},\pi_{1},\pi_{2})$
from $({\mathfrak{A}}_{1},{\mathsf{L}}_{1})$ to
$({\mathfrak{A}}_{2},{\mathsf{L}}_{2})$ is a quantum compact metric space
$({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$ and quantum isometries
$\pi_{j}:({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})\rightarrow({\mathfrak{A}}_{j},{\mathsf{L}}_{j})$
for $j=1,2$.

The _extent_ ${\chi\left({\tau}\right)}$ of the tunnel $\tau$ is defined as

${\chi\left({\tau}\right)}=\max_{j\in\\{1,2\\}}{\mathsf{Haus}_{{\mathsf{mk}_{{\mathsf{L}}_{\mathfrak{D}}}}}}({\mathscr{S}}({\mathfrak{D}}),\\{\varphi\circ\pi_{j}:\varphi\in{\mathscr{S}}({\mathfrak{A}}_{j})\\})\text{.}$

######  Notation 1.6.

Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ be two quantum compact metric
spaces. The class of all tunnels from
$({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ to
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ is denoted by

${\text{\calligra
Tunnels}\,\left[\left({{\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}}\right)\stackrel{{\scriptstyle}}{{\longrightarrow}}\left({{\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}}\right)\right]}{}\text{.}$

We emphasize that, by Convention (1.3), if

$\tau=({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}},\ldots)\in{\text{\calligra
Tunnels}\,\left[\left({{\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}}\right)\stackrel{{\scriptstyle}}{{\longrightarrow}}\left({{\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}}\right)\right]}$

then $({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$ is an $(A,B)$-quantum
compact metric space where $A$ and $B$ are fixed throughout the construction
of the propinquity.

######  Definition 1.7.

The propinquity between two quantum compact metric spaces
$({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ is:

${\mathsf{\Lambda}^{\ast}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}))=\inf\left\\{{\chi\left({\tau}\right)}:\tau\in{\text{\calligra
Tunnels}\,\left[\left({{\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}}\right)\stackrel{{\scriptstyle}}{{\longrightarrow}}\left({{\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}}\right)\right]}\right\\}\text{.}$

By exploiting the Leibniz property, the Gromov–Hausdorff propinquity provides
an analogue of the Gromov–Hausdorff distance to noncommutative geometry [17,
15, 18, 14], namely, a complete metric over the class of quantum compact
metric spaces which is zero between isometrically isomorphic quantum compact
metric spaces. Moreover, it induces the same topology as the usual
Gromov–Hausdorff distance of the class of compact metric spaces (with the
identification between a compact metric space $(X,d)$ and the quantum compact
metric space $(C(X),{\mathsf{L}})$ with ${\mathsf{L}}$ the usual Lipschitz
seminorm).

Definition 1.2 has proven helpful, and was the foundation for new metrics
between certain higher structures over quantum compact metric spaces [19, 20,
21], including spectral triples [22].

However, Rieffel’s work on convergence of modules over quantum compact metric
spaces required a strengthening of the Leibniz property by requiring, in
addition, that Lip norms be well behaved with respect to the inverse map [30],
as follows.

######  Definition 1.8 ([32, Definition 2.1]).

A quantum compact metric space $({\mathfrak{A}},{\mathsf{L}})$ is _$C$
-strongly Leibniz_, for some $C\geqslant 1$, if for all
$a\in{\operatorname*{dom}({{\mathsf{L}}})}\cap\mathrm{GL}({\mathfrak{A}})$;

  1. (1)

$a^{-1}\in{\operatorname*{dom}({{\mathsf{L}}})}$, and

  2. (2)

${\mathsf{L}}\left(a^{-1}\right)\leqslant
C\left\|{a^{-1}}\right\|_{{\mathfrak{A}}}^{2}{\mathsf{L}}(a)\text{.}$

We say that ${\mathsf{L}}$ is _$C$ -strongly Leibniz_ when it meets these
conditions.

As seen above, even when estimating the propinquity between two strongly
Leibniz quantum compact metric spaces, the tunnels involved in the computation
of the propinquity need not themselves use strongly Leibniz quantum compact
metric spaces. However, the propinquity is defined with some flexibility,
which enables one to restrict the class of quantum compact metric spaces
involved in its construction. Thus, if one wishes to only work with strongly
Leibniz quantum compact metric spaces, then we may apply the methods of [17]
to obtain such a “strongly Leibniz propinquity,” which provides a natural
framework, for example, for Rieffel’s work on modules.

######  Notation 1.9.

Let $C\geqslant 1$ and let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ be two $C$-strongly Leibniz
quantum compact metric spaces. The class of all tunnels from
$({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ to
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ of the form
$({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}},\pi_{\mathfrak{A}},\pi_{\mathfrak{B}})$,
where $({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$ is also $C$-strongly
Leibniz, is denoted

${\text{\calligra
Tunnels}\,\left[\left({{\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}}\right)\stackrel{{\scriptstyle
SL_{C}}}{{\longrightarrow}}\left({{\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}}\right)\right]}{}\text{.}$

######  Definition 1.10.

The _$C$ -strongly Leibniz propinquity_ ${\mathsf{\Lambda}^{\ast}_{SL_{C}}}$
on the class $\mathcal{SL}_{C}$ of all $C$-strongly Leibniz quantum compact
metric spaces is defined by setting

${\mathsf{\Lambda}^{\ast}_{SL_{C}}}\left(({\mathfrak{A}}_{1},{\mathsf{L}}_{1}),({\mathfrak{A}}_{2},{\mathsf{L}}_{2})\right)=\inf\left\\{{\chi\left({\tau}\right)}:\tau\in{\text{\calligra
Tunnels}\,\left[\left({{\mathfrak{A}}_{1},{\mathsf{L}}_{1}}\right)\stackrel{{\scriptstyle
SL_{C}}}{{\longrightarrow}}\left({{\mathfrak{A}}_{2},{\mathsf{L}}_{2}}\right)\right]}\right\\}$

for any two $({\mathfrak{A}}_{1},{\mathsf{L}}_{1})$,
$({\mathfrak{A}}_{2},{\mathsf{L}}_{2})$ in $\mathcal{SL}_{C}$. If $C=1$, we
denote ${\mathsf{\Lambda}^{\ast}_{SL_{C}}}$ by
${\mathsf{\Lambda}^{\ast}_{SL}}$.

Of course, for any $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$,
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ in $\mathcal{SL}_{C}$, the
following inequality holds:

(1.1)
${\mathsf{\Lambda}^{\ast}_{SL_{C}}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}))\geqslant{\mathsf{\Lambda}^{\ast}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}))\text{.}$

Thus, it is clear that the strongly Leibniz property is a symmetric function
which is zero exactly between fully quantum isometric quantum compact metric
spaces. We prove in this paper that the strongly Leibniz propinquity is indeed
a _complete metric_ up to full quantum isometry.

We then provide an application of the completeness of the strongly Leibniz
propinquity to inductive limits of strongly Leibniz quantum compact metric
spaces. As discussed in [1], a natural question in noncommutative metric
geometry is to relate the important categorical notion of the limit of
inductive sequences of C*-algebras with the notion of convergence for the
propinquity. The first work in this direction, found in [3], constructed a
$(2,0)$-Leibniz Lip-norm on unital AF algebras with a faithful tracial state:
starting from a unital AF algebra ${\mathfrak{A}}$ with some choice of a
faithful trace $t$ and some sequence $({\mathfrak{A}}_{n})_{n\in{\mathds{N}}}$
of finite-dimensional C*-subalgebras of ${\mathfrak{A}}$ such that
${\mathfrak{A}}$ is the closure of
$\bigcup_{n\in{\mathds{N}}}{\mathfrak{A}}_{n}$, we use the existence of a
unique surjective $t$-preserving conditional expectation
$\mathds{E}_{n}:{\mathfrak{A}}\rightarrow{\mathfrak{A}}_{n}$ for each
$n\in{\mathds{N}}$ to define Lip-norms by

$\forall
a\in{\mathfrak{sa}\left({{\mathfrak{A}}}\right)}\quad{\mathsf{L}}(a)=\sup\left\\{\dim{{\mathfrak{A}}_{n}}\left\|{a-\mathds{E}_{n}(a)}\right\|_{{\mathfrak{A}}}:n\in{\mathds{N}}\right\\},$

allowing for the value $\infty$. In [3], the authors prove that

$\lim_{n\rightarrow\infty}{\mathsf{\Lambda}^{\ast}}(({\mathfrak{A}}_{n},{\mathsf{L}}),({\mathfrak{A}},{\mathsf{L}}))=0,$

and then establish continuity results for the propinquity of the classes of
Effros–Shen algebras and UHF algebras, naturally parametrized by the Baire
space.

However, it is natural to try to construct Lip-norms on inductive limits of
quantum compact metric spaces (not necessarily finite dimensional), under
appropriate conditions, without starting with an L-seminorm on the limit, but
rather, by exploiting the completeness of the propinquity. We refer to [1] for
examples. In this paper, we use the completeness of the strongly Leibniz
propinquity to construct strongly Leibniz L-seminorms on certain inductive
limits of strongly Leibniz quantum compact metric spaces and apply these
results to AF algebras, thus obtaining new quantum metrics on some AF
algebras. These new quantum metrics inherit their strong Leibniz properties
from the work of Rieffel on the strongly Leibniz property of seminorms built
from the standard deviations [33], which we call Frobenius–Rieffel seminorms,
and their careful study in finite dimensions in [2]. We obtain continuity
results for the Effros–Shen algebras and the UHF algebras, as parametrized by
the Baire space, in the spirit of [3], but with our new, strongly Leibniz
L-seminorms.

##  2\. The Strongly Leibniz Gromov–Hausdorff Propinquity

The dual Gromov–Hausdorff propinquity [16, 18, 14] is defined in a flexible
manner and permits one to restrict the choice of tunnels in the construction
in order to work in a specific class of quantum compact metric spaces. We
apply this flexibility to define a specialization of the propinquity to the
class of strongly Leibniz quantum compact metric spaces [31], whose definition
we recalled in Definition 1.8. As it dominates the propinquity, it has the
desired coincidence property, and it is obviously symmetric. We begin by
proving that it satisfies the triangle inequality.

######  Lemma 2.1.

Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$,
$({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ and
$({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$ be in $\mathcal{SL_{C}}$, and
let
$\pi_{\mathfrak{A}}:({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})\twoheadrightarrow({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$
and
$\pi_{\mathfrak{B}}:({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})\twoheadrightarrow({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$
be quantum isometries. For all $\varepsilon>0$, if we set, for all
$(a,b)\in{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{A}}})}\oplus{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{B}}})}$,

${\mathsf{T}}(a,b)=\max\left\\{{\mathsf{L}}_{\mathfrak{A}}(a),{\mathsf{L}}_{\mathfrak{B}}(b),\frac{1}{\varepsilon}\left\|{\pi_{\mathfrak{A}}(a)-\pi_{\mathfrak{B}}(b)}\right\|_{{\mathfrak{D}}}\right\\},$

then ${\mathsf{T}}$ is a $C$-strongly Leibniz L-seminorm on
${\mathfrak{A}}\oplus{\mathfrak{B}}$.

###### Proof.

By [18, Theorem 3.1], $({\mathfrak{A}}\oplus{\mathfrak{B}},{\mathsf{T}})$ is a
quantum compact metric space. It is thus sufficient to prove that
${\mathsf{T}}$ is $C$-strongly Leibniz.

Let
$(a,b)\in{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{A}}})}\oplus{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{B}}})}={\operatorname*{dom}({{\mathsf{T}}})}$
such that $(a,b)\in\mathrm{GL}({\mathfrak{A}}\oplus{\mathfrak{B}})$. Thus
$a\in{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{A}}})}\cap\mathrm{GL}({\mathfrak{A}})$
and
$b\in{\operatorname*{dom}({{\mathsf{L}}_{\mathfrak{B}}})}\cap\mathrm{GL}({\mathfrak{B}})$.
Since ${\mathsf{L}}_{\mathfrak{A}}$ and ${\mathsf{L}}_{\mathfrak{B}}$ are
$C$-strongly Leibniz, we conclude that
$(a,b)^{-1}=(a^{-1},b^{-1})\in{\operatorname*{dom}({{\mathsf{T}}})}$ and

${\mathsf{L}}_{\mathfrak{A}}(a^{-1})\leqslant
C\left\|{a^{-1}}\right\|_{{\mathfrak{A}}}^{2}{\mathsf{L}}_{\mathfrak{A}}(a)\quad\text{
and }\quad{\mathsf{L}}_{\mathfrak{B}}(b^{-1})\leqslant
C\left\|{b^{-1}}\right\|_{{\mathfrak{B}}}^{2}{\mathsf{L}}_{\mathfrak{B}}(b)\text{.}$

On the other hand,

$\displaystyle\left\|{\pi_{\mathfrak{A}}(a^{-1})-\pi_{\mathfrak{B}}(b^{-1})}\right\|_{{\mathfrak{D}}}$
$\displaystyle=\left\|{\pi_{\mathfrak{A}}(a^{-1})(\pi_{\mathfrak{B}}(b)-\pi_{\mathfrak{A}}(a))\pi_{\mathfrak{B}}(b^{-1})}\right\|_{{\mathfrak{D}}}$
$\displaystyle\leqslant\left\|{a^{-1}}\right\|_{{\mathfrak{A}}}\left\|{\pi_{\mathfrak{A}}(a)-\pi_{\mathfrak{B}}(b)}\right\|_{{\mathfrak{D}}}\left\|{b^{-1}}\right\|_{{\mathfrak{B}}}$
$\displaystyle\leqslant\left\|{a^{-1}}\right\|_{{\mathfrak{A}}}\left\|{b^{-1}}\right\|_{{\mathfrak{B}}}\varepsilon{\mathsf{T}}(a,b)$
$\displaystyle\leqslant\varepsilon\left\|{(a^{-1},b^{-1})}\right\|_{{\mathfrak{A}}\oplus{\mathfrak{B}}}^{2}{\mathsf{T}}(a,b)\text{.}$

Therefore, ${\mathsf{T}}(a^{-1},b^{-1})\leqslant
C\left\|{(a^{-1},b^{-1})}\right\|_{{\mathfrak{A}}\oplus{\mathfrak{B}}}^{2}{\mathsf{T}}(a,b)$,
as needed. ∎

Thus, as discussed in [16, 18, 14], we deduce the following lemma.

######  Lemma 2.2.

For all $C\geqslant 1$, the $C$-strongly Leibniz propinquity
${\mathsf{\Lambda}^{\ast}_{SL_{C}}}$ is a metric on $\mathcal{SL_{C}}$.

Our purpose is to prove that the $C$-strongly Leibniz propinquity is complete
as well. Since the propinquity is complete by Equation (1.1), we already have
a description of the limit of any Cauchy sequence in
$\left(\mathcal{SL}_{C},{\mathsf{\Lambda}^{\ast}_{SL_{C}}}\right)$ from [16].
What remains to be shown is that the limit for the metric
${\mathsf{\Lambda}^{\ast}}$ of a Cauchy sequence for the metric
${\mathsf{\Lambda}^{\ast}_{SL_{C}}}$ is indeed $C$-strongly Leibniz; moreover,
we have to check the tunnels constructed in [16] are $C$-strongly Leibniz in
the current setting.

In this section, we fix $C\geqslant 1$, and we assume that we are given a
sequence $({\mathfrak{A}}_{n},{\mathsf{L}}_{n})_{n\in{\mathds{N}}}$ of
$C$-strongly Leibniz quantum compact metric spaces such that for each
$n\in{\mathds{N}}$, there exists a tunnel
$(\tau_{n})_{n\in{\mathds{N}}}=({\mathfrak{D}}_{n},{\mathsf{T}}_{n},\pi_{n},\rho_{n})_{n\in{\mathds{N}}}$
from $({\mathfrak{A}}_{n},{\mathsf{L}}_{n})$ to
$({\mathfrak{A}}_{n+1},{\mathsf{L}}_{n+1})$ such that
${\chi\left({\tau_{n}}\right)}\leqslant\frac{1}{2^{n}}$ and ${\mathsf{T}}_{n}$
is $C$-strongly Leibniz.

Following [16, Section 6], we set

${\mathfrak{S}}=\left\\{(d_{n})_{n\in{\mathds{N}}}\in\prod_{n\in{\mathds{N}}}{\mathfrak{D}}_{n}:\forall
n\in{\mathds{N}}\ \ \ \rho_{n}(d_{n})=\pi_{n+1}(d_{n+1})\text{ and
}\sup_{n\in{\mathds{N}}}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}<\infty\right\\}\text{,}$

and

${\mathfrak{F}}={\raisebox{1.99997pt}{${\mathfrak{S}}$}\left/\raisebox{-1.99997pt}{$\left\\{(d_{n})_{n\in{\mathds{N}}}\in{\mathfrak{S}}:\lim_{n\rightarrow\infty}d_{n}=0\right\\}$}\right.}\text{.}$

Let $q:{\mathfrak{S}}\twoheadrightarrow{\mathfrak{F}}$ be the canonical
surjection, which is a *-epimorphism.

We record the following well-known computation for the quotient norm on
${\mathfrak{F}}$.

######  Lemma 2.3.

For all $a\in{\mathfrak{F}}$ and for all
$(d_{n})_{n\in{\mathds{N}}}\in{\mathfrak{S}}$, if
$q((d_{n})_{n\in{\mathds{N}}})=a$, then

$\left\|{a}\right\|_{{\mathfrak{F}}}=\limsup_{n\rightarrow\infty}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}\text{.}$

###### Proof.

Note that
$\left\|{a}\right\|_{{\mathfrak{F}}}\leqslant\left\|{d}\right\|_{{\mathfrak{S}}}$
whenever $q(d)=a$. Thus, if $N\in{\mathds{N}}$ and
$d^{N}=(\underbracket{0,\ldots,0}_{N\text{ times}},d_{N},d_{N+1},\ldots)$,
then $q(d^{N})=a$, and thus

$\left\|{a}\right\|_{{\mathfrak{F}}}\leqslant\left\|{d^{N}}\right\|_{{\mathfrak{S}}}=\sup_{n\geqslant
N}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}\text{.}$

Therefore,
$\left\|{a}\right\|_{{\mathfrak{F}}}\leqslant\limsup_{n\rightarrow\infty}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}$.

Now, let $\varepsilon>0$. By definition of the norm on the quotient C*-algebra
${\mathfrak{F}}$, there exists $e=(e_{n})_{n\in{\mathds{N}}}\in{\mathfrak{S}}$
such that $q(e)=a$ and
$\left\|{e}\right\|_{{\mathfrak{S}}}-\varepsilon\leqslant\left\|{a}\right\|_{{\mathfrak{F}}}\leqslant\left\|{e}\right\|_{{\mathfrak{S}}}$.
Also
$\lim_{n\rightarrow\infty}\left\|{e_{n}-d_{n}}\right\|_{{\mathfrak{D}}_{n}}=0$.
Thus, there exists $N\in{\mathds{N}}$ such that
$\left\|{e_{n}-d_{n}}\right\|_{{\mathfrak{D}}_{n}}<\varepsilon$ for all
$n\geqslant N$.

Therefore,

$\limsup_{n\rightarrow\infty}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}\leqslant\limsup_{n\rightarrow\infty}\left\|{e_{n}}\right\|_{{\mathfrak{D}}_{n}}+\varepsilon\leqslant\left\|{e}\right\|_{{\mathfrak{S}}}+\varepsilon\leqslant\left\|{a}\right\|_{{\mathfrak{F}}}+2\varepsilon\text{.}$

As $\varepsilon>0$ is arbitrary, we conclude:

$\limsup_{n\rightarrow\infty}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}\leqslant\left\|{a}\right\|_{{\mathfrak{F}}}\text{.}\qed$

For any
$d=(d_{n})_{n\in{\mathds{N}}}\in{\mathfrak{sa}\left({{\mathfrak{S}}}\right)}$,
we let

$\mathsf{S}(d)=\sup_{n\in{\mathds{N}}}{\mathsf{T}}_{n}(d_{n}),$

allowing for the value $\infty$. For all
$a\in{\mathfrak{sa}\left({{\mathfrak{F}}}\right)}$, we define

$\mathsf{Q}(a)=\inf\left\\{\mathsf{S}(d):q(d)=a\right\\}\text{,}$

again, allowing $\infty$. By [16], the seminorm $\mathsf{Q}$ is a Leibniz
L-seminorm on ${\mathfrak{F}}$. The key fact established in [16] is that
$({\mathfrak{A}}_{n},{\mathsf{L}}_{n})_{n\in{\mathds{N}}}$ converges to
$({\mathfrak{F}},\mathsf{Q})$ for the propinquity. We now show that, under our
conditions, $\mathsf{Q}$ is also $C$-strongly Leibniz.

######  Lemma 2.4.

The L-seminorm $\mathsf{Q}$ is $C$-strongly Leibniz.

###### Proof.

Let $a\in{\operatorname*{dom}({\mathsf{Q}})}\cap\mathrm{GL}({\mathfrak{F}})$
and $\varepsilon>0$. There exists
$d=(d_{n})_{n\in{\mathds{N}}}\in{\operatorname*{dom}({\mathsf{S}})}$ such that
$q(d)=a$, and
$\mathsf{S}(d)-\varepsilon\leqslant\mathsf{Q}(a)\leqslant\mathsf{S}(d)$.

Let
$e=(e_{n})_{n\in{\mathds{N}}}\in{\mathfrak{sa}\left({{\mathfrak{S}}}\right)}$
such that $q(e)=a^{-1}$. By definition, since
$q(ed)=q(de)=1_{{\mathfrak{F}}}$, we conclude that

$\lim_{n\rightarrow\infty}\left\|{d_{n}e_{n}-1_{n}}\right\|_{{\mathfrak{D}}_{n}}=0\text{,}$

and thus, there exists $N\in{\mathds{N}}$ such that
$\left\|{d_{n}e_{n}-1_{n}}\right\|_{{\mathfrak{D}}_{n}}<1$ for all $n\geqslant
N$. Therefore, $d_{n}e_{n}$ is invertible in ${\mathfrak{D}}_{n}$ for all
$n\geqslant N$. Consequently, if $n\geqslant N$, then
$d_{n}(e_{n}(d_{n}e_{n})^{-1})=1_{n}$. Since $d$ and $e$ are self-adjoint,
$\left\|{e_{n}d_{n}-1_{n}}\right\|_{{\mathfrak{D}}_{n}}<1$, and thus
$e_{n}d_{n}\in\mathrm{GL}({\mathfrak{D}}_{n})$ for $n\geqslant N$; moreover
$((e_{n}d_{n})^{-1}e_{n})d_{n}=1_{n}$. Thus, for all $n\geqslant N$,
$d_{n}\in\mathrm{GL}({\mathfrak{D}}_{n})$.

Now let $h=(h_{n})_{n\in{\mathds{N}}}\in{\mathfrak{S}}$ be defined by setting,
for all $n\in{\mathds{N}}$:

$h_{n}=\begin{cases}1_{n}\text{ if $n<N$, }\\\ d_{n}\text{ if $n\geqslant
N$.}\end{cases}$

By construction, $h\in\mathrm{GL}({\mathfrak{S}})$ and $q(h)=a$. Moreover,
$q(h^{-1})=a^{-1}$ and

$h^{-1}=\left(1_{0},\ldots,1_{N-1},d_{N}^{-1},d_{N+1}^{-1},\ldots\right)\text{.}$

Since ${\mathsf{T}}_{n}$ is $C$-strongly Leibniz,
$h_{n}^{-1}\in{\operatorname*{dom}({{\mathsf{T}}_{n}})}$ for all
$n\in{\mathds{N}}$, and

${\mathsf{T}}_{n}(h_{n}^{-1})\leqslant
C\left\|{h_{n}^{-1}}\right\|_{{\mathfrak{D}}_{n}}^{2}{\mathsf{T}}_{n}(h_{n})\text{.}$

Moreover, $\mathsf{S}(h)\leqslant\mathsf{S}(d)$ (since
${\mathsf{T}}_{n}(1_{n})=0$), so
$\mathsf{S}(h)\leqslant\mathsf{Q}(a)+\varepsilon$. By Lemma (2.3),
$\left\|{a^{-1}}\right\|_{{\mathfrak{F}}}=\limsup_{n\rightarrow\infty}\left\|{h^{-1}_{n}}\right\|_{{\mathfrak{D}}_{n}}$.
Thus, there exists $N^{\prime}\in{\mathds{N}}$ such that, if $n\geqslant
N^{\prime}$, then

$\sup_{n\geqslant
N^{\prime}}\left\|{d_{n}^{-1}}\right\|_{{\mathfrak{D}}_{n}}-\varepsilon\leqslant\left\|{a^{-1}}\right\|_{{\mathfrak{F}}}\leqslant\sup_{n\geqslant
N^{\prime}}\left\|{d_{n}^{-1}}\right\|_{{\mathfrak{D}}_{n}}\text{.}$

Let $N^{\prime\prime}=\max\\{N,N^{\prime}\\}$ and define $g\in{\mathfrak{S}}$
by $g=(g_{n})_{n\in{\mathds{N}}}$ with

$\forall n\in{\mathds{N}}\quad g_{n}=\begin{cases}1_{n}\text{ if
$n<N^{\prime\prime}$,}\\\ d_{n}\text{ if $n\geqslant
N^{\prime\prime}$.}\end{cases}$

Once again, note that $q(g)=a$ and
$\mathsf{S}(g)\leqslant\mathsf{S}(h)\leqslant\mathsf{Q}(a)+\varepsilon$.
Moreover, $g\in\mathrm{GL}({\mathfrak{S}})$, with $q(g^{-1})=a^{-1}$.

Thus,

$\displaystyle\mathsf{Q}(a^{-1})$ $\displaystyle\leqslant\mathsf{S}(g^{-1})$
$\displaystyle\leqslant\sup_{n\in{\mathds{N}}}{\mathsf{T}}_{n}(g_{n}^{-1})$
$\displaystyle\leqslant\sup_{n\geqslant
N^{\prime\prime}}{\mathsf{T}}_{n}(d_{n}^{-1})$ $\displaystyle\leqslant
C\sup_{n\geqslant
N^{\prime\prime}}\left\|{d_{n}^{-1}}\right\|_{{\mathfrak{D}}_{n}}^{2}\sup_{n\geqslant
N^{\prime\prime}}{\mathsf{T}}_{n}(d_{n})$ $\displaystyle\leqslant
C\left(\left\|{a^{-1}}\right\|_{{\mathfrak{F}}}+\varepsilon\right)^{2}\mathsf{S}(g)$
$\displaystyle\leqslant
C\left(\left\|{a^{-1}}\right\|_{{\mathfrak{F}}}+\varepsilon\right)^{2}\left(\mathsf{Q}(a)+\varepsilon\right)\text{.}$

Since $\varepsilon>0$ is arbitrary,

$\mathsf{Q}(a^{-1})\leqslant
C\left\|{a^{-1}}\right\|_{{\mathfrak{F}}}^{2}\mathsf{Q}(a)\text{.}$

Thus, $\mathsf{Q}$ is strongly Leibniz. ∎

We now deduce the following theorem.

######  Theorem 2.5.

Let $C\geqslant 1$. The $C$-strongly Leibniz propinquity on $\mathcal{SL}_{C}$
is complete.

###### Proof.

We maintain the notations introduced above. We have seen that
$({\mathfrak{F}},\mathsf{Q})$ is a $C$-strongly Leibniz quantum compact metric
space.

For each $N\in{\mathds{N}}$, let

${\mathfrak{S}}_{N}=\left\\{(d_{n})_{n\geqslant N}\in\prod_{n\geqslant
N}{\mathfrak{D}}_{n}:\forall n\geqslant
N\quad\rho_{n}(d_{n})=\pi_{n+1}(d_{n+1}),\sup_{n\geqslant
N}\left\|{d_{n}}\right\|_{{\mathfrak{D}}_{n}}<\infty\right\\}\text{.}$

We also let $\Pi_{N}((d_{n})_{n\geqslant
N})=\pi_{N}(d_{N})\in{\mathfrak{A}}_{N}$ for all $(d_{n})_{n\geqslant
N}\in{\mathfrak{S}}_{N}$. The C*-algebra ${\mathfrak{F}}$ is naturally
*-isomorphic to

${\raisebox{1.99997pt}{${\mathfrak{S}}_{N}$}\left/\raisebox{-1.99997pt}{$\\{(d_{n})_{n\geqslant
N}\in{\mathfrak{S}}_{N}:\lim_{n\rightarrow\infty}d_{n}=0\\}$}\right.}\text{;}$

we let $q_{N}:{\mathfrak{S}}_{N}\twoheadrightarrow{\mathfrak{F}}$ be the
associated canonical surjection.

We also let $\mathsf{S}_{N}:(d_{n})_{n\geqslant N}\mapsto\sup_{n\geqslant
N}{\mathsf{T}}_{n}(d_{n})$ (allowing the value $\infty$). By [16],
$({\mathfrak{S}}_{N},\mathsf{S}_{N})$ is a quantum compact metric space.

In [16], the fourth author proved that

$\forall
a\in{\mathfrak{sa}\left({{\mathfrak{F}}}\right)}\quad\mathsf{Q}(a)=\inf\left\\{\mathsf{S}_{N}(d):d\in{\mathfrak{sa}\left({{\mathfrak{S}}_{N}}\right)},q_{N}(d)=a\right\\}$

and $\Pi_{N}$ is a quantum isometry to
$({\mathfrak{A}}_{N},{\mathsf{L}}_{N})$. In other words,
$({\mathfrak{S}}_{N},\mathsf{S}_{N},\Pi_{N},q_{N})$ is a tunnel from
$({\mathfrak{A}}_{N},{\mathsf{L}}_{N})$ to $({\mathfrak{F}},\mathsf{Q})$ with
extent at most $\frac{1}{2^{N}}$.

It suffices to prove that $\mathsf{S}_{N}$ is $C$-strongly Leibniz. This is
immediate by definition: if $d=(d_{n})_{n\geqslant
N}\in\mathrm{GL}({\mathfrak{S}}_{N})\cap{\operatorname*{dom}({\mathsf{S}_{N}})}$,
then

${\mathsf{T}}_{n}(d_{n}^{-1})\leqslant
C\left\|{d_{n}^{-1}}\right\|_{{\mathfrak{D}}_{n}}^{2}{\mathsf{T}}_{n}(d_{n})$

for all $n\geqslant N$, and thus

$\mathsf{S}_{N}(d^{-1})\leqslant
C\left\|{d^{-1}}\right\|_{{\mathfrak{S}}_{N}}^{2}\mathsf{S}_{N}(d)\text{.}$

This completes our proof. ∎

##  3\. Inductive limits of strongly Leibniz quantum compact metric spaces

In [1, Section 2], the first author constructed quantum compact metric spaces
on inductive limits such that the given inductive sequence converged to the
inductive limit in propinquity. In this section, we specialize these results
to strongly Leibniz quantum compact metric spaces using the results of the
previous section. As a main application of this section and article, we find
strongly Leibniz L-seminorms on AF-algebras that allow for explicit estimates
in the strongly Leibniz propinquity and convergence of the Effros–Shen
algebras now in the class of strongly Leibniz compact quantum metric spaces.

######  Theorem 3.1.

Fix $C\geqslant 1$. Let
${\mathfrak{A}}=\overline{\cup_{n\in{\mathds{N}}}{\mathfrak{A}}_{n}}^{\|\cdot\|_{\mathfrak{A}}}$
be a unital C*-algebra such that $({\mathfrak{A}}_{n})_{n\in{\mathds{N}}}$ is
a non-decreasing sequence of unital C*-algebras of ${\mathfrak{A}}$. Assume
that $({\mathfrak{A}}_{n},{\mathsf{L}}_{n})_{n\in{\mathds{N}}}$ is a
$C$-strongly Leibniz quantum compact metric space for all $n\in{\mathds{N}}$.
Let $(\beta(j))_{j\in{\mathds{N}}}$ be a summable sequence in $(0,\infty)$.

If for all $n\in{\mathds{N}}$

  1. (1)

${\mathsf{L}}_{n+1}(a)\leqslant{\mathsf{L}}_{n}(a)$ for all
$a\in{\mathfrak{A}}_{n}$, and

  2. (2)

for all $a\in{\mathfrak{A}}_{n+1},{\mathsf{L}}_{n+1}(a)\leqslant 1$, there
exists $b\in{\mathfrak{A}}_{n},{\mathsf{L}}_{n}(b)\leqslant 1$ such that

$\|a-b\|_{\mathfrak{A}}<\beta(n),$

then there exists a $C$-strongly Leibniz seminorm ${\mathsf{L}}$ on
${\mathfrak{A}}$ such that $({\mathfrak{A}},{\mathsf{L}})$ is a quantum
compact metric space where

${\mathsf{\Lambda}^{\ast}_{SL_{C}}}(({\mathfrak{A}}_{n},{\mathsf{L}}_{n}),({\mathfrak{A}},{\mathsf{L}}))\leqslant
4\sum_{j=n}^{\infty}\beta(j)$

for all $n\in{\mathds{N}}$, and thus

$\lim_{n\to\infty}{\mathsf{\Lambda}^{\ast}_{SL_{C}}}(({\mathfrak{A}}_{n},{\mathsf{L}}_{n}),({\mathfrak{A}},{\mathsf{L}}))=0.$

###### Proof.

Since the tunnels of (1) from [1, Theorem 2.15] are $C$-strongly Leibniz by
Lemma 2.1, this result follows immediately from [1, Theorem 2.15] and Lemma
2.4. ∎

We turn our attention to the AF setting.

######  Definition 3.2 ( [4, Definition 1.5.9 and Tomiyama’s Theorem
1.5.10]).

Let ${\mathfrak{A}}$ be a unital C*-algebra and let
${\mathfrak{B}}\subseteq{\mathfrak{A}}$ be a unital C*-subalgebra. A linear
map $E:{\mathfrak{A}}\rightarrow{\mathfrak{B}}$ is a conditional expectation
if $E(b)=b$ for all $b\in{\mathfrak{B}}$,
$\|E(a)\|_{\mathfrak{A}}\leqslant\|a\|_{\mathfrak{A}}$ for all
$a\in{\mathfrak{A}}$, and $E(bab^{\prime})=bE(a)b^{\prime}$ for all
$a\in{\mathfrak{A}},b,b^{\prime}\in{\mathfrak{B}}$.

A conditional expectation is faithful if $E(a^{*}a)=0$ implies $a=0$.

######  Theorem-Definition 3.3 ( [33, Section 5] and [24]).

Let ${\mathfrak{A}}$ be a unital C*-algebra and let
${\mathfrak{B}}\subseteq{\mathfrak{A}}$ be a unital C*-subalgebra. If
$E:{\mathfrak{A}}\rightarrow{\mathfrak{B}}$ is a faithful conditional
expectation, then

$\|a\|_{E}=\sqrt{\|E(a^{*}a)\|_{\mathfrak{A}}}$

defines a norm on ${\mathfrak{A}}$ called the Frobenius–Rieffel norm
associated to $E$.

A quick application of the C*-identity shows that
$\|\cdot\|_{E}\leqslant\|\cdot\|_{\mathfrak{A}}$. We now place strongly
Leibniz L-seminorms on all unital AF-algebras equipped with a faithful tracial
state that allow for explicit approximations from the finite-dimensional
C*-subalgebras. The following construction comes from the first and last
author’s work in [3], where they used the C*-norms instead of the
Frobenius–Rieffel norms. However, the L-seminorms of [3] are only known to be
quasi-Leibniz with $A=2$ and $B=0$, and we do not know if they are strongly
Leibniz for any $C\geqslant 1$. Yet, due to a suggestion of Rieffel (to the
first author at the Fall 2016 AMS Western section at University of Denver) to
replace the C*-norms with his norms from [33], we now have the following
strongly Leibniz L-seminorms that still have the desired convergence results
of [3].

######  Theorem 3.4.

Let
${\mathfrak{A}}=\overline{\cup_{n\in{\mathds{N}}}{\mathfrak{A}}_{n}}^{\|\cdot\|_{\mathfrak{A}}}$
be a unital AF algebra equipped with a faithful tracial state $\tau$ such that
${\mathfrak{A}}_{0}={\mathds{C}}1_{\mathfrak{A}}$. Set
${\mathcal{U}}=({\mathfrak{A}}_{n})_{n\in{\mathds{N}}}$ and let
$(\beta(n))_{n\in{\mathds{N}}}$ be a summable sequence of positive real
numbers. For each $n\in{\mathds{N}}$, let

$E_{n}^{\tau}:{\mathfrak{A}}\rightarrow{\mathfrak{A}}_{n}$

be the unique $\tau$-preserving faithful conditional expectation. For
$n\in{\mathds{N}}\setminus\\{0\\}$, let $\kappa_{n}>0$ such that

$\kappa_{n}\|a\|_{{\mathfrak{A}}}\leqslant\|a\|_{E_{n}^{\tau}}$

for all $a\in{\mathfrak{A}}_{n+1}$, and set
$\kappa=(\kappa_{n})_{n\in{\mathds{N}}}$. For each
$n\in{\mathds{N}}\setminus\\{0\\}$, let

${\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(a)=\max_{m\in\\{0,1,\ldots,n-1\\}}\frac{\max\left\\{\|a-E_{m}^{\tau}(a)\|_{E_{m}^{\tau}},\|a^{*}-E_{m}^{\tau}(a^{*})\|_{E_{m}^{\tau}}\right\\}}{\kappa_{m}\beta(m)}$

for all $a\in{\mathfrak{A}}_{n}$, and let
${\mathsf{L}}_{{\mathcal{U}}_{0},\beta}^{\tau,\kappa}=0$.

Then
$({\mathfrak{A}}_{n},{\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa})$ is
a strongly Leibniz quantum compact metric space (with $(A,B,C)=(1,0,1)$) for
all $n\in{\mathds{N}}$, and there exists a seminorm
${\mathsf{L}}_{{\mathcal{U}},\beta}^{\tau,\kappa}$ such that
$({\mathfrak{A}},{\mathsf{L}}_{{\mathcal{U}},\beta}^{\tau,\kappa})$ is a
strongly Leibniz quantum compact metric space (with $(A,B,C)=(1,0,1)$) where

${\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{n},{\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}),({\mathfrak{A}},{\mathsf{L}}_{{\mathcal{U}},\beta}^{\tau,\kappa}))\leqslant
4\sum_{j=n}^{\infty}\beta(j)$

for all $n\in{\mathds{N}}$, and thus

$\lim_{n\to\infty}{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{n},{\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}),({\mathfrak{A}},{\mathsf{L}}_{{\mathcal{U}},\beta}^{\tau,\kappa}))=0.$

###### Proof.

Let $n\in{\mathds{N}}$. By construction, for $a\in{\mathfrak{A}}_{n},$
${\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(a)=0$ if and only if
$a=\mu 1_{{\mathfrak{A}}_{n}}$ for some $\mu\in{\mathds{C}}$. Hence, since
${\mathfrak{A}}_{n}$ is finite-dimensional,
${\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}$ is an L-seminorm on
${\mathfrak{A}}_{n}$. Furthermore,
$({\mathfrak{A}}_{n},{\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa})$ is
a strongly Leibniz compact quantum metric space (with $A=1,B=0,C=1$) by [33,
Proposition 5.4 and Theorem 5.5].

By construction, for all $n\in{\mathds{N}}$ and $a\in{\mathfrak{A}}$, we have
that

${\mathsf{L}}_{{\mathcal{U}}_{n+1},\beta}^{\tau,\kappa}(a)={\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(a).$

Thus, (1) of Theorem 3.1 is satisfied.

For (2), let $a\in{\mathfrak{A}}_{n+1}$ such that
${\mathsf{L}}_{{\mathcal{U}}_{n+1},\beta}^{\tau,\kappa}(a)\leqslant 1$.
Consider $E_{n}^{\tau}(a)\in{\mathfrak{A}}_{n}$. We have that
$\|a-E_{n}^{\tau}(a)\|_{E_{n}}\leqslant\kappa_{n}\beta(n)$. Thus,

$\kappa_{n}\|a-E_{n}^{\tau}(a)\|_{\mathfrak{A}}\leqslant\|a-E_{n}^{\tau}(a)\|_{E_{n}}\leqslant\kappa_{n}\beta(n),$

so $\|a-E_{n}^{\tau}(a)\|_{\mathfrak{A}}\leqslant\beta(n).$ Note that if
$n=0$, then
${\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(E_{n}^{\tau}(a))=0\leqslant
1$ and we are done, so for the rest of the proof assume that $n\geqslant 1$.

We show that
${\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(E_{n}^{\tau}(a))\leqslant
1$. Note that

$\displaystyle{\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(E_{n}^{\tau}(a))$
$\displaystyle\quad=\max_{m\in\\{0,1,\ldots,n-1\\}}\frac{\max\left\\{\|E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))\|_{E_{m}^{\tau}},\|E_{n}^{\tau}(a^{*})-E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))\|_{E_{m}^{\tau}}\right\\}}{\kappa_{m}\beta(m)}.$

Let $m\in\\{0,1,\ldots,n\\}.$ Now $E_{m}^{\tau}\circ
E_{n}^{\tau}=E_{m}^{\tau}$ by the proof of [3, Theorem 3.5], $E_{m}$ is
positive by [4, Theorem 1.5.10 (Tomiyama)], and
$E_{m}({\mathfrak{A}})\subseteq E_{n}({\mathfrak{A}})$. Thus,

$\displaystyle
E_{m}^{\tau}([E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))]^{*}[E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))])$
$\displaystyle\quad=E_{m}^{\tau}([E_{n}^{\tau}(a^{*})-E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))][E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))])$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{n}^{\tau}(a)-E_{n}^{\tau}(a^{*})E_{m}^{\tau}(E_{n}^{\tau}(a))-E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{n}^{\tau}(a)$
$\displaystyle\quad\quad\quad+E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a)))$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{n}^{\tau}(a))-E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{m}^{\tau}(E_{n}^{\tau}(a)))-E_{m}^{\tau}(E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{n}^{\tau}(a))$
$\displaystyle\quad\quad\quad+E_{m}^{\tau}(E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a)))$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{n}^{\tau}(a))-E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a))-E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a))$
$\displaystyle\quad\quad\quad+E_{m}^{\tau}(E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a)))$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{n}^{\tau}(a))-E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)-E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)$
$\displaystyle\quad\quad\quad+E_{m}^{\tau}(E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a)))$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{n}^{\tau}(a))-2E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)+E_{m}^{\tau}(E_{m}^{\tau}(E_{n}^{\tau}(a^{*}))E_{m}^{\tau}(E_{n}^{\tau}(a)))$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*})E_{n}^{\tau}(a))-2E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)+E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a).$

Similarly,

$\displaystyle E_{m}^{\tau}((a-E_{m}^{\tau}(a))^{*}(a-E_{m}^{\tau}(a)))$
$\displaystyle\quad=E_{m}^{\tau}(a^{*}a)-2E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)+E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)$
$\displaystyle\quad=E_{m}^{\tau}(E_{n}^{\tau}(a^{*}a))-2E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a)+E_{m}^{\tau}(a^{*})E_{m}^{\tau}(a).$

By [4, Proposition 1.5.7],
$E_{n}^{\tau}(a^{*}a)-E^{\tau}_{n}(a^{*})E^{\tau}_{n}(a)\geqslant 0$ and hence

$E_{m}^{\tau}(E_{n}^{\tau}(a^{*}a))-E_{m}^{\tau}(E^{\tau}_{n}(a^{*})E^{\tau}_{n}(a))\geqslant
0$

since $E_{m}^{\tau}$ is a conditional expectation. Thus,

$\displaystyle
E_{m}^{\tau}([E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))]^{*}[E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))])$
$\displaystyle\quad\quad\quad\leqslant
E_{m}^{\tau}((a-E_{m}^{\tau}(a))^{*}(a-E_{m}^{\tau}(a))).$

Since
$E_{m}^{\tau}([E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))]^{*}[E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))])\geqslant
0$, we gather

$\displaystyle\|E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))\|_{E_{m}^{\tau}}^{2}$
$\displaystyle=\|E_{m}^{\tau}([E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))]^{*}[E_{n}^{\tau}(a)-E_{m}^{\tau}(E_{n}^{\tau}(a))])\|_{{\mathfrak{A}}}$
$\displaystyle\leqslant\|E_{m}^{\tau}((a-E_{m}^{\tau}(a))^{*}(a-E_{m}^{\tau}(a)))\|_{{\mathfrak{A}}}$
$\displaystyle=\|a-E_{m}^{\tau}(a)\|_{E_{m}^{\tau}}^{2}.$

Therefore, repeating this process with $a^{*}$ in place of $a$, we conclude
that

${\mathsf{L}}_{{\mathcal{U}}_{n},\beta}^{\tau,\kappa}(E_{n}^{\tau}(a))\leqslant{\mathsf{L}}_{{\mathcal{U}}_{n+1},\beta}^{\tau,\kappa}(a)\leqslant
1.$

The proof is complete by Theorem 3.1. ∎

In [2], some equivalence constants $\kappa_{n}$ were obtained explicitly on
direct sums of matrix algebras. The next results ensure that equivalence
constants obtained in this way translate to the inductive limit.

######  Proposition 3.5.

Let $({\mathfrak{A}}_{n},\alpha_{n})_{n\in{\mathds{N}}}$ be an inductive
sequence of C*-algebras (see [23, Section 6.1]) such that:

  1. (1)

${\mathfrak{A}}_{0}={\mathds{C}}$ and
${\mathfrak{A}}_{n}=\bigoplus_{k=1}^{n_{n}}{\mathfrak{M}}_{d_{n,k}}({\mathds{C}})$
for all $n\in{\mathds{N}}\setminus\\{0\\}$, where
$d_{n,k}\in{\mathds{N}}\setminus\\{0\\}$ for each
$n\in{\mathds{N}}\setminus\\{0\\}$ and $k\in\\{1,2,\ldots,n_{n}\\}$;

  2. (2)

$\alpha_{n}:{\mathfrak{A}}_{n}\rightarrow{\mathfrak{A}}_{n+1}$ is a unital
*-monomorphism for all $n\in{\mathds{N}}$;

  3. (3)

the inductive limit ${\mathfrak{A}}=\underrightarrow{\lim}\
({\mathfrak{A}}_{n},\alpha_{n})_{n\in{\mathds{N}}}$ is equipped with a
faithful tracial state $\tau$.

For each $n\in{\mathds{N}}$, let
$\alpha^{(n+1)},\alpha^{(n)}:{\mathfrak{A}}_{n}\rightarrow{\mathfrak{A}}$ be
the canonical unital *-monomorphisms satisfying

(3.1) $\alpha^{(n+1)}\circ\alpha_{n}=\alpha^{(n)}.$

Note that
${\mathfrak{A}}=\overline{\cup_{n\in{\mathds{N}}}\alpha^{(n)}({\mathfrak{A}}_{n})}^{\|\cdot\|_{\mathfrak{A}}}$
and
$\alpha^{(n)}({\mathfrak{A}}_{n})\subseteq\alpha^{(n+1)}({\mathfrak{A}}_{n+1})$
and $\alpha^{(0)}({\mathfrak{A}}_{0})={\mathds{C}}1_{\mathfrak{A}}$ (see [23,
Section 6.1]). For each $n\in{\mathds{N}}$, let

$E^{\tau}_{n}:{\mathfrak{A}}\rightarrow\alpha^{(n)}({\mathfrak{A}}_{n})$

denote the unique $\tau$-preserving faithful conditional expectation onto
$\alpha^{(n)}({\mathfrak{A}}_{n}).$ For each $n\in{\mathds{N}}$, let

(3.2) $\tau_{n}=\tau\circ\alpha^{(n)},$

which is a faithful tracial state on ${\mathfrak{A}}_{n}$, and let

$E^{\tau_{n+1}}_{n+1,n}:{\mathfrak{A}}_{n+1}\rightarrow\alpha_{n}({\mathfrak{A}}_{n})$

be the unique $\tau_{n+1}$-preserving faithful conditional expectation onto
$\alpha_{n}({\mathfrak{A}}_{n})$. Let $\kappa_{n}>0$ such that

$\kappa_{n}\|a\|_{{\mathfrak{A}}_{n+1}}\leqslant\|a\|_{E^{\tau_{n+1}}_{n+1,n}}$

for all $a\in{\mathfrak{A}}_{n+1}$.

Then, for all $n\in{\mathds{N}}$,

$E^{\tau}_{n}\circ\alpha^{(n+1)}=\alpha^{(n+1)}\circ E^{\tau_{n+1}}_{n+1,n},$

and moreover,

$\kappa_{n}\|a\|_{\mathfrak{A}}\leqslant\|a\|_{E^{\tau}_{n}}$

for all $a\in\alpha^{(n+1)}({\mathfrak{A}}_{n+1})$.

###### Proof.

Let $n\in{\mathds{N}}$ and let $B_{n}$ denote the set of matrix units for
${\mathfrak{A}}_{n}$. By [3, Expression (4.1)] for both conditional
expectations $E^{\tau}_{n}$ and $E^{\tau_{n+1}}_{n+1,n}$, we have for all
$a\in{\mathfrak{A}}_{n+1}$

$\displaystyle E^{\tau}_{n}(\alpha^{(n+1)}(a))$ $\displaystyle=\sum_{e\in
B_{n}}\frac{\tau(\alpha^{(n+1)}(a)\alpha^{(n)}(e^{*}))}{\tau(\alpha^{(n)}(e^{*})\alpha^{(n)}(e))}\alpha^{(n)}(e)$
$\displaystyle=\sum_{e\in
B_{n}}\frac{\tau(\alpha^{(n+1)}(a)\alpha^{(n)}(e^{*}))}{\tau(\alpha^{(n)}(e^{*})\alpha^{(n)}(e))}\alpha^{(n+1)}(\alpha_{n}(e))$
$\displaystyle=\alpha^{(n+1)}\left(\sum_{e\in
B_{n}}\frac{\tau(\alpha^{(n+1)}(a)\alpha^{(n)}(e^{*}))}{\tau(\alpha^{(n)}(e^{*})\alpha^{(n)}(e))}\alpha_{n}(e)\right)$
$\displaystyle=\alpha^{(n+1)}\left(\sum_{e\in
B_{n}}\frac{\tau(\alpha^{(n+1)}(a)\alpha^{(n+1)}(\alpha_{n}(e^{*})))}{\tau(\alpha^{(n+1)}(\alpha_{n}(e^{*})\alpha_{n}(e))}\alpha_{n}(e)\right)$
$\displaystyle=\alpha^{(n+1)}\left(\sum_{e\in
B_{n}}\frac{\tau_{n+1}(a\alpha_{n}(e^{*}))}{\tau_{n+1}(\alpha_{n}(e^{*})\alpha_{n}(e))}\alpha_{n}(e)\right)$
$\displaystyle=\alpha^{(n+1)}(E^{\tau_{n+1}}_{n+1,n}(a))$

and so

$\displaystyle E^{\tau}_{n}\circ\alpha^{(n+1)}=\alpha^{(n+1)}\circ
E^{\tau_{n+1}}_{n+1,n}.$

Let $a\in\alpha^{(n+1)}({\mathfrak{A}}_{n+1})$. Thus, there exists a unique
$a_{n+1}\in{\mathfrak{A}}_{n+1}$ such that $a=\alpha^{(n+1)}(a_{n+1})$. We
have

$\displaystyle\|a\|_{E^{\tau}_{n}}^{2}$
$\displaystyle=\left\|E^{\tau}_{n}\left(\alpha^{(n+1)}(a_{n+1})^{*}\alpha^{(n+1)}(a_{n+1})\right)\right\|_{\mathfrak{A}}$
$\displaystyle=\left\|E^{\tau}_{n}\left(\alpha^{(n+1)}(a_{n+1}^{*}a_{n+1})\right)\right\|_{\mathfrak{A}}$
$\displaystyle=\left\|\alpha^{(n+1)}\left(E^{\tau_{n+1}}_{n+1,n}(a_{n+1}^{*}a_{n+1})\right)\right\|_{\mathfrak{A}}$
$\displaystyle=\left\|E^{\tau_{n+1}}_{n+1,n}(a_{n+1}^{*}a_{n+1})\right\|_{{\mathfrak{A}}_{n+1}}$
$\displaystyle\geqslant\kappa_{n}^{2}\|a_{n+1}\|_{{\mathfrak{A}}_{n+1}}^{2}$
$\displaystyle=\kappa_{n}^{2}\|a\|_{\mathfrak{A}}^{2}.$

Therefore,

$\kappa_{n}\|a\|_{\mathfrak{A}}\leqslant\|a\|_{E^{\tau}_{n}}$

as desired. ∎

The convergence of the Effros–Shen algebras in [3] relied on a continuous
field of L-seminorms on the finite-dimensional subalgebras of the inductive
sequence and, although the L-seminorms of Theorem 3.4 have a similar structure
to those of [3, Theorem 3.5], we need two important facts to ensure that the
L-seminorms of Theorem 3.4 also form a continuous field of L-seminorms in an
appropriate sense. The first, Proposition 3.6, takes care of the fact that we
switch the C*-norm for Frobenius–Rieffel norms and the second fact takes care
of the continuity of the equivalence constants, which was already proven in
[2, Theorem 5.2] for particular equivalence constants that were calculated
explicitly.

######  Proposition 3.6.

Let
$N\in{\mathds{N}},n_{1},n_{2},\ldots,n_{N}\in{\mathds{N}}\setminus\\{0\\}$,
and ${\mathfrak{A}}=\oplus_{k=1}^{N}{\mathfrak{M}}_{n_{k}}({\mathds{C}})$. Let
$M\in{\mathds{N}}\setminus\\{0\\},m_{1},m_{2},\ldots,m_{M}\in{\mathds{N}}\setminus\\{0\\}$,
and ${\mathfrak{B}}=\oplus_{k=1}^{M}{\mathfrak{M}}_{m_{k}}({\mathds{C}})$.
Assume that there exists a unital *-monomorphism
$\alpha:{\mathfrak{B}}\rightarrow{\mathfrak{A}}$. For each
$n\in{\mathds{N}}\cup\\{\infty\\}$, let $\mathbf{v}^{n}\in(0,1)^{N}$ and let
$\tau_{\mathbf{v}^{n}}$ be the faithful tracial state defined for all
$a=(a_{1},a_{2},\ldots,a_{N})\in{\mathfrak{A}}$ by

$\tau_{\mathbf{v}^{n}}(a)=\sum_{k=1}^{N}\frac{v^{n}_{k}}{n_{k}}\mathrm{Tr}(a_{k}),$

where $\mathrm{Tr}$ denotes the trace of a matrix. Let
$E^{\tau_{\mathbf{v}^{n}}}:{\mathfrak{A}}\rightarrow\alpha({\mathfrak{B}})$
denote the unique $\tau_{\mathbf{v}^{n}}$-preserving faithful conditional
expectation onto $\alpha({\mathfrak{B}})$.

If $(\mathbf{v}^{n})_{n\in{\mathds{N}}}$ converges to $\mathbf{v}^{\infty}$
coordinate-wise, then
$(\|\cdot\|_{E^{\tau_{\mathbf{v}^{n}}}})_{n\in{\mathds{N}}}$ converges to
$\|\cdot\|_{E^{\tau_{\mathbf{v}^{\infty}}}}$ uniformly on any compact set of
$({\mathfrak{A}},\|\cdot\|_{\mathfrak{A}})$ and thus converges pointwise on
${\mathfrak{A}}$.

###### Proof.

Let $B$ be the set of matrix units for ${\mathfrak{B}}$. Fix
$a=(a_{1},a_{2},\ldots,a_{N})\in{\mathfrak{A}}$. By [3, Expression (4.1)], for
each $n\in{\mathds{N}}\cup\\{\infty\\}$,

$E^{\tau_{\mathbf{v}^{n}}}(a)=\sum_{e\in
B}\frac{\tau_{\mathbf{v}^{n}}(a\alpha(e^{*}))}{\tau_{\mathbf{v}^{n}}(\alpha(e)\alpha(e^{*}))}\alpha(e).$

The condition that $(\mathbf{v}^{n})_{n\in{\mathds{N}}}$ converges to
$\mathbf{v}^{\infty}$ coordinate-wise is equivalent to weak* convergence of
$(\tau_{\mathbf{v}^{n}})_{n\in{\mathds{N}}}$ to $\tau_{\mathbf{v}^{\infty}}$.
Thus, by continuity of addition and scalar multiplication,
$(E^{\tau_{\mathbf{v}^{n}}}(a))_{n\in{\mathds{N}}}$ converges to
$E^{\tau_{\mathbf{v}^{\infty}}}(a)$ with respect to
$\|\cdot\|_{\mathfrak{A}}$.

Now, we prove uniform convergence on any compact set of
$({\mathfrak{A}},\|\cdot\|_{\mathfrak{A}})$. Let $C\subset{\mathfrak{A}}$ be
compact with respect to $\|\cdot\|_{\mathfrak{A}}$. Let $\varepsilon>0$. By
compactness, there exist $N\in{\mathds{N}}$ and $a_{1},a_{2},\ldots,a_{N}\in
C$ such that

$C\subseteq\bigcup_{k=1}^{N}\\{a\in{\mathfrak{A}}:\|a-a_{k}\|_{\mathfrak{A}}<\varepsilon/3\\}.$

By pointwise convergence, choose $N^{\prime}\in{\mathds{N}}$ such that
$\|E^{\tau_{\mathbf{v}^{n}}}(a_{k})-E^{\tau_{\mathbf{v}^{\infty}}}(a_{k})\|_{\mathfrak{A}}<\varepsilon/3$
for all $n\geqslant N^{\prime}$ and $k\in\\{1,2,\ldots,N\\}$. Let $n\geqslant
N^{\prime}$ and let $a\in C$. Then there exists $k\in\\{1,2,\ldots,N\\}$ such
that $\|a-a_{k}\|_{\mathfrak{A}}<\varepsilon/3$. Thus,

$\displaystyle\|E^{\tau_{\mathbf{v}^{n}}}(a)-E^{\tau_{\mathbf{v}^{\infty}}}(a)\|_{\mathfrak{A}}$
$\displaystyle\leqslant\|E^{\tau_{\mathbf{v}^{n}}}(a)-E^{\tau_{\mathbf{v}^{n}}}(a_{k})\|_{\mathfrak{A}}+\|E^{\tau_{\mathbf{v}^{n}}}(a_{k})-E^{\tau_{\mathbf{v}^{\infty}}}(a_{k})\|_{\mathfrak{A}}$
$\displaystyle\quad\quad+\|E^{\tau_{\mathbf{v}^{\infty}}}(a_{k})-E^{\tau_{\mathbf{v}^{\infty}}}(a)\|_{\mathfrak{A}}$
$\displaystyle<\|E^{\tau_{\mathbf{v}^{n}}}(a-a_{k})\|_{\mathfrak{A}}+\frac{\varepsilon}{3}+\|E^{\tau_{\mathbf{v}^{\infty}}}(a-a_{k})\|_{\mathfrak{A}}$
$\displaystyle\leqslant\|a-a_{k}\|_{\mathfrak{A}}+\frac{\varepsilon}{3}+\|a-a_{k}\|_{\mathfrak{A}}$
$\displaystyle<\varepsilon,$

where Definition 3.2 is used in the penultimate inequality.

Hence, by the reverse triangle inequality and uniform continuity of the square
root function,
$(\sqrt{\|E^{\tau_{\mathbf{v}^{n}}}(\cdot)\|_{\mathfrak{A}}})_{n\in{\mathds{N}}}$
converges to $\sqrt{\|E^{\tau_{\mathbf{v}^{\infty}}}(\cdot)\|_{\mathfrak{A}}}$
uniformly on any compact set of $({\mathfrak{A}},\|\cdot\|_{\mathfrak{A}})$.
As singletons are compact, we have pointwise convergence on ${\mathfrak{A}}$.
∎

We now focus on the Effros–Shen algebras and begin with their definition found
in [8]. Let $\theta\in{\mathds{R}}$ be irrational. There exists a unique
sequence of integers $(r^{\theta}_{n})_{n\in{\mathds{N}}}$ with
$r^{\theta}_{n}>0$ for all $n\in{\mathds{N}}\setminus\\{0\\}$ such that

$\theta=\lim_{n\to\infty}r_{0}^{\theta}+\cfrac{1}{r^{\theta}_{1}+\cfrac{1}{r^{\theta}_{2}+\cfrac{1}{r^{\theta}_{3}+\cfrac{1}{\ddots+\cfrac{1}{r^{\theta}_{n}}}}}}.$

When $\theta\in(0,1)$, we have that $r^{\theta}_{0}=0$. The sequence
$(r^{\theta}_{n})_{n\in{\mathds{N}}_{0}}$ is the continued fraction expansion
of $\theta$ [10].

Next, we define the finite-dimensional C*-subalgebras of the Effros–Shen
algebras. For each $n\in{\mathds{N}}$, define

$p_{0}^{\theta}=r_{0}^{\theta},\quad p_{1}^{\theta}=1\quad\text{ and }\quad
q_{0}^{\theta}=1,\quad q_{1}^{\theta}=r^{\theta}_{1},$

and set

$p_{n+1}^{\theta}=r^{\theta}_{n+1}p_{n}^{\theta}+p_{n-1}^{\theta}$

and

$q_{n+1}^{\theta}=r^{\theta}_{n+1}q_{n}^{\theta}+q_{n-1}^{\theta}.$

The sequence $\left(p_{n}^{\theta}/q_{n}^{\theta}\right)_{n\in\mathbb{N}_{0}}$
of convergents $p^{\theta}_{n}/q^{\theta}_{n}$ converges to $\theta$. In fact,
for each $n\in{\mathds{N}}$,

$\frac{p_{n}^{\theta}}{q_{n}^{\theta}}=r_{0}^{\theta}+\cfrac{1}{r^{\theta}_{1}+\cfrac{1}{r^{\theta}_{2}+\cfrac{1}{r^{\theta}_{3}+\cfrac{1}{\ddots+\cfrac{1}{r^{\theta}_{n}}}}}}.$

We now define the C*-algebras with which we endow Frobenius–Rieffel norms. Let
${\mathfrak{A}}_{\theta,0}=\mathbb{C}$ and, for each $n\in\mathbb{N}_{0}$, let

${\mathfrak{A}}_{\theta,n}={\mathfrak{M}}_{q_{n}^{\theta}}({\mathds{C}})\oplus{\mathfrak{M}}_{q_{n-1}^{\theta}}({\mathds{C}}).$

These form an inductive sequence with the maps

(3.3) $\alpha_{\theta,n}:a\oplus
b\in{\mathfrak{A}}_{\theta,n}\mapsto\mathrm{diag}\left(a,\ldots,a,b\right)\oplus
a\in{\mathfrak{A}}_{\theta,n+1},$

where there are $r^{\theta}_{n+1}$ copies of $a$ on the diagonal in the first
summand of ${\mathfrak{A}}_{\theta,n+1}$. This is a unital *-monomorphism by
construction. For $n=0$,

$\alpha_{\theta,0}:\lambda\in{\mathfrak{A}}_{\theta,0}\mapsto\mathrm{diag}(\lambda,\ldots,\lambda)\oplus\lambda\
\in{\mathfrak{A}}_{\theta,1}.$

The Effros–Shen algebra associated to $\theta$ is the inductive limit (see
[23, Section 6.1])

${\mathfrak{A}}_{\theta}=\underrightarrow{\lim}\
({\mathfrak{A}}_{\theta,n},\alpha_{\theta,n})_{n\in{\mathds{N}}}.$

There exists a unique faithful tracial state $\tau_{\theta}$ on
${\mathfrak{A}}_{\theta}$ such that for each
$n\in{\mathds{N}}\setminus\\{0\\}$, $\tau_{\theta,n}$ (see Expression (3.2))
is defined for each $(a,b)\in{\mathfrak{A}}_{\theta,n}$ by

$\tau_{\theta,n}(a,b)=t(\theta,n)\frac{1}{q_{n}^{\theta}}\mathrm{Tr}(a)+(1-t(\theta,n))\frac{1}{q_{n-1}^{\theta}}\mathrm{Tr}(b),$

where

$t(\theta,n)=(-1)^{n-1}q_{n}^{\theta}(\theta
q_{n-1}^{\theta}-p_{n-1}^{\theta})\in(0,1)$

(see [3, Lemma 5.5]).

For each $n\in{\mathds{N}}\setminus\\{0\\}$, define

$\kappa_{\theta,n}=\sqrt{\frac{\theta
q^{\theta}_{n}-p^{\theta}_{n}}{\left(\theta
q^{\theta}_{n-2}-p^{\theta}_{n-2}\right)r^{\theta}_{n}(r^{\theta}_{n}+1)^{2}}}$

as in [2, Theorem 5.2] and let
$\kappa_{\theta}=(\kappa_{\theta,n})_{n\in{\mathds{N}}}$,

$\beta_{\theta}(n)=\frac{1}{\dim({\mathfrak{A}}_{\theta,n})}=\frac{1}{(q_{n}^{\theta})^{2}+(q_{n-1}^{\theta})^{2}},$
$\mathcal{U}^{\theta}_{n}=\alpha^{(n)}_{\theta}({\mathfrak{A}}_{\theta,n})$

as in Expression (3.1). For each
$a\in\alpha^{(n)}_{\theta}({\mathfrak{A}}_{\theta,n})$, let

(3.4)
${\mathsf{L}}_{\theta,n}(a)={\mathsf{L}}^{\tau_{\theta},\kappa_{\theta}}_{\mathcal{U}^{\theta}_{n},\beta_{\theta}}(a)=\max_{m\in\\{0,1,\ldots,n-1\\}}\frac{\max\left\\{\|a-E_{m}^{\tau_{\theta}}(a)\|_{E_{m}^{\tau_{\theta}}},\|a^{*}-E_{m}^{\tau_{\theta}}(a^{*})\|_{E_{m}^{\tau_{\theta}}}\right\\}}{\kappa_{\theta,m}\beta_{\theta}(m)}$

as in Theorem 3.4.

######  Theorem 3.7.

If $\theta\in(0,1)\setminus{\mathds{Q}}$, then there exists a strongly Leibniz
L-seminorm ${\mathsf{L}}_{\theta}$ on ${\mathfrak{A}}_{\theta}$ with
$A=1,B=0,C=1$ such that

(3.5)
${\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta},{\mathsf{L}}_{\theta}),({\mathfrak{A}}_{\theta,n},{\mathsf{L}}_{\theta,n}))\leqslant
4\sum_{j=n}^{\infty}\beta_{\theta}(j)$

for all $n\in{\mathds{N}}$, and thus

$\lim_{n\to\infty}{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta},{\mathsf{L}}_{\theta}),({\mathfrak{A}}_{\theta,n},{\mathsf{L}}_{\theta,n}))=0.$

###### Proof.

Let $n\in{\mathds{N}}$. By [2, Theorem 5.2] and Proposition 3.5,

$\kappa_{\theta,n}\|a\|_{{\mathfrak{A}}_{\theta}}\leqslant\|a\|_{E^{\tau_{\theta}}_{n}}$

for all $a\in\alpha^{(n+1)}_{\theta}({\mathfrak{A}}_{\theta,n+1})$. The proof
is complete by Theorem 3.4. ∎

We now prove our main result about the Effros–Shen algebras.

######  Theorem 3.8.

The map

$\theta\in(0,1)\setminus{\mathds{Q}}\longmapsto({\mathfrak{A}}_{\theta},{\mathsf{L}}_{\theta}),$

where ${\mathsf{L}}_{\theta}$ is defined in Expression (3.5), is continuous
with respect to the usual topology on $(0,1)\setminus{\mathds{Q}}$ and the
topology induced by ${\mathsf{\Lambda}^{\ast}_{SL}}$.

###### Proof.

Let $(\theta(n))_{n\in{\mathds{N}}}$ be a sequence in
$(0,1)\setminus{\mathds{Q}}$ that converges to
$\theta(\infty)\in(0,1)\setminus{\mathds{Q}}$. Let $\varepsilon>0$. By [10],
$(1/q_{m}^{\theta})_{m\in{\mathds{N}}}$ is square summable for all
$\theta\in(0,1)\setminus{\mathds{Q}}$. Moreover, if $\Phi=\phi-1$, where
$\phi$ is the golden ratio, then $q_{m}^{\theta(n)}\geqslant q_{m}^{\Phi}$ for
all $m\in{\mathds{N}}$ and $n\in{\mathds{N}}\cup\\{\infty\\}$. Choose
$N_{1}\in{\mathds{N}}$ such that

$4\sum_{j=N_{1}}^{\infty}\frac{1}{(q_{j}^{\Phi})^{2}+(q_{j-1}^{\Phi})^{2}}<\frac{\varepsilon}{3}.$

Then Theorem 3.7 ensures that

${\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n)},{\mathsf{L}}_{\theta(n)}),({\mathfrak{A}}_{\theta(n),N_{1}},{\mathsf{L}}_{\theta(n),N_{1}}))\leqslant
4\sum_{j=N_{1}}^{\infty}\frac{1}{(q_{j}^{\Phi})^{2}+(q_{j-1}^{\Phi})^{2}}<\frac{\varepsilon}{3}$

for all $n\in{\mathds{N}}\cup\\{\infty\\}$.

By [3, Proposition 5.10], choose $N_{2}\in{\mathds{N}}$ such that

$q_{N_{1}}^{\theta(n)}=q_{N_{1}}^{\theta(\infty)}\quad\text{ and }\quad
q_{N_{1}-1}^{\theta(n)}=q_{N_{1}-1}^{\theta(\infty)}$

for all $n\geqslant N_{2}$. Therefore,
${\mathfrak{A}}_{\theta(n),N_{1}}={\mathfrak{A}}_{\theta(\infty),N_{1}}$ for
all $n\geqslant N_{2}$. By the proof of [3, Lemma 5.12] along with Proposition
3.6 and [2, Theorem 5.2], we have for all
$a\in{\mathfrak{A}}_{\theta(\infty),N_{1}}$ that

$\lim_{m\to\infty}{\mathsf{L}}_{\theta(N_{2}+m),N_{1}}\circ\alpha_{\theta(N_{2}+m)}^{(N_{1})}(a)={\mathsf{L}}_{\theta(\infty),N_{1}}\circ\alpha_{\theta(\infty)}^{(N_{1})}(a).$

Thus, by the same proof as [3, Lemma 5.13]

$\lim_{n\to\infty}{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n),N_{1}},{\mathsf{L}}_{\theta(n),N_{1}}),({\mathfrak{A}}_{\theta(\infty),N_{1}},{\mathsf{L}}_{\theta(\infty),N_{1}}))=0.$

Therefore, we may choose $N_{3}\geqslant N_{2}$ such that

${\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n),N_{1}},{\mathsf{L}}_{\theta(n),N_{1}}),({\mathfrak{A}}_{\theta(\infty),N_{1}},{\mathsf{L}}_{\theta(\infty),N_{1}}))<\frac{\varepsilon}{3}$

for all $n\geqslant N_{3}.$ Hence, if $n\geqslant N_{3}$, then

$\displaystyle{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n)},{\mathsf{L}}_{\theta(n)}),({\mathfrak{A}}_{\theta(\infty)},{\mathsf{L}}_{\theta(\infty)}))$
$\displaystyle\quad\leqslant{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n)},{\mathsf{L}}_{\theta(n)}),({\mathfrak{A}}_{\theta(n),N_{1}},{\mathsf{L}}_{\theta(n),N_{1}}))$
$\displaystyle\quad\quad\quad+{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n),N_{1}},{\mathsf{L}}_{\theta(n),N_{1}}),({\mathfrak{A}}_{\theta(\infty),N_{1}},{\mathsf{L}}_{\theta(\infty),N_{1}}))$
$\displaystyle\quad\quad\quad+{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(\infty),N_{1}},{\mathsf{L}}_{\theta(\infty),N_{1}}),({\mathfrak{A}}_{\theta(\infty)},{\mathsf{L}}_{\theta(\infty)}))$
$\displaystyle\quad<\frac{\varepsilon}{3}+{\mathsf{\Lambda}^{\ast}_{SL}}(({\mathfrak{A}}_{\theta(n),N_{1}},{\mathsf{L}}_{\theta(n),N_{1}}),({\mathfrak{A}}_{\theta(\infty),N_{1}},{\mathsf{L}}_{\theta(\infty),N_{1}}))+\frac{\varepsilon}{3}$
$\displaystyle\quad<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\varepsilon$

by the triangle inequality. ∎

Thus, we see that the equivalence constants found in [2] were vital in this
continuity result. There is nothing that guarantees that any equivalence
constant would provide the same result. However, Proposition 3.10 shows that
we can also obtain continuity of the map in Theorem 3.8 using the sharpest
equivalence constants, which are guaranteed to exist for finite-dimensional
spaces. Now, we do not know if the equivalence constants of [2, Theorem 5.2]
are sharp, but we chose to present the proof of Theorem 3.8 using these
constants since they provided continuity with explicit L-seminorms rather than
L-seminorms that are built using unknown sharp constants. Thus, the purpose of
Proposition 3.10 is to show that if one cannot calculate explicit equivalence
constants that provide continuity, then at least, one can achieve continuity
with the existence of sharp equivalence constants. First, we prove a lemma.

######  Lemma 3.9.

Let $(C,d)$ be a compact metric space. Let $(f_{n})_{n\in{\mathds{N}}}$ be a
sequence of real-valued continuous functions on $X$, and let
$f:X\rightarrow{\mathds{R}}$ be continuous.

If $(f_{n})_{n\in{\mathds{N}}}$ converges to $f$ uniformly, then $(\min_{x\in
C}f_{n}(x))_{n\in{\mathds{N}}}$ converges to $\min_{x\in C}f(x)$ and
$(\max_{x\in C}f_{n}(x))_{n\in{\mathds{N}}}$ converges to $\max_{x\in C}f(x)$.

###### Proof.

Since $C$ is compact, $\inf_{x\in C}f(x)=\min_{x\in C}f(x)=\min f$ and
$\inf_{x\in C}f_{n}(x)=\min_{x\in C}f_{n}(x)=\min f_{n}$ for all
$n\in{\mathds{N}}$. Let $\varepsilon>0$. There exists an $N\in\mathbb{N}$ such
that for $n\geqslant N$, we have $|f_{n}(x)-f(x)|<\varepsilon/2$ for all $x\in
C$. Then for $n\geqslant N$,

$f(x)-\varepsilon/2<f_{n}(x)<f(x)+\varepsilon/2$

for all $x\in C$. We take the infimum of this inequality to obtain

$\min f-\varepsilon/2\leqslant\min f_{n}\leqslant\min f+\varepsilon/2,$

which implies $|\min f-\min f_{n}|\leqslant\varepsilon/2<\varepsilon$.

A similar argument establishes the result for $\min$ replaced with $\max$. ∎

######  Proposition 3.10.

Let $(V,\|\cdot\|)$ be a finite-dimensional normed vector space. Let
$(\|\cdot\|_{n})_{n\in{\mathds{N}}}$ be a sequence of norms on $V$ converging
uniformly on the unit ball of $(V,\|\cdot\|)$ to a norm $\|\cdot\|_{\infty}$
on $V$.

If for each $n\in{\mathds{N}}\cup\\{\infty\\}$ we have

$\kappa_{n}\|\cdot\|\leqslant\|\cdot\|_{n}\leqslant\lambda_{n}\|\cdot\|$

where $\kappa_{n}>0,\lambda_{n}>0$ are sharp, then
$(\kappa_{n})_{n\in{\mathds{N}}}$ converges to $\kappa_{\infty}$ and
$(\lambda_{n})_{n\in{\mathds{N}}}$ converges to $\lambda_{\infty}$.

###### Proof.

Let $n\in{\mathds{N}}\cup\\{\infty\\}$. First, note that $a\in
V\mapsto\|a\|_{n}$ is continuous with respect to $\|\cdot\|$ since the norms
are equivalent. Thus, since $\\{a\in V:\|a\|=1\\}$ is compact by finite
dimensionality, $\\{\|a\|_{n}\in{\mathds{R}}:a\in V,\|a\|=1\\}$ is compact.
Hence

$\inf\\{\|a\|_{n}\in{\mathds{R}}:a\in
V,\|a\|=1\\}=\min\\{\|a\|_{n}\in{\mathds{R}}:a\in V,\|a\|=1\\}>0$

as $\|a\|=1$ implies that $a\neq 0$ and thus $\|a\|_{n}>0$. Since $\kappa_{n}$
is sharp,

$\kappa_{n}=\min\\{\|a\|_{n}\in{\mathds{R}}:a\in V,\|a\|=1\\}.$

Therefore, by Lemma 3.9, $(\kappa_{n})_{n\in{\mathds{N}}}$ converges to
$\kappa_{\infty}$ since the unit sphere of $(V,\|\cdot\|)$ is compact by
finite dimensionality. The remaining result follows similarly. ∎

Thus, combining this result with Proposition 3.6, we also have a proof of
Theorem 3.8 using the sharp constants for $\kappa^{\theta}_{n}$ in Expression
3.4 rather than the explicit ones of [2, Theorem 5.2].

For our final result, we present convergence of UHF algebras with respect to
convergence of their multiplicity sequences. Unlike the Effros–Shen case,
where the continuity result relied on continuity of the equivalence constants
in some appropriate sense, convergence in UHF algebras occurs regardless of
which equivalence constants are chosen. First, we detail the metric space that
we use to describe convergence of the multiplicity sequences and the standard
construction of the class of UHF algebras.

######  Definition 3.11.

The _Baire space_ ${\mathscr{N}}$ is the set
$({\mathds{N}}\setminus\\{0\\})^{\mathds{N}}$ endowed with the metric
$\mathsf{d}$ defined, for any two $(x(n))_{n\in{\mathds{N}}}$,
$(y(n))_{n\in{\mathds{N}}}$ in ${\mathscr{N}}$, by

$d_{\mathscr{N}}\left((x(n))_{n\in{\mathds{N}}},(y(n))_{n\in{\mathds{N}}}\right)=\begin{cases}0\
\ \ \ \text{ if $x(n)=y(n)$ for all $n\in{\mathds{N}}$},\\\ \\\
2^{-\min\left\\{n\in{\mathds{N}}:x(n)\not=y(n)\right\\}}\ \ \ \text{
otherwise}\text{.}\end{cases}$

Next, we define UHF algebras in a way that suits our needs. Given
$(\beta(n))_{n\in{\mathds{N}}}\in{\mathscr{N}}$, let

$\boxtimes\beta(n)=\begin{cases}1&\text{ if }n=0,\\\
\prod_{j=0}^{n-1}(\beta(j)+1)&\text{ otherwise}.\end{cases}$

For each $n\in{\mathds{N}}$, define a unital *-monomorphism by

$\mu_{\beta,n}:a\in{\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}})\longmapsto\mathrm{diag}(a,a,\ldots,a)\in{\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}}),$

where there are $\beta(n)+1$ copies of $a$ in $\mathrm{diag}(a,a,\ldots,a)$.
Set $\mathsf{uhf}((\beta(n))_{n\in{\mathds{N}}})=\underrightarrow{\lim}\
({\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}}),\mu_{\beta,n})_{n\in{\mathds{N}}}$.
The map

$(\beta(n))_{n\in{\mathds{N}}}\in{\mathscr{N}}\longmapsto\mathsf{uhf}((\beta(n))_{n\in{\mathds{N}}})$

is a surjection onto the class of all UHF algebras up to *-isomorphism by [7,
Chapter III.5].

For each $n\in{\mathds{N}}$, let

$\gamma_{\beta}(n)=\frac{1}{\dim({\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}}))},$

and let

$\rho_{\beta}$

be the unique faithful tracial state on $uhf((\beta(n))_{n\in{\mathds{N}}})$,
and set

$\mathcal{V}^{\beta}_{n}=\mu^{(n)}_{\beta}({\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}}))$

as in Expression (3.1).

Next, let $\rho_{\beta,n+1}$ denote the unique faithful tracial state on
${\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}})$. Fix
$\lambda^{\beta}_{n}>0$ such that

(3.6)
$\lambda^{\beta}_{n}\|a\|_{{\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}})}\leqslant\|a\|_{E^{\rho_{\beta,n}}}$

for all $a\in{\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}})$, where
$E^{\rho_{\beta,n}}:{\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}})\rightarrow\mu_{\beta,n}({\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}}))$
is the unique faithful $\rho_{\beta,n+1}$-preserving conditional expectation
onto $\mu_{\beta,n}({\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}}))$. Here
we note that $\lambda^{\beta}_{n}$ is neither explicit nor necessarily the
sharp constant and we assume that $\lambda^{\beta}_{n}$ only depends on
${\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}})$, which is allowed since
$\rho_{\beta,n+1}$ is the unique faithful tracial state on
${\mathfrak{M}}_{\boxtimes\beta(n+1)}({\mathds{C}})$. Let
$\lambda^{\beta}=(\lambda^{\beta}_{n})_{n\in{\mathds{N}}}$.

For each
$a\in\mu^{(n)}_{\beta}({\mathfrak{M}}_{\boxtimes\beta(n)}({\mathds{C}}))$, let

(3.7)
${\mathsf{L}}^{\mathscr{N}}_{\beta,n}(a)={\mathsf{L}}^{\rho_{\beta},\lambda}_{\mathcal{V}^{\beta}_{n},\gamma_{\beta}}(a)$

as in Theorem 3.4.

######  Theorem 3.12.

The map

$\beta\in{\mathscr{N}}\longmapsto(\mathsf{uhf}((\beta(n))_{n\in{\mathds{N}}}),{\mathsf{L}}^{\mathscr{N}}_{\beta}),$

where ${\mathsf{L}}^{\mathscr{N}}_{\beta}$ is defined in Theorem 3.4 using the
L-seminorms defined in Expression (3.7), is continuous with respect to the
Baire space and the topology induced by ${\mathsf{\Lambda}^{\ast}_{SL}}$.

###### Proof.

The majority of this proof is complete by the proof of [3, Theorem 4.9]. All
that remains is continuity of the equivalence constants, but this follows
similarly as the proof of [3, Theorem 4.9]. Indeed, if
$d_{\mathscr{N}}(\beta,\eta)<\frac{1}{2^{n}}$, then for all $k\leqslant n,$ we
have $\lambda^{\beta}_{k}=\lambda^{\eta}_{k}$ since

  1. (1)

$\rho_{\beta,k+1}=\rho_{\eta,k+1}$,

  2. (2)

${\mathfrak{M}}_{\boxtimes\beta(k+1)}({\mathds{C}})={\mathfrak{M}}_{\boxtimes\eta(k+1)}({\mathds{C}})$,

  3. (3)

$\mu_{\beta,k}({\mathfrak{M}}_{\boxtimes\beta(k)}({\mathds{C}}))=\mu_{\eta,k}({\mathfrak{M}}_{\boxtimes\eta(k)}({\mathds{C}}))$,
and

  4. (4)

$E^{\rho_{\beta,k}}=E^{\rho_{\eta,k}}$,

which is all the information required to fix $\lambda^{\beta}_{k}$ and
$\lambda^{\eta}_{k}$. ∎

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