Inductive limits of C*-algebras and compact quantum metrics spaces
Konrad Aguilar






Inductive limits and quantum metrics]Inductive limits of C*-algebras and compact quantum metrics spaces

School of Mathematical and Statistical Sciences 
 Arizona State University 
  901 S. Palm Walk, Tempe, AZ 85287-1804
konrad.aguilar@asu.edu




[2010]Primary:  46L89, 46L30, 58B34.




Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a compact quantum metric of Rieffel, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of  on compact quantum metric spaces. This allows us to place new quantum metrics on all unital AF algebras that extend our previous work with  on unital AF algebras with faithful tracial state.  As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov-Hausdorff propinquity topology.

[
    Konrad Aguilar
    December 30, 2023
=====================








§ INTRODUCTION AND BACKGROUND

Noncommutative Metric Geometry introduced by Rieffel <cit.> and motivated by work of Connes <cit.> provides a solid framework for producing continuous families of quantum metric spaces, which are, in part, C*-algebras. This was done by producing a distance on the class of quantum metric spaces <cit.>, which is analogous to the Gromov-Hausdorff distance between compact metric spaces.  The first example of a continuous family in this metric was quantum tori with respect to their standard parametric space due to Rieffel <cit.>. Recently,  developed a distance on certain classes of quantum metric spaces,  called the Gromov-Hausdorff propinquity <cit.>, which allows one to capture the C*-algebraic structure as well as the quantum metric structure while also establishing continuity results about Hilbert C*-modules, group actions, and vector bundles associated to quantum metric spaces <cit.>.

This article focuses on the categorical notion of convergence of C*-algebras given by  inductive sequences of C*-algebras and their inductive limits, which  is a crucial topic of research in C*-algebras due, in part, to the Elliott classification program that began in <cit.> where Elliott addressed the classification of inductive limits whose inductive sequences comprised of finite-dimensional C*-algebras. Due to the inception of Noncommutative Metric Geometry, the question of when this  categorical notion of convergence passes to a metric convergence arose naturally. But, first, to discuss convergence of C*-algebras, one must first equip the C*-algebras with the quantum metrics. There are many C*-algebras that have been equipped with quantum metrics including: certain Group C*-algebras <cit.>, quantum tori and fuzzy tori <cit.>, noncommutative solenoids <cit.>, quantum Podleś sphere with Dabrowski-Sitarz spectral triple <cit.>, and many more including untial AF algebras <cit.>. Now, in the case of unital AF algebras equipped with faithful tracial states, in <cit.>, quantum metrics were placed in such a way that any inductive sequence that formed the given unital AF algebra converges to the AF algebra in the metric sense of propinquity so that the authors were able to also establish convergence of classes of AF algebras such as UHF algebras <cit.> and Effros-Shen  algebras <cit.>.  However, the types of quantum metrics built in the faithful tracial state case to acheive such convergence results  have not been extended to all unital AF algebras until this article (see Theorem <ref> and Proposition <ref>) even though there are quantum metrics on all unital AF algebras as mentioned above. And, we apply these findings to capture the entire Fell topology on the ideal space on any given unital AF algebra in the propinquity topology by way of a continuous map in Theorem <ref>, which was not possible before since unital AF algebras may have ideals without faithful tracial states. This shows that the propinquity topology is diverse enough to capture this vital topology.

 However, our approach to produce these new quantum metrics on unital AF algebras is inspired by answering a more general question about quantum metrics on inductive limits that need not necessarily be AF. In every case of quantum metrics on inductive limits, the quantum metric is built directly on the inductive limit, which indirectly builds quantum metrics on the terms of the inductive sequence.  In our experience and as seen in this article, it  seems that in order to push results about quantum metrics on inductive limits to new avenues, one must have the ability to build quantum metrics on the inductive limit from suitable quantum metrics on the C*-algebras of the inductive sequence just as the C*-norm on an inductive limit of C*-algebras is constructed from the the C*-norms on the C*-algebras of the inductive sequence.  Thus, in this paper, we accomplish this task under suitable sufficient conditions to obtain our new quantum metrics on AF algebras. The main fact we use is that the Dual Gromov-Hausdorff Propinqiuity <cit.> is complete on certain classes of compact quantum metric spaces, and thus bestows a method for forming limits of quantum metric spaces.  Of course, this limit may not be an inductive limit of quantum metric spaces in any categorical sense, but the idea is to combine the categorical notion of inductive limit of C*-algebras with the metric limit formed by completeness with respect to propinquity. This is done in Theorem <ref> by using Cauchy sequences from inductive sequences of C*-algebras in dual propinquity that exploit the C*-inductive limit structure. We now introduce some background before we move onto our main results.

A compact quantum  metric space (,) is an ordered pair where  is a unital C*-algebra with unit 1_  and  is a seminorm over  defined on   whose domain = { a ∈ :  (a) < ∞} is a unital dense subspace of  over  such that:


  * { a ∈ : (a) = 0 } = _,

  * the  defined, for all two states φ, ψ∈(), by:

    (φ, ψ) = sup{ |φ(a) - ψ(a)| : a∈, (a) ≤ 1 }

metrizes the weak* topology of (), and

  * the seminorm  is lower semi-continuous with respect to ·_.

If (,) is a  compact quantum metric space, then we call the seminorm  a Lip-norm and we denote the diameter of the compact metric space ((), ) by .





A (C,D)-quasi-Leibniz compact quantum metric space (,), for some C ∈, C≥ 1 and D ∈,D≥ 0, is  compact quantum metric space such that
the seminorm  is  (C,D)-quasi-Leibniz, i.e. for all a,b ∈:

    max{(a ∘ b ), ({a,b}) }≤ C(a_(b) + b_(a)) + D (a)(b),

where a∘ b = ab+ba/2 is the Jordan product and {a,b}=ab-ba/2i is the Lie product.


When C=1, D=0, we call  a Leibniz Lip-norm.  When we do not specify C and D, we call (,) a quasi-Leibniz  compact quantum metric space.

Next, we define the  notion of isomorphism for compact quantum metric spaces, formerly called an isometric isomorphism.

Let (, _), (, _) be two compact quantum metric spaces.  A *-isomorphism π: → is a quantum isometry if _∘π = _.




This definition is correct since for a *-isomorphism π: → to be a quantum isometry, it is necessary and sufficient for its dual map to be an isometry between the state spaces with respect to the associated quantum metrics (see <cit.>).



Before we state the theorem that 's dual propinquity is a complete metric up to quantum isometry, we first define a crucial object used to provide estimates for the dual propinquity.


Fix C, ≥ 1, D≥ 0.  Let (_1, _1), (_2, _2) be (C,D)-quasi-Leibniz compact quantum metric spaces. A (C,D)-tunnel τ from (_1, _1) to  (_2, _2) is a 4-tuple τ=(, _, π_1, π_2), where (, _) is a (C,D)-quasi-Leibniz compact quantum metric space and for all j ∈{1, 2}, the map π_j: →_j is a unital *-epimorphism such that for all a ∈_j:

    _j (a)=inf{_ (d) : d∈ and π_j(d)=a}.


We take the following theorem from <cit.> since it adapts the dual propinquity from <cit.> to the quasi-Leibniz case.

Fix C ≥ 1, D ≥ 0.  There exists a pseudo-metric on the class of (C,D)-quasi-Leibniz compact quantum metric spaces called the dual Gromov-Hausdorff propinquity, denoted by , such that for any two (C,D)-quasi-Leibniz compact quantum metric spaces (, _), (, _)


  * ((, _), (, _))=0 if and only if (, _) and (, _) are quantum isometric, and thus  is a metric up to the equivalence relation of quantum isometry,

  * ((, _), (, _)) ≤ 2 λ(τ), 

where τ is any (C,D)-tunnel from (, _) to (, _) and λ(τ) is the length of the tunnel (see <cit.>),

and furthermore  is a complete metric on the class of (C,D)-quasi-Leibniz compact quantum metric spaces up to quantum isometry.

We note futher that for any fixed (C,D),  the dual propinquity is a noncommutative analogue of the Gromov-Hausdorff distance on the class of compact metric spaces up to isometry since one can embed this class homeomorphically into the class of (C,D)-quasi-Leibniz compact quantum metric up to quantum isometry with respect to the topology induced by  (see <cit.>).

Although we do not provide the definition of the length of a tunnel in this background, we provide a useful tool for bestowing estimates of the lengths of certain tunnels that arise in this paper. This tool was key in defining the quantum Gromov-Hausdorff propinquity, which is another metric on the class of (C,D)-quasi-Leibniz compact quantum metric up to quantum isometry that is likely not complete, but still has many applications, which includes finding estimates for the dual propinquity.

Let ,  be two unital C*-algebras.  A bridge γ from  to  is a 4-tuple γ=(, ω, π_, π_) such that


  *  is a unital C*-algebra and ω∈,

  * the set _1(ω)= {ψ∈() : ∀ d ∈, ψ(d)=ψ(ω d)=ψ(d ω)} is non-empty, in which case ω is called the pivot, and 

  * π_ : → and π_: → are unital *-monomorphisms. 





A bridge always exists between two unital C*-algebras, ,. Consider any C*-algebra formed the algebraic tensor product ⊙, denoted by .  Then take 1_=1_⊙ 1_ as the pivot (here _1(ω)=()), and a∈↦ a ⊙ 1_ and b ∈↦ 1_⊙ b as the unital *-monomorphisms. 



The next lemma produces a characterization of lengths of the types of bridges that appear in this article, which follows immediately from definition. But, first, we introduce a definition for the types of bridges that appear  in this article. 

Let  be a unital C*-algebra, and let ⊆ be a unital C*-subalgebra of .  We call the 4-tuple (, 1_, ι, id_) the evident bridge from  to , where ι: → is the inclusion mapping and id_: → is the identity map. 


Let (, _), (, _) be two  compact quantum metric spaces.  If a bridge γ from  to  is of the form γ=(, 1_, π_, π_), then the length of the bridge is

    λ(γ |_, _)=
       max{[ sup_a ∈, _(a) ≤ 1{inf_b ∈, _(b)≤ 1{π_(a)-π_(b)_}},;  sup_b ∈, _(b) ≤ 1{inf_a ∈, _(a)≤ 1{π_(a)-π_(b)_}} ]}

In particular, this holds for evident bridges.

Next, we see how lengths of bridges can be used to estimate lengths of certain tunnels. We note that the length of any bridge between two compact quantum metric spaces is finite (see the discussion preceding <cit.>).

Fix C≥ 1, D ≥ 0. Let (, _), (, _) be two (C, D)-quasi-Leibniz compact quantum metric spaces.  Let γ=(, ω, π_, π_) be a bridge from  to . Fix any r>λ(γ|_, _), where λ(γ|_, _) is the length of the bridge γ. 

If we define for all (a,b) ∈⊕

    ^r_γ|_, _(a,b)=max{_(a), _(b), π_(a)ω - ωπ_(b)_/r}

and we let p_: (a,b) ∈⊕→ a ∈ and p_ : (a,b) ∈⊕→ b ∈ denote the canonical surjections, then τ= (⊕, ^r_γ|_, _, p_, p_) is a (C,D)-tunnel from (, _) to  (, _) with length λ(τ) ≤ r, and 

    ((, _), (, _)) ≤ 2r.


This allows us to define:

Fix C≥ 1, D ≥ 0.  Let (, _), (, _) be two (C, D)-quasi-Leibniz compact quantum metric spaces.  Let γ=(, ω, π_, π_) be a bridge from  to . We call the (C,D)-tunnel (⊕, ^r_γ|_, _, p_, p_) from (, _) to  (, _) of Theorem <ref> the (r, γ|_, _)-evident tunnel associated to the bridge γ, Lip-norms _, _,  and r >λ(γ|_, _).




§ QUANTUM METRICS ON INDUCTIVE LIMITS BY CAUCHY SEQUENCES


We now give a name to when a Cauchy sequence of quasi-Leibniz compact quantum metric spaces coming from an inductive sequence of C*-algebras does produce the inductive limit in the propinquity limit. In this section, we provide natural sufficient conditions for producing such a result in Theorem <ref> by harnessing the indcutive sequence structure. The following definition aims to describe when a Cauchy sequence of quasi-Leibniz compact quantum metric spaces is compatible with C*-algebraic structure of an inductive limt.
First, we provide a convention for inductive sequences and limits of C*-algebras. 

A unital C*-algebra =∪_n ∈_n ^·_ is a  unital  inductive limit of C*-algebras if (_n)_n ∈ is a non-decreasing sequence of unital C*-subalgebras of .

The above convention captures all unital inductive limits of C*-algebras up to *-isomorphism, so there is no loss of generality by making such a convention <cit.>.

Fix C ≥ 1, D ≥ 0. Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras.  If ((_n, __n))_n ∈ is a Cauchy sequence in dual propinquity of (C,D)-quasi-Leibniz compact quantum metric spaces with limit (, _), then we call the sequence   ((_n, __n))_n ∈ an -C*-convergent sequence if  is *-isomorphic to .

From this definition, we see that the inductive limit  itself will be a (C, D)-quasi-Leibniz compact quantum metric space and a limit to the given Cauchy seqeunce in dual propinquity, and summarize this in:

Fix C ≥ 1, D ≥ 0. Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras such that ((_n, __n))_n ∈ is a Cauchy sequence of (C,D)-quasi-Leibniz compact quantum metric spaces  with dual propinquity limit (, _).

 If ((_n, __n))_n ∈ is a -C*-convergent for some *-isomorphism π: →, then ^_ := _∘π is a (C,D)-quasi-Leibniz Lip-norm on     such that  

    ((, _), ( , ^_)) =0  and  lim_n →∞((_n, __n), ( , ^_)) = 0.



It is routine to check that ^_ is a (C,D)-quasi-Leibniz Lip-norm on  such that  ((, _), ( , ^_)) =0 since π is a quantum isometry by construction.  The convergence result follows from the triangle inequality.

A key observation is that: it can be the case that an inductive limit of an inductive sequence of C*-algebras equipped with quantum metrics, which happen to produce a Cauchy sequence in dual propinquity, need not be *-isomorphic to the C*-algebra of the limit in the dual propinquity. Let us now provide an example of non-C*-convergent sequence built from the CAR algebra <cit.> to motivate sufficient conditions that produce C*-convergent sequences. We note that the following example is motivated by work in <cit.>.
[A non-C*-convergent sequence]
Consider the inductive limit, the CAR algebra _2^∞= ∪_n ∈(_2^∞)_n^·__2^∞ <cit.>, where (_2^∞)_n ≅ M_2^n() for all n ∈.

For each n ∈, define _(_2^∞)_n (a) = ((_2^∞)_n)·a-τ(a)1__2^∞__2^∞ for all a ∈ (_2^∞)_n, where τ is the unique faithful tracial state on _2^∞. By <cit.>, the pair ((_2^∞)_n, _(_2^∞)_n ) is a (2,0)-quasi Leibniz compact quantum metric space.

Now, consider σ_n : λ∈↦λ 1_(_2^∞)_n∈ (_2^∞)_n and σ_n() with its unique Lip-norm _, the 0-seminorm, which is also (2,0)-quasi-Leibniz. Clearly σ_n() ≇_2^∞.  Fix n ∈, consider the evident bridge from σ_n() to (_2^∞)_n (Definition <ref>). Next, let λ∈σ_n() (so _(λ)= 0 ≤ 1), then λ-λ__2^∞=0 where _(_2^∞)_n (λ)=0.  Now, let a∈ (_2^∞)_n such that _(_2^∞)_n (a)  ≤ 1, then σ_n(τ(a))-a__2^∞≤ 1/((_2^∞)_n) where _(σ_n(τ(a))=0 ≤ 1.  Thus, by Lemma <ref>, the length of the bridge is less than or equal to 1/((_2^∞)_n). Hence, by  and Theorem <ref>, we have that 

    ^*((, _), ((_2^∞)_n, _(_2^∞)_n )) ≤4/((_2^∞)_n)

and thus lim_n →∞^*((, _), ((_2^∞)_n, _(_2^∞)_n ))=0. Thus, the  Cauchy sequence  ( ((_2^∞)_n, _(_2^∞)_n )) is not _2^∞-C*-convergent.

The above example is interesting on its own as it reflects the fact that matrices can approximate any unital commutative C*-algebra in the Gromov-Hausdorff propinquity <cit.>. However, this is not suitable for our current pursuits.  Now, back to our main goal, it should be noted that there do exist Lip-norms _n' on (_2^∞)_n such that the sequence (((_2^∞)_n,_n'))_n ∈ is Cauchy in  dual propinquity and is _2^∞-C*-con­vergent by <cit.>.
We would now like to provide some sufficient conditions to build C*-convergent  sequences.  The above example can motivate such conditions.  Indeed, consider the inclusion mapping ι_n : (_2^∞)_n → (_2^∞)_n+1 and let a ∈ (_2^∞)_n.  We have 

    _(_2^∞)_n+1(ι_n(a))   = ((_2^∞)_n+1)·a-τ(a)1__2^∞__2^∞
       = ((_2^∞)_n+1)/((_2^∞)_n)_(_2^∞)_n(a).
 Therefore _(_2^∞)_n+1(ι_n(a))≰_(_2^∞)_n(a).  Thus, the canonical embeddings are not contractive with respect to the Lip-norms, which is not very compatible with the notion of an inductive limit. For instance, the embeddings for an inductive sequence of C*-algebras are contractive with respect to the C*-norms. Thus, requiring contractivity seems to be a desirable condition along with the fact that the Lip-norms on AF algebras in <cit.> are all contractive. Furthermore, the Lip-norms in <cit.> allowed for explicit upper bounds on the lengths of the evident bridges of Definition <ref> in dual propinquity. From this, the authors  showed convergence of AF algebras and not just convergence of the inductive sequences that formed the AF algebras. However, in  <cit.>, these methods only worked for AF algebras with faithful tracial states. Thus, as a consequence of this paper and Section <ref>  and the following definition, we will have similar convergence results for all AF algebras with or without faithful tracial state by building Lip-norms on inductive limits from the Lip-norms on the terms of the inductive sequence.  Of course, the Lip-norms on the inductive limits of <cit.> were built explicitly on the inductive limit, so our Lip-norms in this paper would be redundant in the faithful tracial state case and up to passing to a subsequence these inductive sequences satisfy the following definition automatically, and of course C*-convergent with respect to the given inductive limits. Thus, another main purpose of this paper is bestow a method to construct Lip-norms on inductive limits from Lip-norms on  certain inductive sequences without knowledge of any quantum metric structure on the inductive limit itself.

Fix C ≥ 1, D ≥ 0. Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras.  Let ((_n, __n))_n ∈ be a sequence of (C,D)-quasi-Leibniz compact quantum metric spaces. We the call the inductive limit  an ((_n, __n))_n ∈-propinquity approximable inductive limit if the following hold for each n ∈:


  * there exists a (C, D)-quasi-Leibniz Lip-norm __n for _n such that __n is defined on _n and  { a∈_n :__n(a)< ∞} is a dense *-subalgebra of _n,

  * it holds that if a ∈_n, then __n+1(a) ≤__n(a), and

  * there exists a sequence (β(j))_j ∈⊂ (0, ∞) such that ∑_j=0^∞β(j) < ∞ and the length of evident bridge γ_n,n+1=(_n+1, 1_, ι_n,n+1, id_n+1) of Definition (<ref>),  satisfies

    λ(γ_n,n+1|__n, __n+1) ≤β(n),
 and we denote the associated (2β(n), γ_n,n+1|__n, __n+1)-evident tunnel of Definition <ref> by τ_n,n+1.


We note that the above definition satisifies the notion of an inductive sequence in a certain category of compact quantum metric spaces defined in <cit.>, but so does the sequence in Example <ref>. Thus, the above definition is an attempt to provide further criteria to allow the inductive sequence to form some sort of limit even though it is not an inductive limit. In particular,
 the purpose of this definition is to focus on a situation that is capable of allowing for convergence of inductive limits themselves by providing explicit estimates from the inductive sequences as seen in Theorem <ref>. Approximability occured in and was motivated by the work in <cit.>.  However, we recall that this only occured in the case for AF algebras equipped with faithful tracial states. Approximable quantum metrics have not yet been provided for AF algebras outside the faithful tracial state case even by the Lip-norms on all AF algebras built from quotient norms in <cit.> and spectral triples in <cit.>. The fact that the inductive sequences converge to the inductive limit in these cases came from a compactness argument, which did not provide explicit estimates assocaited to evident bridges (see <cit.>).  Yet, there are still advantages to these quantum metrics as they preserve more algebraic structure than the ones of this paper and <cit.> since they are strongly Leibniz of <cit.> and have domains that preserve taking inverses; hence, these Lip-norms of  <cit.> still require more investigation, which lies outside the context of this paper.

We  now begin the journey to show that Definition <ref> gifts C*-convergent sequences. We begin with:

Fix C ≥ 1, D ≥ 0. Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras. 

If  is ((_n, __n))_n ∈-propinquity approximable for some sequence of (C,D)-quasi-Leibniz compact quantum metric spaces and summable (β(j))_j ∈⊂ (0, ∞), then:

 
  * the sequence ((_n, __n))_n ∈ is Cauchy in dual propinquity, where for  n ∈
 
    ((_n, __n), ( _n+1, __n+1)) ≤ 2 λ(τ_n,n+1) ≤ 4 β(n),  and

 
  * given any (C,D)-quasi-Leibniz limit (, _) up to quantum isometry  of the Cauchy sequence ((_n, __n))_n ∈, it holds that 
 
    ((_n, __n), (, _)) ≤ 4 ∑_j=n^∞β(j)    for all n ∈.




The inequalities of (1) follow immediately from Theorem <ref>.  The fact that ((_n, __n))_n ∈ is Cauchy follows from the fact that (β(j))_j ∈ is summable. Conclusion (2) follows from the triangle inequality.

Thus, if we find a limit  (, _) of ((_n, __n))_n ∈ such that ≅, then ((_n, __n))_n ∈ will be -C*-convergent.  Thankfully,  provided a succinct construction of a limit of  certain Cauchy sequences in dual propinquity in <cit.>, which   used to prove that the dual propinquity is complete.  Hence, we now provide a summary of the construction.
[<cit.>]
Fix C ≥ 1, D ≥ 0. Let ((_n, __n))_n ∈ be sequence of (C, D)-quasi-Leibniz compact quantum metric spaces. For each n ∈, let  τ__n=(_n, ^n, π_n, ω_n) be a (C,D)-tunnel from (_n, __n) to (_n+1, __n+1) and denote τ__=(τ__n)_n ∈.

We consider :=∏_n ∈_n as the unital C*-algebra of bounded sequences.

Define the seminorm

    S_0 : (d_n)_n ∈∈∏_n ∈_n⟼sup{^n(d_n) : n ∈}

and 

    K_0:={ d=(d_n)_n ∈∈|  (∀ n ∈)  π_n+1(d_n+1)=ω_n(d_n) }
       L_0 := { d=(d_n)_n ∈∈K_0 | S_0(d) < ∞}

 We denote Re(d)=(d+d^*)/2 and Im(d)=(d-d^*)/(2i) for all d∈∏_n ∈_n.
Define

    S_0:={d=(d_n)_n ∈∈|Re(d),Im(d) ∈L_0 } and  G_0:=S_0^·_.



Next, define the subspace

    I_0 := { (d_n)_n ∈∈G_0 |lim_n →∞d_n__n = 0 },

and

    ^τ__:=G_0/I_0.
  

Now, let q: G_0 →^τ__ be the quotient map and define for all a ∈^τ__

    _τ__(a):=inf{ S_0(d): d ∈G_0 and  q(d)=a }.


With this, we state the key result of <cit.> that leads immediately to the proof of completeness of the dual propinquity. 

 Fix C ≥ 1, D ≥ 0.  Let ((_n, __n))_n ∈ be sequence of (C, D)-quasi-Leibniz compact quantum metric spaces. For each n ∈, let  τ__n=(_n, ^n, π_n, ω_n) be a (C,D)-tunnel from (_n, __n) to (_n+1, __n+1).
 
 Using Notation <ref>, if  ∑_n=0^∞λ(τ__n) < ∞, then:


  * G_0 is a unital C*-subalgebra of , 

  * I_0 is a closed two-sided ideal of G_0, and thus ^τ__ is a unital C*-algebra, 

  * (^τ__, _τ__) is a (C,D)-quasi-Leibniz compact quantum metric space, 

  * the sequence ((_n, __n))_n ∈ is Cauchy in dual propinquity, and 

    lim_n →∞( (_n, __n),(^τ__, _τ__))=0.



Next, we gather more detail about the objects of Notation <ref> in the setting of approximable inductive limits. But, first we state some hypotheses we will use in the next few theorems and definitions.

Fix C ≥ 1, D ≥ 0.  Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras.  Let  be ((_n, __n))_n ∈-propinquity approximable for some sequence of (C,D)-quasi-Leibniz compact quantum metric spaces and summable (β(j))_j ∈⊂ (0, ∞).

  Let τ__n=τ_n,n+1 of Notation <ref> be the   (2β(n), γ_n,n+1 |__n,  __n+1)-evident tunnel from (_n, __n) to (_n+1, __n+1) of Definition <ref> for each n ∈.


If we assume Hypothesis <ref>, then from Notation <ref>


  * K_0= { ((a_n^n, a^n_n+1))_n ∈∈∏_n ∈_n ⊕_n+1| (∀ n ∈) a^n_n+1=a^n+1_n+1}, 

  * 1_=1_G_0=((1_, 1_))_n ∈ and  for all d=((a_n^n, a^n_n+1))_n ∈∈K_0, we have

    S_0(d) = sup_n ∈{max{__n(a_n^n ), a^n_n-a^n+1_n+1_/2β(n)}},


  * and lim_n →∞( (_n, __n),(^τ__, _τ__))=0, where for n ∈,

    ((_n, __n),(^τ__, _τ__)) ≤ 4 ∑_j=n^∞β(j)
    .

 


Apply Definition <ref> to Notation <ref> to obtain conclusions (1) and (2). Conclusion (3) follows immediately from Proposition <ref> and Theorem <ref>.

The above proposition shows us that ^τ__ is beginning to look  like an inductive limit of C*-algebras itself given Definition <ref>. Thus, next, we begin building our *-isomorphism from  onto ^τ__.  To do this, we follow the standard method for providing *-isomorphisms from inductive limits by universality <cit.>.


Assuming Hypothesis <ref>, 
define ψ_0: _0 → by

    ψ_0(a_0)=((a_0,a_0))_n ∈,

and 
for n ∈∖{0}, define ψ_n : _n → by 

    ψ_n(a_n)=((0,0), …, (0,0), (0,a_n),(a_n, a_n),(a_n,a_n), … ),

where (0,a_n) ∈_n-1=_n-1⊕_n.



Assuming Hypothesis <ref>, the map ψ_n : _n → is a *-monomorphism such that ψ_n(_n) ⊆_0 for all n ∈.


Fix n ∈. The fact that ψ_n is a *-monomorphism is clear. Let a ∈_n such that __n(a) < ∞.  If a ∈1_, then S_0(ψ_n(a))=0< ∞. So, assume a ∉1_.  Thus __n(Re(a))< ∞,  __n(Im(a))< ∞ by  (1) of Definition <ref>. In particular, since ψ_n(_n) ⊆K_0 by Proposition <ref> , we have S_0(Re(ψ_n(a)))= max{__n (Re(a)), Re(a)_/(2β(n-1))} < ∞ and similarly S_0(Im(ψ_n(a)))< ∞ by Proposition <ref> and (2) of Definition <ref>.  Therefore, by (1) of Definition <ref>

    ψ_n(_n)    = ψ_n({a ∈_n : __n(a) < ∞}^·_)
       ⊆ψ_n( {a ∈_n : __n(a) < ∞})^·_
       ⊆S_0^·_= _0.

by continuity.

This lemma allows us to define:

Assuming Hypothesis <ref>, for each n ∈, by Lemma <ref> we may define
 ψ^(n) : _n →^τ__ by 

    ψ^(n) :=q ∘ψ_n

where q : _0 →^τ__ is the quotient map.



Assuming Hypothesis <ref>, the map
ψ^(n):_n →^τ__ is a unital *-mono­morphism for each n ∈. Furthermore ψ^(n)=ψ^(n+1) on _n for all n ∈.


Fix n ∈. The map  ψ^(n)  is clearly a  *-homomorphism.  For unital, we note that ψ_n(1_)-1_∈I_0, and thus ψ^(n)(1_)=1_^τ__.  For injectivity, assume a,b ∈_n and ψ^(n)(a)=ψ^(n)(b).  Thus ψ_n(a)-ψ_n(b) ∈I_0. Hence 0 = lim_n →∞a-b_ = a-b_ 
which implies a=b. Finally, let a ∈_n⊆_n+1. Then, again we have  ψ_n(a)-ψ_n+1(a) ∈I_0, and thus ψ^(n)(a)=ψ^(n+1)(a).

Next, we prove the main theorem of this section, where we see all the notions introduced thus far come together.



Fix C ≥ 1, D ≥ 0.  Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras. 

If  is ((_n, __n))_n ∈-propinquity approximable for some sequence of (C,D)-quasi-Leibniz compact quantum metric spaces and summable (β(j))_j ∈⊂ (0, ∞), then the sequence ((_n, __n))_n ∈ is -C*-convergent with respect to  (^τ__, _τ__), where:


  * τ__=(τ_n,n+1)_n ∈ are the evident tunnels from (3) of Definition <ref>,  

  * there exists a unital *-isomorphism ψ: →^τ__ such that  ψ=ψ^(n) on _n for all n ∈ of Definition <ref>, and

  * if we define _:= _τ__∘ψ, then (, _) is a (C, D)-quasi-Leibniz compact quantum metric space such that ∪_n ∈__n⊆_ with 

    ^* ((_n, _n), (, _)) ≤ 4 ∑_j=n^∞β(j)    and   lim_n →∞^* ((_n, _n), (, _))=0.




The fact that there exists a unital *-monomorphism ψ: →^τ__ such that  ψ=ψ^(n) on _n for all n ∈ follows from <cit.> and Lemma <ref>. 

Next, we show ψ()=^τ__. Let a+I_0 ∈^τ__. Let ε>0.  There exists b=(b_n)_n ∈=((b_n^n, b^n_n+1))_n ∈∈S_0 such that a+I_0 - b+I_0_ < ε/2 by density.  Thus Re(b), Im(b) ∈L_0. Hence, there exists r∈, r>0 such that S_0(Re(b))≤ r, S_0(Im(b))≤ r.  There exists N ∈, N>1 such that 2r ·∑_j=N^∞β(j) < ε/4.  

Next, note that Re(b)=((Re(b_n^n), Re(b^n_n+1)))_n ∈. Define c=((c_n^n, c^n_n+1))_n ∈∈ in the following way:  

    (c_n^n, c^n_n+1)=(0,0)     : 0 ≤ n ≤ N-2
    
    (0,Re(b^N-1_N))     : n=N-1 
    
    (Re(b_n^n), Re(b^n_n+1))    : n ≥ N.
Therefore c ∈L_0 and c-Re(b) ∈I_0, which implies that c+I_0=Re(b) + I_0 ∈^τ__.

Now, consider ψ_N(Re(b^N-1_N)) and recall that Re(b^N-1_N)=Re(b^N_N) and that Re(b^n_n+1)=Re(b^n+1_n+1) for all n ∈ by Proposition <ref>.  Therefore 

    ψ_N(Re(b^N-1_N))-c_=sup_k ∈Re(b^N_N)-Re(b^N+k+1_N+k+1)_.

 Since S_0(Re(b))≤ r< ∞, we have that Re(b_n^n)- Re(b^n+1_n+1)_≤ r · 2 β(n) for all n ∈ by Proposition <ref>. Hence for all k ∈

    Re(b^N_N)-Re(b^N+k+1_N+k+1)_≤ 2 r ·∑_j=N^N+kβ(j) ≤  2 r ·∑_j=N^∞β(j) < ε/4.

Thus ψ_N(Re(b^N-1_N))-c_≤ε/4.  Similarly, we may find d ∈L_0 with Im(b)+I_0 =d+I_0  such that ψ_N(Im(b^N-1_N))-d_≤ε/4. Note that c+id ∈S_0 with b+I_0= (c+id)+I_0 and that ψ_N(b^N-1_N)- (c+id) _≤ε/2. Therefore, since ψ(b^N-1_N) =ψ^(N)(b^N-1_N) ) as b^N-1_N ∈_N, we gather 

    a+I_0 - ψ(b^N-1_N) _^τ__   ≤a+I_0 - b+I_0_^τ__+ ψ(b^N-1_N) - b+I_0_^τ__
        
    < ε/2 + ψ(b^N-1_N) - (c+id)+I_0_^τ__
       ≤ε/2+ψ_N(b^N-1_N)- (c+id) _
       ≤ε/2+ε/2 =ε

by definition of quotient norm. In particular, the set ψ() is dense in ^τ__.  As ψ is an isometry and  is complete, it must be the case that ψ()=^τ__. 

Now, assume that a ∈∪_n ∈__n, then there exists N ∈, N>1 such that a ∈_N and  __N(a)< ∞. Thus by Proposition <ref>, 

    _(a)   =_^τ__∘ψ(a)= _^τ__∘ψ^(N)(a) ≤ S_0(ψ_N(a))
       = max{__N(a), a_/(2β(N-1))}< ∞.

The remaining follows from Proposition <ref> and Proposition <ref>.

Next, we see how our approximable inductive limits are well-suited for providing convergent sequences of inductive limits by reducing the problem of showing convergence to the terms of the inductive sequence, which will be applied in Theorem <ref>.

Fix C ≥ 1, D ≥ 0.  For each k ∈∪{∞},   let ^k=∪_n ∈^k_n ^·_ be a unital   inductive limit of C*-algebras such that  ^k is ((^k_n, _^k_n))_n ∈-propinquity approximable for some sequence of (C,D)-quasi-Leibniz compact quantum metric spaces and summable (β^k(j))_j ∈⊂ (0, ∞), and let (^k, _^k) be the limit of ((^k_n, _^k_n))_n ∈ in dual propinquity given by (3) of Theorem <ref>.

If:


  * there exists a summable (β(j))_j ∈⊂ (0, ∞) such that β^k(j) ≤β(j) for all k ∈∪{0},j ∈, and

  * for every n ∈, it holds thats 

    lim_k →∞( (^k_n,_^k_n), (_n^∞, __n^∞))=0,


then

    lim_k →∞( (^k,_^k), (^∞, _^∞))=0.



Let ε>0. There exists N ∈ such that 4∑_j=N^∞β(j) < ε/3.  Thus by Theorem <ref> and the triangle inequality, for each k ∈,

    ( (^k,_^k), (^∞, _^∞))
       ≤( (^k,_^k), (_N^k, __N^k))+(  (_N^k, __N^k),  (_N^∞, __N^∞))
          +((_N^∞, __N^∞), (^∞, _^∞))
       ≤ 4∑_j=N^∞β^k(j)+(  (_N^k, __N^k),  (_N^∞, __N^∞)) +4∑_j=N^∞β^∞(j)
        < (2ε/3) + (  (_N^k, __N^k),  (_N^∞, __N^∞)).

Therefore

    lim sup_k →∞( (^k,_^k), (^∞, _^∞)) ≤ (2ε/3)< ε,

which completes the proof as ε>0 was arbitrary.

In this process of building a Lip-norm on the inductive limit, we also created new Lip-norms on the terms on the inductive sequence itself.  We close this section with some comparisons in the next proposition that also serves to explain further the structure of these new Lip-norms.  We also consider the case when the inductive limit already comes equipped with a certain kind of Lip-norm.

Fix C ≥ 1, D ≥ 0.  Let =∪_n ∈_n ^·_ be a unital   inductive limit of C*-algebras. Let  be  ((_n, __n))_n ∈-propinquity approximable for some sequence of (C,D)-quasi-Leibniz compact quantum metric spaces and summable (β(j))_j ∈⊂ (0, ∞), and let β(-1):=∞ with the convention that (1/∞)=0.


If (^τ__, _τ__), ψ, and  (, _) are as in Theorem <ref>, then


  * for each n ∈, the pair (_n, _τ__∘ψ^(n)) is a (C,D)-quasi-Leibniz compact quantum metric space such that

    _(a)=_τ__∘ψ^(n)(a) ≤max{1, _n__n/β(n-1)}·__n(a)

 for all  a ∈_n, and

  * if there exists a (C,D)-quasi-Leibniz Lip-norm _' on  defined and lower semi-continuous on all of  such that _'=__n on _n for each n ∈, then

    __n(a) ≤_τ__∘ψ^(n)(a) ≤max{1, _n__n/β(n-1)}·__n(a)

for all  a ∈_n, equivalently

    _'(a) ≤_(a) ≤max{1, _n__n/β(n-1)}·_'(a)

for all  a ∈_n.



Fix n ∈.  Let a ∈__n. If n=0, then the conclusions are clear since the only Lip-norm on  is the 0-seminorm.  So, assume that n ≥ 1. Then by Proposition <ref>, we have that

    _τ__∘ψ^(n)(a) ≤ S_0(ψ_n(a))= max{__n(a), a_/β(n-1)} < ∞.

Therefore, the domain of _τ__∘ψ^(n) is dense in _n. Now, fix a state μ∈(^τ__).  We have that {d ∈^τ__|_τ__(d) ≤1  and μ(d)=0} is compact by <cit.>.  Thus the set {d ∈ψ^(n)(_n)|_τ__(d) ≤1  and μ(d)=0} is compact.  Therefore, as _τ__∘ψ^(n) is lower semi-continuous on _n and vanishes only on scalars as ψ^(n) is a unital *-monomorphism by Lemma <ref>, we have that (_n, _τ__∘ψ^(n)) is a (C,D)-quasi-Leibniz compact quantum metric space by <cit.>  since ψ^(n) is a *-homomorphism.

Next, fix a ∈__n. As Lip-norms vanish on scalars, we have for any λ∈ by Proposition <ref> and since ψ^(n) is unital,

    _τ__∘ψ^(n)(a)   =_τ__∘ψ^(n)(a-λ1_)  ≤ S_0(ψ_n(a-λ1_))
        = max{__n(a-λ1_), a-λ1__/β(n-1)}
        =max{__n(a), a-λ1__/β(n-1)}.

Thus, as λ∈ was arbitrary, we have 

    _τ__∘ψ^(n)(a)    ≤max{__n(a), inf_λ∈{a-λ1__}/β(n-1)}
       ≤max{__n(a), (_n__n)/β(n-1)) __n(a)}
        =  max{1, (_n__n)/β(n-1)) }__n(a)

by <cit.>.  This completes (1).

For (2), fix a ∈_n.  Next, let d=((d_k^k, d_k+1^k))_k ∈∈I_0.  Thus, the sequence (d^k_k)_k ∈⊂∪_l ∈_l converges to 0. And, the sequence (a -d^k+n_k+n)_k ∈⊂∪_l ∈_l converges to a.  Therefore, since _' is lower semi-continuous on , we have

    _'(a) ≤lim inf_k →∞_'(a-d^k+n_k+n)= lim inf_k →∞__k+n(a-d^k+n_k+n) ≤ S_0(ψ_n(a)-d)

by Proposition <ref>.  Thus _'(a) ≤_τ__∘ψ^(n)(a) as d∈I_0 was arbitrary.













§ ALL UNITAL AF ALGEBRAS ARE PROPINQUITY APPROXIMABLE


In this section, we use our results in the previous section to show that any unital AF algebra  is propinquity approximable with regard to any non-decreasing sequence of unital finite-dimensional C*-subaglebras (_n)_n ∈ such that, we have  =∪_n ∈_n^·_.  This was already shown to be the case in <cit.> only for unital AF algebras equipped with faithful tracial states. This allowed for investigation of UHF algebras <cit.> and Effros-Shen algebras <cit.>, but left out critical examples of AF algebras like the unitalization of the compact operators on a separable Hilbert space <cit.> and the Boca-Mundici AF algebra <cit.>. Now, all unital AF algebras have been shown to be quasi-Leibniz compact quantum metric spaces <cit.>, yet outside the faithful tracial state case, these quantum metrics do not provide approximability. And, approximability is desirable since it allows one to prove theorems about convergence of sequences of  unital AF algebras. We display this directly in this section by showing that convergence of ideals of unital AF algebras in the Fell topology provides convergence of their unitizations in the dual propinquity topology in Theorem <ref> with the new quasi-Leibniz quantum metrics on all unital AF algebras introduced in this section in Theorem <ref>.  To build these and quantum metrics, we simply use the techniques of the previous section and <cit.> along with the observation that all finite-dimensional C*-algebras have faithful tracial states, and we find that our construction generalizes the Lip-norms of <cit.> in Proposition <ref>.



We call a unital C*-algebra of the form =∪_n ∈_n^·_, a unital AF algebra if (_n)_n ∈ is a non-decreasing  sequence of unital finite-dimensional C*-subalgebras such that _0=1_.

We provide the definition of conditional expectations for convenience.
 Let  be a unital C*-algebra and let ⊆ be a unital C*-subalgebra. A linear map E : → such that E()= is a conditional expectation if 


  * E(b)=b for all b ∈, 

  * E is  contractive completely positive <cit.>, and

  * E(bxb')=bE(x)b' for all x ∈, b,b' ∈.


Next, we give notation for the conditional expectations for the finite-dimen­sional C*-algebras of a given AF algebra.

Let =∪_n ∈_n^·_ be a unital AF algebra. For each n ∈,    let E_n+1,n: _n+1→_n be a condition expectation, which always exists by <cit.>  since  finite-dimensional C*-algebras have faithful tracial states and are von Neumann algebras.  If n>m≥ 0, then denote

    E_n,m:=E_m+1,m∘⋯∘ E_n,n-1: _n →_m,

and if n=m, then E_n,m:=id_n, the identity map on _n. 








Let =∪_n ∈_n^·_ be a unital AF algebra. Denote 𝒰=(_n)_n ∈. Let (β(j))_j ∈⊂ (0, ∞) be summable.  For each n ∈,  let E_n+1,n: _n+1→_n be a contitional expectation onto _n.

If for each n ∈∖{0}, we define for all a ∈_n,

    __n, E^β(a):=max_m ∈{0, …,  n-1}{a-E_n,m(a)_/β(m)}

using Notation <ref> and __0, E^β is the 0-seminorm on _0,  then


  * for each n ∈, the pair (_n, __n, E^β) is a (2,0)-quasi-Leibniz comact quantum metric space such that _n__n, E^β≤ 2β(0),

  *  is ((_n, __n, E^β))_n ∈-approximable with respect to (β(j))_j ∈⊂ (0, ∞), and 

  * the (2, 0)-quasi-Leibniz Lip-norm _𝒰, E^β on  given by (2) and Theorem <ref> satisfies ∪_n ∈_n⊆_𝒰, E^β, where

    lim_n →∞((_n, __n, E^β), (, _𝒰, E^β))=0

and 

    ((_n, __n, E^β), (, _𝒰, E^β)) ≤ 4 ∑_j=n^∞β(j)    for each  n ∈,

where for each n ∈, it holds that  

    _𝒰, E^β(a) ≤max{1,2 β(0)/β(n-1)}·__n, E^β(a)
 for all a ∈_n, where β(-1):=∞ with the convention that (1/∞)=0. 

In particular, every unital AF algebra  is propinquity approximable, and  is propinquity approximable by any increasing sequence (_n)_n ∈ of finite-dimensional unital C*-subalgebras such that _0=1_ and =∪_n ∈_n^·_, and if k ∈ (1, ∞), for each j ∈, then we may choose β(j)≤ 1/(_j)^k.


Fix n ∈∖{0}.  Note that E_n,0(_n) ⊆_0=1_ and _0 ⊆_m for all m ∈. Thus __n, E^β(a)=0 if and only if a ∈1_. Furthermore { a ∈_n |__n(a) < ∞}=_n and __n, E^β is lower semi-continuous.  Also __n, E^β is (2,0)-quasi-Leibniz by <cit.>.

 Denote σ : λ1_∈_0 ↦λ∈ and consider μ_n:=σ∘ E_n,0, which is a state on _n.  Let a ∈_n such that __n, E^β(a) ≤ 1 and μ_n(a)=0.  Thus E_n,0(a)=0, and so a_=a-E_n,0(a)_≤β(0).  Hence the set {a ∈_n |__n, E^β(a) ≤ 1  and μ_n(a)=0} is totally bounded by finite-dimensionality. Therefore  (_n, __n, E^β) is a (2,0)-quasi-Leibniz compact quantum metric space by <cit.>. To estimate the diameter, we note that for any a ∈_n with __n, E^β(a) ≤ 1 and any μ, ν∈(_n), 
 
    |μ(a)-ν(a)|   =|μ(a-μ_n(a)1_)-(ν(a-μ_n(a)1_)| = |(μ-ν)(a-μ_n(a)1_)| 
       ≤ 2 a-μ_n(a)1__ = 2a-E_n,0(a)≤ 2 β(0).

 
 Thus (1) is proven. Now, Fix n ∈.  Assume a ∈_n.  Then E_n+1,n(a)=a and so E_n+1,m(a)=E_n,m(a) for all m ∈{0, …, n}. Therefore

    __n+1, E^β(a)= __n, E^β(a)  for all   a ∈_n.



 Next, we estimate lengths of the evident bridges. Fix n ∈.  Let a ∈_n such that __n, E^β(a) ≤ 1. Then __n+1, E^β(a)= __n, E^β(a) ≤ 1 by Equation (<ref>) and a-a_=0. Now, let a ∈_n+1 such that __n+1, E^β(a) ≤ 1.  
 Fix m ∈{0, …, n-1}. Note that E_n+1,n(a) ∈_n and  consider 
 
    E_n+1,n(a)-E_n,m(E_n+1,n(a))_   =  E_n+1,n(a)-E_n+1,m(a)_
        = E_n+1,n(a)-E_n+1,n(E_n+1,m(a))_
         =E_n+1,n(a-E_n+1,m(a))_≤a-E_n+1,m(a)_,

 where we used the fact that _m ⊆_n in line 2.  Thus 
 
    __n, E^β(E_n+1,n(a)) ≤__n+1, E^β(a) ≤ 1.

  Furthermore, the assumption that __n+1, E^β(a) ≤ 1 implies that a-E_n+1,n(a) _≤β(n). Finally, note that if a∈_n+1, then E_n+1,n(a) ∈_n by positivity of conditional expectations. Therefore by Lemma <ref>, we have that the length of the evident bridge from Definition <ref> satisfies  
  
    λ(γ_n,n+1|__n, E^β, __n+1, E^β) ≤β(n).
  
 
 Thus (2) is proven, and (3) follows immediately from Theorem <ref> and Proposition <ref>.  The remaining statement follows from the fact that there exist conditional expectations on any finite-dimensional C*-algebra onto any C*-subalgebra by <cit.>, and the fact that for any increasing sequence (_n)_n ∈  such that _0=1_, it holds that (_j) ≥ j for all j ∈.

We note that for certain AF algebras, the beta sequence can be given as β(n)=1/(_n).  For example, this is the case of UHF algebras, Effros-Shen algebras, and the unitalization of the compact operators as long as the increasing sequence _n is chosen appropriately.  However, for the C*-algebra of continuous functions on the one-point compactification on the natural numbers C(), one could take the increasing sequence to be _n ≅^n+1, in which case taking the dimension to a power greater than 1 is necessary to produce a summable sequence.

Next, we see that in the case of a unital AF algebra with a faithful tracial state, the Lip-norms of Theorem <ref> can recover the Lip-norms of <cit.> up to quantum isometry.  Thus, we provide a generalization of the results of <cit.> in the case that (β(j))_j ∈ is summable, and as a consequence, we  achieve convergence in the quantum Gromov-Hausdorff propinquity of   <cit.> in the faithful tracial state case.

Let =∪_n∈_n^·_ be a unital AF algebra equipped with  a faithful tracial state μ.  Denote 𝒰=(_n)_n ∈, and let (β(j))_j ∈⊂ (0, ∞) be summable.  Let

    _𝒰, μ^β(a)=sup_n ∈a-E_n(a)_/β(n)

be the (2,0)-quasi-Leibniz Lip-norm on  of <cit.>, where E_n: →_n is the unique μ-preserving conditional expectation onto _n.

If for each n ∈, we define E_n+1,n:=E_n|__n+1: _n+1→_n and let __n,E^β, _𝒰,E^β be the associated (2,0)-quasi-Leibniz Lip-norms on _n,, respectively, from Theorem <ref>, then:


  * for each n ∈, we have _𝒰, μ^β(a)=__n,E^β(a) for all a∈_n, 

  * for each n ∈ and all a ∈_n, it holds that

    _𝒰,μ^β(a) ≤_𝒰, E^β(a) ≤max{1, 2β(0)/β(n-1)}_𝒰,μ^β(a),

where β(-1):=∞ with the convention that (1/∞)=0,

  * there exists a quantum isometry π : → and so

    ((, _𝒰, μ^β),(, _𝒰, E^β))=0,


  * and for each n ∈, 

    ((_n, __n, E^β),(, _𝒰, E^β)) ≤ 2 ((_n, __n, E^β),(, _𝒰, E^β)) ≤ 2β(n),

and thus lim_n →∞((_n, __n, E^β),(, _𝒰, E^β))=0, where  is the quantum Gromov-Hausdorff propinquity of <cit.>.
 


Fix n ∈∖{0}. By the proof of <cit.>, it holds tht E_n ∘ E_n-1=E_n-1.  Therefore E_n,m=E_m for all m ∈{0, …, n-1}.  Thus (1) is proven. Conclusions (2) and (3) follow from Theorem <ref> and <cit.> and the fact that limits are unique up to quantum isometry in the dual propinquity. Conclusion (4) follows from <cit.> and <cit.>.


We can now apply our findings to discover new continuous families of unital AF algebras since we now do not require faithful tracial states.  We will form a continuous map from the ideal space of any unital AF algebra into the dual propinquity space by mapping an ideal to its unitization with carefully chosen Lip-norms  from Theorem <ref>. So, although we cannot discuss convergence of ideals themselves since ideals may not contain a unit, we can now in the very least capture the Fell topology by considering the unitization of the ideals.  This was not possible before due to the fact that  not every unital AF algebra has a faithful tracial state.  However, since an ideal may or may not contain unit, we have to be careful by what we mean by the "unitization" as these seem to differ when considering a unital versus a non-unital C*-algebra.  Yet, thankfully, there does exist a unitization process of C*-algebras that does not distinguish between the unital and non-unital case, which allows us to proceed much more smoothly in our construction of Lip-norms.  We present this unitization now.  The only known reference that we know of this is from <https://mathoverflow.net/questions/210025/unitization-process-of-unital-and-non-unital-c-algebras> due to user UwF. Thus, we provide details for the proof found in the above link. 

Let  be a C*-algebra with or without unit. The unitization of , denoted , is the vector space ⊕ equipped with


  * adjoint defined by (a,λ)^*:=(a^*, λ) for all a ∈, λ∈, 

  * multiplication defined by (a, λ)(b, μ):=(ab+μ a+λ b,λμ)  for all a,b ∈, λ∈, 

  * and norm defined by 
    (a,λ)_:=max{sup_b ∈, b_≤ 1{ab+λ b_},   |λ| },

for all a ∈, λ∈,

where  equipped with these operations and norm is a unital C*-algebra with unit 1_=(0,1) and a_=(a,0)_ for all a ∈ and  is *-isomorphic to the maximal ideal of co-dimension one of  given by {(a, λ) ∈| a ∈, λ=0}.

Furthermore:


  (4) if  is a C*-subalgebra of , then  is the unital  C*-subalgebra of  given by {(b,λ) ∈| b ∈, λ∈}, 

  (5) if  is non-unital, then (a,λ)_=sup_b ∈, b_≤ 1{ab+λ b_} for all a ∈, λ∈, and

  (6) if  is unital, then (a,λ)_=max{a+λ 1__, |λ|}  for all a ∈, λ∈, in which case the direct sum C*-algebra  ⊕ given by coordinate-wise operations and max norm is *-isomorphic to  by the *-isomorphism 

    (a, λ) ∈⊕⟼(a-λ1_, λ) ∈.




It is routine to check  that  is a *-algebra with respect to the defined operations. Let (⊕) be the unital Banach algebra of bounded operators from the direct sum C*-algebra ⊕ (given by coordinate-wise operations and max norm) to itself. Denote the unit of  (⊕) by 1_(⊕). For each (a, λ)∈⊕, let L_(a,λ)∈(⊕) denote the left multiplication operator, that is L_(a,λ) (b,μ)=(ab, λμ) for all (b, μ) ∈⊕.  By construction, the map

    (a, λ) ∈⟼ L_(a,λ)+λ 1_(⊕)∈(⊕)

is a linear multiplicative function. Now, we show it is injective.  Assume that (a, λ) ∈ such that L_(a,λ)+λ 1_(⊕)=0.  Thus (L_(a,λ)+λ 1_(⊕))(0,1)= (0, λ)  (0,0)=(0,λ) λ=0.  Hence (0,0)=(L_(a,λ)+λ 1_(⊕))(a^*,0)=L_(a,0)(a^*,0)=(aa^*,0)  aa^*=0  a=0. For all (a, λ) ∈, we gather that

    L_(a,λ)+λ 1_(⊕)_(⊕)
         =sup_(b,μ) ∈⊕, (b,μ) _⊕≤ 1{(L_(a,λ)+λ 1_(⊕))(b, μ) _⊕}
        =sup_b ∈, μ∈, b_≤ 1, |μ|≤ 1{(ab+λ b, λμ) _⊕}
        = sup_b ∈, μ∈, b_≤ 1, |μ|≤ 1{max{ab+λ b_, |λμ|}}
        = sup_b ∈, b_≤ 1{max{ab+λ b_, |λ |}}=(a,λ)_.

Therefore ·_ defines a Banach algebra norm on . The fact that this norm satisfies the C*-identity follows the same argument as the proof of the standard unitization <cit.>.

For (4), cleary the set {(b,λ) ∈| b ∈, λ∈} is a unital C*-subalgebra of  that equals . Since they have the same algebraic operations and adjoint, their norms also agree since C*-norms are unique.  

Conclusion (5) follows similiary since the expression on the right-hand side defines a C*-norm on , when  is non-unital by <cit.>.

Finally, for (6), when  is unital, we have that for b ∈, b_≤ 1

    ab+λ b_ =(a+λ1_)b_≤a+λ 1__,

and since 1_∈, 1__=1, we have the desired equality for the norm.  It is routine to show that the map in (6) is a *-isomorphism.


Although we do not define the Fell topology on the ideals of a C*-algebra, we provide some results about the topology. For a more detailed summary see <cit.>.
[<cit.>] Let  be a C*-algebra. Let Ideal() denote the set of norm-closed two-sided ideals of  including {0},.  The Fell topology on Ideal() is a compact Hausdorff topology such that a net (I_μ)_μ∈Δ⊆Ideal() converges to I ∈Ideal() if and only if for each a ∈ the net (a+I_μ_/I_μ)_μ∈Δ⊆ converges to a+I_/I∈ with respect to the usual topology on .

On primitive ideals, the Fell topology is stronger than the Jacobson topology <cit.>, and is in fact built from the closed subsets of the primitive ideals in the Jacobson topology. The Fell topology is a general construction of a topology on closed subsets of a given topology introduced by Fell in  <cit.>.  The next summarizes a topology introduced in <cit.> on inductive limits of C*-algebras built as an inverse limit topology from the Fellt topologies on the ideals of the C*-algebras of the inductive seqeunce, and this topology happens to agree with the Fell  topology in the case of AF algebras and is in general stronger than the Fell topology.

Let =∪_n ∈_n^·_ be an inductive limit of C*-algebras (not necessarily unital). Denote 𝒰=(_n)_n ∈.  The following hold.


  * For each n ∈, the map ι_n+1_i: J ∈Ideal(_n+1) ↦ J ∩_n ∈Ideal(_n) is continuous with respect to the associated Fell topologies on Ideal(_n+1) and Ideal(_n).

  * The inverse limit Ideal()_∞⊆∏_n ∈Ideal(_n) with its topology  given by the inverse limit sequence (Ideal(_n), Fell, ι_n+1_i)_n ∈ (<cit.>) provides a topology on Ideal() denoted Fell_i(𝒰) by the initial topology of the injection

    J ∈Ideal() ↦ (J ∩_n)_n ∈∈Ideal()_∞.


  * The topology Fell_i(𝒰) is stronger than Fell on Ideal(), and if  is AF, then Fell_i(𝒰)=Fell and is metrized by the metric defined for each I,J ∈Ideal() by

    𝗆_i(𝒰)(I,J)=
    0     : I ∩_n=J∩_n  for all  n ∈
    
    2^-min{n ∈: I ∩_n ≠ J ∩_n}    : otherwise.



Now, we are ready to prove our result about ideal convergence. 

Let =∪_n ∈_n^·_ be a unital AF algebra. Denote 𝒰=(_n)_n ∈. Using  Theorem-Definition <ref>, for each n ∈∖{0} and each J ∈Ideal(_n+1), fix a conditional expectation E^J_n+1,n: J→J ∩_n,  and for each J ∈Ideal(_1), fix a conditional expectation E^J_1,0 : I→1_I. Fix a summable sequence (β(j))_j ∈⊂ (0, ∞).

Next, for every I∈Ideal(), set 𝒰_I=( I_n)_n ∈, where for n ∈∖{0}, we set I_n=I ∩_n and I_0=1_I,  and note that I=∪_n ∈I_n^·_I, where 𝒰_I is an non-decreasing sequence of finite-dimensional unital C*subalgebras of I by Theorem-Definition <ref>.   For each, n ∈∖{0}, define for all (a, λ) ∈I_n, 

    ^β_I_n,E^I(a, λ):=max_m ∈{0, …,n-1}{(a, λ)- E^I ∩_m+1_m+1,m∘⋯∘ E^I∩_n_n,n-1(a, λ)_I/β(m)},

and let ^β_I_0,E^I be the zero seminorm on I_0.


If  for every I∈Ideal(), we define ^β_𝒰_I,E^I as in Theorem <ref>,
then, the map 

    ℱ_: I ∈Ideal() ⟼(I, ^β_𝒰_I,E^I) ∈qLcqms_(2,0)

is continuous, where Ideal() is equipped with the Fell topology and qLcqms_(2,0) is the class of (2,0)-quasi-Leibniz compact quantum metric spaces equipped with the topology induced by the dual propinquity .  Furthermore, if β(j)≤ 2^-j-5 for all j ∈, then the map ℱ_ is Lipschitz with respect to the metric on ideals in (3) of Proposition <ref> and the dual propinquity.


Let (I(n))_n ∈∪{∞}⊆Ideal() be a sequence that converges to I(∞) with respect to the Fell topology of Theorem-Definition <ref>. We will use Theorem <ref>. Fix M ∈∖{0}. By Proposition <ref>, there exists N ∈ such that 𝗆_i(𝒰)(I(n), I(∞)) < 2^-(M+1) for all n ≥ N.  Therefore, for all n ≥ N, it holds that I(n) ∩_k=I(∞) ∩_k and I(n)_k=I(∞)_k for all k ∈{0, …, M}. Therefore, by construction, we have that if n ≥ N, then 

    ^β_I(n)_M,E^I(n)(a, λ)
       = max_m ∈{0, …,M-1}{(a, λ)- E^I(n) ∩_m+1_m+1,m∘⋯∘ E^I(n)∩_M_M,M-1(a, λ)_I(n)/β(m)}
       = max_m ∈{0, …,M-1}{(a, λ)- E^I(n) ∩_m+1_m+1,m∘⋯∘ E^I(n)∩_M_M,M-1(a, λ)_I(n)_M/β(m)}
       = max_m ∈{0, …,M-1}{(a, λ)- E^I(∞) ∩_m+1_m+1,m∘⋯∘ E^I(∞)∩_M_M,M-1(a, λ)_I(∞)_M/β(m)}
       = max_m ∈{0, …,M-1}{(a, λ)- E^I(∞) ∩_m+1_m+1,m∘⋯∘ E^I(∞)∩_M_M,M-1(a, λ)_I(∞)/β(m)}
        =^β_I(∞)_M,E^I(∞)(a,λ)

 for all (a, λ) ∈I(n)_M=I(∞)_M by (4) of Theorem-Definition <ref>.  In particular, the identity map is a quantum isometry between ( I(n)_M,^β_I(n)_M,E^I(n)) and (I(∞)_M,^β_I(∞)_M,E^I(∞)) for all n ≥ N.  Hence

    lim_n →∞(( I(n)_M,^β_I(n)_M,E^I(n)), (I(∞)_M,^β_I(∞)_M,E^I(∞)))=0.

The same result is immediate for M=0.
Thus, as M ∈ was arbitrary, we have that

    lim_n →∞(( I(n),^β_𝒰_I(n),E^I(n)), (I(∞),^β_𝒰_I(∞),E^I(∞)))=0

by Theorem <ref>. Hence, the desired map is continuous.  

Next, assume that β(j) ≤ 2^-j-5 for all j ∈.  Fix I,J ∈Ideal() such that I ≠ J.  Therefore, there exists n ∈ such that 𝗆_i(𝒰)(I,J)=2^-n. Assume that n ≥ 1. By the same argument as above and Theorem <ref>, we have

    (( I,^β_𝒰_I,E^I), (J,^β_𝒰_J,E^J))
       ≤(( I,^β_𝒰_I,E^I), (I_n-1,^β_I_n-1,E^I)) + ((I_n-1,^β_I_n-1,E^I), (J_n-1,^β_J_n-1,E^J))
          + (( J,^β_𝒰_J,E^J), (J_n-1,^β_J_n-1,E^J))
       ≤ 4∑_j=n-1^∞ 2^-j-5 +0 + 4∑_j=n-1^∞ 2^-j-5= (1/4) (4· 2^-n)= 𝗆_i(𝒰)(I,J).

The case n=0 follows similarly. 

We now make a remark about a certain choice we made in this section. Given a unital AF algebra =∪_n ∈_n^·_ with a non-decreasing sequence of unital finite-dimensional C*-subalgebras (_n)_n ∈ of ,   we could have used the conditional expectations from _n+1 onto  _n  of Theorem <ref> to build conditional expectations from  onto _n  that behave much like the ones in the faithful tracial state case of Proposition <ref>.  This would build Lip-norms on  that allow for propinquity approximability with respect to quantum propinquity and not just dual propinquity in the case of any unital AF algebra, which is not that much of an advantage since the dual propinquity is preferred anyway as it's complete.  However, to extend the conditional expectations in such a way to accomplish this,  one would essentailly be using the injectivity of finite-dimensional C*-algebras <cit.>. Although the main example of this paper is unital AF algebras, the goal of this paper is to provide a more general framework for future work on providing Lip-norms on inductive limits that allow for not only convergence of inductive sequences to inductive limits but also convergence of sequences of inductive limits, and injectivity of C*-algebras is certainly not a common property especially among the standard classes of inductive limits studied. Hence, we chose to present our work in this section without using the injectivity of finite-dimensional C*-algebras to set the stage for future work.

We conclude this paper by noting that the results in <cit.> are all true in this more general setting by Proposition <ref>.  One just simply needs replace the convergence conditions on the sequence (β(n))_n ∈ with summability conditions when working in the non-faithful tracial state case.  For instance, the compactness results of <cit.> and the convergence of families of Lip-norms on a fixed unital AF algebra result of <cit.> are true wih this minor change, and we note that as a result <cit.> provides continuous families of Lip-norms on the unitization of the compact operators by varying the  sequence (β(n))_n ∈ in a suitable topology. 

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