Bunce-Deddens algebras as quantum Gromov-Hausorff distance limits of circle algebras

Noncommutative analogues of the Gromov-Hausdorff distance [31, 19, 16, 21, 18] allow for the discussion of limits for certain sequences unital C*-algebras endowed with a notion of a quantum metric, in a manner which generalizes the notion of convergence of compact metric spaces in the sense of the Edwards-Gromov-Hausdorff distance [9]. Of course, in C*-algebra theory, a very common notion of limit is given by inductive limits of inductive sequences, as used to great success in classification theory, among others. In particular, inductive limits of finite dimensional C*-algebras, called AF-algebras, can be seen as the beginning of the research on the classification of C*-algebras. Reconciling metric convergence and inductive limits for AF algebras has been the topic of a previous work from the first two authors [3], followed by several developments by the second author [2, 1]. A next, crucial chapter in the theory of classification, was the study of inductive limits of circle algebras, called A${\mathds{T}}$-algebras. A well-known example of an A${\mathds{T}}$-algebra is given by the Bunce-Deddens algebras [6], and the present work proposes to see how metric convergence can be reconciled with the notion of inductive limit for this particular family of A${\mathds{T}}$-algebras. Moreover, we also prove that the function which maps an element of the Baire space — a sequence of natural numbers — to its associated Bunce-Deddens algebra is a continuous map for Rieffel’s quantum Gromov-Hausdorff distance.

The first noncommutative analogue of the Gromov-Hausdorff distance, motivated by questions from mathematical physics, was discovered by Rieffel [31]. It was constructed over the class of quantum metric order unit spaces, which Rieffel called quantum compact metric spaces — though as we shall briefly discuss, this last term’s meaning has evolved in time. The idea behind the definition of an quantum metric order unit space, inspired by an idea of Connes [7], is to generalize the structure of a space of Lipschitz maps, endowed with the Lipschitz seminorm. Rieffel especially noted that a key property of the Lipschitz seminorm on the space of real-valued functions over a compact metric space $X$ is that it induces by duality a distance on the space of Radon probability measures over $X$ which metrizes the weak* topology. This distance is of course the Monge-Kantorovich metric introduced by Kantorovich [11]. This property can be made sense of in the more general, noncommutative context.

The first occurrence of the term quantum compact metric space can be found in Connes’ work on spectral triples [7]. Rieffel proposed a definition for this term based on order unit spaces and seminorms which generalize Lipschitz seminorms on spaces of functions over compact metric spaces. The following definition encapsulates Rieffel’s notion as used in his construction of the quantum Gromov-Hausdorff distance.

We note that the requirement that the unit ball of Lip-norms be closed is not included in [31]. However, as explained in [31], if a seminorm $S$ satisfies (1) and (2) but not (3) in Definition (1.3), then setting:

allowing for $\infty$, gives a quantum metric order unit space $({\mathfrak{A}},{\mathsf{L}})$ with ${\mathsf{mk}_{{\mathsf{L}}}}={\mathsf{mk}_{S}}$ and ${\mathsf{L}}$ is lower semicontinuous on ${\mathfrak{A}}$. So Assumption (3) can always be made, and it simplifies the statement of many theorems.

Rieffel’s quantum Gromov-Hausdorff distance is thus a noncommutative analogue of the Gromov-Hausdorff distance for quantum metric order unit spaces. Its definition follows Edwards and Gromov’s ideas, though of course, the techniques needed to establish the properties of this new metric are quite different.

Rieffel proved in [31] that the distance ${\mathsf{dist}}_{q}$ is a complete pseudo-metric on the class of all quantum metric order unit spaces, which is zero between two quantum metric order unit spaces $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ if and only there exists a positive linear map $\pi:{\mathfrak{A}}\rightarrow{\mathfrak{B}}$ such that ${\mathsf{L}}_{\mathfrak{B}}\circ\pi={\mathsf{L}}_{\mathfrak{A}}$. Several examples of convergence for this metric were derived [14, 31, 29].

In time, it has become apparent that progress in noncommutative metric geometry requires a noncommutative analogue of the Gromov-Hausdorff distance for the class of quantum compact metric spaces defined, not on order unit spaces, but on actual C*-algebras, with the appropriate coincidence property. The Gromov-Hausdorff propinquity [16, 21] provides such an analogue.

For any fixed function $F$ as above, the propinquity, denoted ${\mathsf{\Lambda}^{\ast}_{F}}$, is a complete metric on the class of quantum compact metric spaces with the $F$-Leibniz property, with the property that the propinquity between two such quantum compact metric spaces $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ is null if and only if there exists a *-isomorphism $\pi:{\mathfrak{A}}\rightarrow{\mathfrak{B}}$ such that ${\mathsf{L}}_{\mathfrak{B}}\circ\pi={\mathsf{L}}_{\mathfrak{A}}$, and $\pi$ is called a full quantum isometry [18]. We note that the propinquity is a metric up to full quantum isometry on the class of all $F$-Leibniz quantum compact metric spaces without fixed $F$, which we denote ${\mathsf{\Lambda}^{\ast}}$, but it is not complete in this case. The propinquity, when restricted to “classical” quantum metric spaces, is topologically equivalent to the Gromov-Hausdorff distance. New techniques are needed to prove the properties of the propinquity and utilize it since working with quantum compact metric spaces rather than quantum metric order unit spaces means working with a more rigid structure. On the other hand, the advantages of working with the propinquity become apparent as it allows for the discussion of convergence of modules [13] or convergence of group actions [12].

Many convergence results are known for the propinquity [16, 13, 3, 30]. In particular, [3], the two first authors initiated the study of convergence, in the metric sense, of sequences used to construct C*-algebras by taking inductive limits, by studying the metric properties of AF-algebras. This work was continued in subsequent papers [2]. As part of this particular line of investigation, the first author proved in [1] that the completeness of the propinquity makes it possible to define a quantum metric on the inductive limit of a sequence of C*-algebras endowed with quantum metric structures, as long as the connecting maps satisfy some natural properties.

The present work proposes to endow Bunce-Deddens algebras with quantum metrics using the same method as [1]. Thus, the idea is to introduce certain quantum metrics on circle algebras and a completeness-based argument to obtain a metric on the A${\mathds{T}}$-algebra obtained as the inductive limit of these circle algebras.

There is however a difficulty in proceeding directly along these lines. Indeed, each quantum metric which we introduce on circle algebras does satisfy a form of Leibniz identity, but we can not prove that there is a uniform choice of such a Leibniz property for the entire inductive sequence for a given Bunce-Deddens algebra. This means that unfortunately, we work outside of any class where we know that the propinquity is a metric (it is of course a pseudo-metric).

Thus, in order to use the techniques of [1], we would like to understand how some of the relevant constructions in [19, 15] for the propinquity may remain valid without the Leibniz property assumptions. As we noted, there is no hope to keep the important coincidence property, but this is not directly used in [1]. However, the proof of completeness is central to the argument of [1].

We see in this paper that in fact, once we remove the constraints to work with C*-algebras and quasi-Leibniz Lip-norms, the construction of the propinquity, if mimicked, simply gives an alternate expression for Rieffel’s distance ${\mathsf{dist}}_{q}$. This is a very interesting fact, since it shows that indeed, the efforts placed in devising new techniques when working with the propinquity are exactly due to working with C*-algebras and Lip-norms with some Leibniz property. We stress that this does not mean that the restriction of ${\mathsf{dist}}_{q}$ to quantum compact metric spaces is the propinquity — it is not as it still does not enjoy the appropriate coincidence property. What it means is that by allowing order unit spaces in the construction of the propinquity, we lose what makes the propinquity different from ${\mathsf{dist}}_{q}$. This observation is of independent interest.

Therefore, in this paper, we prove that Bunce-Deddens algebras are limits of their inductive sequences in a metric sense, for Rieffel’s distance ${\mathsf{dist}}_{q}$. There is one more point of subtlety which we must address here. While Rieffel proved that ${\mathsf{dist}}_{q}$ is complete in [31], we need for our proof in this paper a different description of the limit of a Cauchy sequence for ${\mathsf{dist}}_{q}$. The description we seek is essentially the one obtained in [16] for the propinquity. Thus, we spend some efforts carrying the proof of completeness in [16] to ${\mathsf{dist}}_{q}$. This turns out to be quite technical, but is carried out successfully in this paper. This is an example of how our new expression for ${\mathsf{dist}}_{q}$, inspired by the propinquity, can lead to new observations about ${\mathsf{dist}}_{q}$.

Our paper is thus organized as follows. First, in the next section, we open with a general scheme to turn bi-Lipschitz morphisms between C*-algebras to quantum isometries by manipulating the quantum metrics on their codomain, under the assumption that the range of the morphisms is also the range of a conditional expectation on the codomain. This section is very general, and is concluded with a theorem about making Cauchy sequences for ${\mathsf{dist}}_{q}$ out of inductive limits of quantum compact metric spaces. We already observe that in general, this construction suffers from the problem that no quasi-Leibniz property emerges which is common to all the quantum metrics in the Cauchy sequence, forcing us to work with ${\mathsf{dist}}_{q}$ rather than the propinquity. This second section contains the main idea of the construction of quantum metrics on circle algebras used in this paper.

We then apply our second section to the standard inductive sequences defining Bunce-Deddens algebras in the third and fourth section of this paper. We establish that all the needed ingredients required to apply the first section can be constructed for these circle algebras. We conclude with the fact that such an inductive sequence is naturally Cauchy for ${\mathsf{dist}}_{q}$, and thus by completeness, must converge to some quantum metric order unit space. The question, of course, is whether this quantum metric order unit space is the self-adjoint part of a Bunce-Deddens algebra with some quantum metric.

To answer this question, we generalize [1] to ${\mathsf{dist}}_{q}$ under appropriate hypothesis. In the fifth section, we establish a new expression for ${\mathsf{dist}}_{q}$ inspired, as discussed above, by the propinquity. We then see how the proof of completeness for the propinquity gives a new proof of completeness for ${\mathsf{dist}}_{q}$ which, importantly, gives a different expression for the limit of a Cauchy sequence. Of course, our expression for these limits are isomorphic as quantum metric order unit spaces to Rieffel’s, but our new expression makes it possible to relate the limit in the metric sense to the limit in the categorical sense. This matter is explained in the sixth section of the paper, where [1] is ported to our current framework. We then can answer our problem and prove that, indeed, Bunce-Deddens algebras are limits for ${\mathsf{dist}}_{q}$ of their standard inductive sequence. We use this result to also obtain a continuity result for the family of Bunce-Deddens algebras over the Baire space.

We begin with a very simple observation: if a C*-subalgebra of a quantum compact metric space $({\mathfrak{A}},{\mathsf{L}})$, containing the unit of ${\mathfrak{A}}$, is the range of some conditional expectation on ${\mathfrak{A}}$ which is also contractive for ${\mathsf{L}}$, then we can always modify ${\mathsf{L}}$ to make ${\mathfrak{A}}$ and ${\mathfrak{B}}$ arbitrarily close in the sense of the propinquity (though never at distance $0$ unless ${\mathfrak{A}}$ and ${\mathfrak{B}}$ are *-isomorphic, of course).

During this section, we will keep track of the Leibniz conditions on our quantum metrics, precisely because in fact, it will make clear the difficulties we encountered with our construction about inductive limits. For this purpose, we set $F_{2,0}:(x,y,l_{x},l_{y})\in[0,\infty)^{4}\mapsto 2(xl_{y}+yl_{x})$.

We now turn to a result about bi-Lipschitz *-morphisms. There are several equivalent definitions of Lipschitz morphisms [28, 20], which we now recall.

If $\alpha:({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})\rightarrow({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ is a $k$-Lipschitz morphism between two quantum compact metric spaces, then it naturally becomes a contractive morphism from $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ to $({\mathfrak{B}},\frac{1}{k}{\mathsf{L}}_{\mathfrak{B}})$. Yet if $\alpha$ is actually injective and bi-Lipschitz, then it is usually not possible to adjust the quantum metrics to turn $\pi$ into a quantum isometry. However, under the hypothesis that we can find a conditional expectation $\mathds{E}$ of ${\mathfrak{B}}$ onto the range of $\alpha$ in ${\mathfrak{B}}$ such that $\mathds{E}$ is also a Lipschitz linear map, then it is indeed possible to modify ${\mathsf{L}}_{\mathfrak{B}}$ to turn $\alpha$ into a quantum isometry.

We now bring our two previous observations in one theorem which will be key to our construction.

We record that the tunnel constructed in Theorem (2.6) is given as:

for all $(a,b)\in{\mathfrak{sa}\left({{\mathfrak{A}}\oplus{\mathfrak{B}}}\right)}.$

We have now worked out how, given a bi-Lipschitz morphism between two quantum compact metric spaces, it is possible to change the quantum metric on the codomain of this morphism to turn it into a full quantum isometry onto its range and to make its range and its codomain arbitrarily close in the quantum propinquity, at the cost of relaxing the Leibniz inequality.

We now apply this construction repeatedly to an inductive sequence of quantum compact metric spaces whose connecting maps are all bi-Lipschitz morphisms. The problem which arises is that unfortunately, the Leibniz condition of the seminorms we construct typically worsen at each stage. We will address this matter after we prove that we can indeed make a Cauchy sequence for ${\mathsf{dist}}_{q}$ out of any inductive sequence of quantum compact metric spaces where the connecting maps are bi-Lipschitz.

We note that the above theorem highlights the issue we have with the lack of a uniform Leibniz rule for all $n\in{\mathds{N}}$. Indeed, we see that we can calculate distances between any term in our inductive sequence using propinquity since propinquity is a distance on the class of all quantum compact metric spaces equipped with any $F$-Leibniz property as discussed after Definition 1.7. However, we are unable to find a uniform bound on the $F$-Leibniz properties over all $n\in{\mathds{N}}$ (even in the explicit setting of the Bunce-Deddens algebras) and propinquity is only complete over classes of quantum compact metric spaces with uniform $F$-Leibniz property. But, this is where $\mathrm{dist}_{q}$ has an advantage. It is complete on the class of all quantum metric order unit spaces. Although $\mathrm{dist}_{q}$ doesn’t have the optimal coincidence property, it still has this one crucial advantage, and this is why the above theorem ends with $\mathrm{dist}_{q}$ rather than ${\mathsf{\Lambda}^{\ast}}$.

Of course, we want to relate the limit in Theorem (2.7) with the inductive limit of the given sequence. This is achieved by proving two observations:

• •

  the limit of the sequence is described in the proof of the completeness of ${\mathsf{dist}}_{q}$,

• •

  the tunnels used in the proof of Theorem (2.6), which are given by Expression (2.1), are actually at once related to the inductive limit and to the metric limit computations.

Before we move in this direction, we however begin with our core example for Theorem(2.7): the Bunce-Deddens algebras.

We denote the C*-algebra of $n\times n$ matrices over ${\mathds{C}}$ by ${\mathfrak{M}}_{n}({\mathds{C}})$.