Semigroupoid, Groupoid and Group Actions on Limits for the Gromov-Hausdorff Propinquity

SEMIGROUPOID, GROUPOID AND GROUP ACTIONS ON LIMITS FOR THE GROMOV-HAUSDORFF PROPINQUITY

FRÉDÉRIC LATRÉMOLIÈRE

Abstract. The Gromov-Hausdorff propinquity provides an analytical framework motivated by mathematical physics where quasi-Leibniz quantum compact metric spaces may be studied by means of metric approximations. A natural question in this setting, answered in this paper, is whether group actions pass to the limit for this new geometry: if a sequence of quasi-Leibniz quantum compact metric spaces is Cauchy for the propinquity and each entry carries a non-trivial compact group action, and if the resulting sequence of groups converges, does the Gromov-Hausdorff limit carry a non-trivial action of the limit group ? What about actions from groupoids, or inductive limits of groups? We establish a general result addressing all these matters. Our result provides a first example of a structure which passes to the limit of quantum metric spaces for the propinquity, as well as a new method to construct group actions, including from non-locally compact groups seen as inductive limits of compact groups, on unital C*-algebras. We apply our techniques to obtain some properties of closure of certain classes of quasi-Leibniz quantum compact metric spaces for the propinquity.

The noncommutative Gromov-Hausdorff propinquity [15, 12, 18, 14, 17] is a fundamental component of an analytic framework for the study of the geometry of the hyperspace of quantum metric spaces, with sights on applications to mathematical physics. A natural yet delicate question is to determine which properties do pass to the limit of a sequence of quantum metric spaces. Partial answers to this question can for instance help us determine the closure of a class of quantum metric spaces. Most importantly, the propinquity is intended to be used to construct models for physics from finite dimensional [10,1, 30] or other form of approximations [11, 19], with the models obtained as limits, possibly using the fact that the propinquity is complete. Thus, we would like to know which properties of the approximate finite models would indeed carry to the eventual limit. The existence of group actions on limits of sequences of quantum compact metric spaces is thus of prime interest. The question can be informally asked: if a sequence of metrized compact groups act on a Cauchy sequence of quasi-Leibniz quantum compact metric spaces, and if these groups converge in some appropriate sense, then does the limit carry an action of the limit group? Asked in such a manner, the answer is trivially yes — one may involve trivial actions — but it is easy and

Date: March 24, 2018. 2000 Mathematics Subject Classification. Primary: 46L89, 46L30, 58B34. Key words and phrases. Noncommutative metric geometry, Gromov-Hausdorff convergence, Monge-Kantorovich distance, Quantum Metric Spaces, Lip-norms, semigroupoid, semigroup, groupoid and group actions. This work is part of the project supported by the grant H2020-MSCA-RISE-2015-691246- QUANTUM DYNAMICS. 1 2 FRÉDÉRIC LATRÉMOLIÈRE rather natural to impose a form of nontriviality using the underlying quantum metrics. Indeed, we can control the dilation of the automorphisms [13] arising from the actions at each level to ensure that any sort of limit would also be non- trivial. We also note that much efforts have been dedicated to constructing full matrix approximations of certain symmetric spaces [26, 28, 29, 30] with the idea that finite dimensional quantum metric spaces are the noncommutative analogues of finite spaces, though they can carry actions of compact Lie groups such as SU(n), and therefore have highly non-trivial symmetries — thus, offering an option to work with “finite spaces” with continuous symmetries. From this perspective, our present work addresses a sort of dual question: if we do use approximations of a physical model using, say, finite dimensional algebras with some specified symmetries, what are sufficient conditions for these symmetries to still be present at the limit for the propinquity? We establish a formal answer to this problem, in a more general context: rather than only working with groups, we prove the result for an appropriate notion of action of semigroupoids — thus, our result applies to groups actions, groupoid actions on class of C*-algebras, and semigroups actions, among others. We also note that our result can be applied to prove the existence of non-trivial actions of inductive limits of groups, even if this inductive limit is not locally compact. While left for further research, the results of this paper hints at a possible application of techniques from noncommutative metric geometry to the study of actions of non-locally compact groups on C*-algebras. As an application of our work, we exhibit closed classes of quasi-Leibniz quantum compact metric spaces, which is in general a very difficult problem to solve. The key observation is that with our method, the limit of ergodic actions is ergodic, in the sense described in this paper — an observation which is quite powerful. For instance, we can deduce from our work and known facts on ergodic actions of SU(2) that the closure of many classes of quasi-Leibniz quantum compact metric spaces whose quantum metrics are induced by ergodic actions of SU(2) consists only of type I C*-algebras. In contrast, the closure of classes of fuzzy tori whose quantum metric arises from a fixed length function of the tori, and ergodic actions of closed subgroups of the tori, consists only of fuzzy and quantum tori (and from our own prior work, include all of the quantum tori). More generally, our work opens a new direction for finding unital C*-algebras on which compact groups act ergodically, by investigating the closure for the propinquity of finite dimensional C*-algebras on which a given compact group acts ergodically. The study of this problem from a metric property will be the subject of subsequent papers, and represents an exiting potential development for our project. We begin this paper with a review of our theory for the hyperspace of quantum metric spaces. We then prove our main result in the context of actions of countable semigroupoids, appropriately defined. Our result does not per say require an action of a semigroupoid, but rather an approximate form of it. We then turn to various applications of our main result, concerning actions of groups, groupoids, and semigroups.

The following notations will be used throughout this paper. Notation. If (E, d) is a metric space, the diameter sup{d(x, y): x, y ∈ E} of E is denoted by diam (E, d). SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 3

The norm of a normed vector space X is denoted by k · kX by default. The unit of a unital C*-algebra A is denoted by 1A. The state space of a C*-algebra A is denoted by S (A) while the real subspace of self-adjoint elements of A is denoted by sa (A). Given any two self-adjoint elements a, b ∈ sa (A) of a C*-algebra A, the Jordan ab+ba product 2 , i.e. the real part of ab, is denoted by a ◦ b, while the Lie product ab−ba 2i — the imaginary part of ab — is denoted by {a, b}. The notation ◦ will often be used for composition of various maps, but will always mean the Jordan product when applied to elements of a C*-algebra, as is customary. The triple (sa (A), · ◦ ·, {·, ·}) is a Jordan-Lie algebra [9]. Last, for any continuous linear map ϕ : E → F between two normed vector E spaces E and F , the norm of ϕ is denoted by |||ϕ|||F — or simply |||ϕ|||E if E = F .

1. Background Compact quantum metric spaces were introduced by Rieffel [23, 24], inspired by the work of Connes [4], as a noncommutative generalization of algebras of Lipschitz functions over compact metric space [33]. We added to this original approach the requirement that a form of the Leibniz inequality holds for the generalized Lipschitz seminorm in a quantum compact metric space; the upper bound of this inequality is given using a particular type of functions: Definition 1.1. A function F : [0, ∞]4 → [0, ∞] is admissible when: (1) F is weakly increasing if [0, ∞]4 is equipped with the product order, (2) for all x, y, lx, ly > 0 we have xly + ylx 6 F (x, y, lx, ly). The second condition in the above definition of an admissible function ensures that the usual Leibniz inequality is the “best” upper bound one might expect for our results to hold in general. A quasi-Leibniz quantum compact metric space is thus defined as follows: Definition 1.2 ([4, 23, 24, 16]). A F –quasi-Leibniz quantum compact metric space (A, L), where F is an admissible function, is an ordered pair of a unital C*-algebra A and a seminorm L defined on a dense Jordan-Lie subalgebra dom (L) of sa (A) such that:

(1) {a ∈ dom (L): L(a) = 0} = R1A, (2) the Monge-Kantorovich metric mkL deﬁned between any two states ϕ, ψ ∈ S (A) of A by:

mkL(ϕ, ψ) = sup {|ϕ(a) − ψ(a)| : a ∈ dom (L), L(a) 6 1} metrizes the weak* topology of S (A), (3) for all a, b ∈ dom (L) we have:

max {L (a ◦ b) , L ({a, b})} 6 F (kakA, kbkA, L(a), L(b)) ,

(4) L is lower semi-continuous with respect to k · kA. When (A, L) is a quasi-Leibniz quantum compact metric space, the seminorm L is called an L-seminorm. Notation 1.3. If (A, L) is a quasi-Leibniz quantum compact metric space, then (S (A), mkL) is a compact metric space, so it has ﬁnite diameter, which we denote by diam (A, L). 4 FRÉDÉRIC LATRÉMOLIÈRE

Example 1.4 (Classical Metric Spaces). Let (X, d) be a compact metric space, and for all f : X → R, deﬁne: |f(x) − f(y)| L(f) = sup : x, y ∈ X, x 6= y . d(x, y) The pair (C(X), L) of the unital C*-algebra of C-valued continuous functions on X and the seminorm L is a Leibniz quantum compact metric space. Example 1.5 (Noncommutative Examples). Many examples of noncommutative quasi-Leibniz quantum compact metric spaces are known to exist, including quantum tori [23, 25], curved quantum tori [11], AF algebras [1], noncommutative solenoids [13], hyperbolic groups C*-algebras [22], nilpotent group C*-algebras [3], among others. Quasi-Leibniz quantum compact metric spaces are objects for several natural categories. For this paper, we will use three diﬀerent notions of morphisms. We begin with the weakest of these notions, which we will use to state our main theorem at a hopefully reasonable level of generality.

Deﬁnition 1.6. Let (A, LA) and (B, LB) be two quasi-Leibniz quantum compact metric spaces. A continuous linear map ϕ : A → B is Lipschitz when:

∀a ∈ dom (LA) ϕ(a) ∈ dom (LB) and ϕ(1A) = 1B. Remark 1.7. A linear Lipschitz map, in particular, maps self-adjoint elements to self-adjoint elements, since it is continuous and maps the domain of an L-seminorm to self-adjoint elements — while domains of L-seminorms are dense in the self- adjoint part of C*-algebras. The terminology of Deﬁnition (1.6) is explained by the following corollary of our work in [19]:

Proposition 1.8. If (A, LA) and (B, LB) are quasi-Leibniz quantum compact metric spaces, and if ϕ : A → B is a linear map with ϕ(1A) = 1B, then the following are equivalent: (1) ϕ is Lipschitz, (2) there exists k > 0 such that LB ◦ ϕ 6 kLA. Proof. This result follows by [19, Theorem 2.1]. It is straightforward to check that the identity of any quasi-Leibniz quantum compact metric spaces is a Lipschitz linear map, and that the composition of Lip- schitz linear maps is itself Lipschitz, so we have deﬁned a category, as announced. Among all the Lipschitz linear functions, we ﬁnd the Lipschitz morphisms [19]:

Definition 1.9. Let (A, LA) and (B, LB) be quasi-Leibniz quantum compact metric spaces. A Lipschitz morphism ϕ :(A, LA) → (B, LB) is a unital *-morphism which is also Lipschitz linear. Using Lipschitz morphisms as arrows does form a category over the class of quasi-Leibniz quantum compact metric spaces, which is in fact the natural, default category for our theory. To be more explicit, one may argue that going from just linear maps to actual *-morphisms is the key reason to introduce the propinquity, SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 5 versus other versions of the noncommutative Gromov-Hausdorff distance [31,8, 27]: the propinquity was introduced specifically to allow us to work within the category of C*-algebras with their *-morphisms. Moreover, the dual maps induced by unital *-morphisms restricts to functions between state spaces, and these functions are Lipschitz for the Monge-Kantorovich metric if and only if the *-morphism is a Lipschitz morphism, thus completing the geometric picture. The only reason for our digression via the Lipschitz linear maps is to be able to phrase our main theorem at a slightly higher level of generality, as certain examples, such as semigroups of positive linear maps, have appeared in the literature and present potential examples for applications of our present work. There is a natural distance on the space of unital continuous linear maps between quasi-Leibniz quantum compact metric spaces, which generalizes the Monge- Kantorovich length of *-automorphisms discussed in [19]. We will use this distance for two purposes. It will quantify the idea that the composition of two Lipschitz linear maps almost equal to some other Lipschitz linear map — a tool used in the hypothesis of our main theorem — and it will also give us a non-triviality test, in the conclusion of our main theorem.

Deﬁnition 1.10. Let (A, LA) be a quasi-Leibniz quantum compact metric spaces and B be a unital C*-algebra. For any two linear maps ϕ, ψ from A to B which map 1A to 1B, we deﬁne:

LA mkDB (ϕ, ψ) = sup {kϕ(a) − ψ(a)kB : a ∈ dom (LA), LA(a) 6 1} .

LA In particular, if (A, LA) = (B, LB) then we write mk`LA (ϕ) for mkDA (id, ϕ) where id is the identity automorphism of A. In [19, Proposition 5.2], we proved that for any quasi-Leibniz quantum compact metric space (A, L), the function mk`L is a length function on the group of *- automorphisms of A which metrizes the pointwise convergence topology. This result and its proof extends readily to the setting of Deﬁnition (1.10), so we sketch the proof rather brieﬂy:

Proposition 1.11. Let (A, LA) be a quasi-Leibniz quantum compact metric spaces LA and B be a unital C*-algebra. The function mkDB (·, ·) deﬁnes a distance on the set of all continuous linear functions from A to B which map 1A to 1B. Proof. Let µ ∈ S (A). By deﬁnition of the Monge-Kantorovich metric, for all a ∈ dom (LA): ka − µ(a)1AkA 6 diam (A, LA)LA(a). Thus for any two continuous linear maps ϕ, ψ : A → B mapping 1A to 1B, and for all a ∈ dom (LA) with LA(a) 6 1:

kϕ(a) − ψ(a)kB = kϕ(a − µ(a)1A) − ψ(a − µ(a)1A)kB A A 6 diam (A, LA) |||ϕ|||B + |||ψ|||B .

LA Thus mkDB (·, ·) is bounded. The triangle inequality and the symmetry prop- LA erties of mkDB (·, ·) are obvious. LA If mkDB (ϕ, ψ) = 0 then ϕ and ψ agree on the total set {a ∈ sa (A): LA(a) 6 1}, LA and thus by linearity and continuity, they agree on A. Thus mkDB (·, ·) is a distance as claimed. 6 FRÉDÉRIC LATRÉMOLIÈRE

LA The proof that mkDB (·, ·) induces the topology of pointwise convergence follows the same argument as [19, Claim 5.4, Proposition 5.2], as the reader can easily checked (where pointwise convergence topology means the locally convex topology induced by the family of seminorms deﬁned over the space of linear maps from A to B by ϕ 7→ kϕ(a)kB where a ranges over all of A).

Remark 1.12. Deﬁnition (1.10) and Proposition (1.11) would make sense and hold if the codomain was some normed vector space, as long as we work on the set of continuous linear maps which map the unit to some ﬁxed vector. This level of generality is not however needed here.

Now, among all the Lipschitz morphisms, the quantum isometries form a sub- category, and are essential tools in the construction of the propinquity. Moreover, our chosen notion of isomorphism between quasi-Leibniz quantum compact metric spaces will be given by the full quantum isometries — an important observation to understand the geometry of the propinquity. The notion of a quantum isometry is largely due to Rieffel, and essentially expresses that McShane theorem [20] on extension of real-valued Lipschitz functions holds in our noncommutative context. This justifies, a posteriori, the emphasis on self-adjoint elements in Definition (1.2). Our modification of Rieffel’s notion consists of the requirement that quantum isometries be *-morphisms, and in particular, full quantum isometries be *-isomorphisms, rather than positive unital linear maps.

Deﬁnition 1.13. Let (A, LA) and (B, LB) be two quasi-Leibniz quantum compact metric spaces. A quantum isometry π :(A, LA) (B, LB) is a *-epimorphism π : A B such that for all b ∈ sa (B):

LB(b) = inf {LA(a): π(a) = b} .

A full quantum isometry π :(A, LA) → (B, LB) is a *-isomorphism π : A → B such that LB ◦ π = LA. A full quantum isometry is a quantum isometry which is a also a bijection and whose inverse map is a quantum isometry. Quantum isometries are in particular 1-Lipschitz morphisms between quasi-Leibniz quantum compact metric spaces [14] and they are morphisms for a category whose objects are quasi-Leibniz quantum compact metric spaces [31]. Our work focuses on the construction of geometries on the hyperspace of all quasi-Leibniz quantum compact metric spaces, or phrased a little more formally, on defining metrics on the class of all quasi-Leibniz quantum compact metric spaces, or some subclass therein. There are many subtleties involved in these constructions; our focus will be on the dual Gromov-Hausdorff propinquity as introduced in [12, 18]. The dual Gromov-Hausdorff propinquity is a complete distance, up to full quantum isometry, on the class of F –quasi-Leibniz quantum compact metric spaces, which induces the topology of the Gromov-Hausdorff distance [5,6] on the subclass of all classical metric spaces. Its construction is summarized as follows. We first generalize the idea of an isometric embedding of two quasi-Leibniz quantum compact metric spaces into a third one. SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 7

Deﬁnition 1.14 ([12]). Let (A, LA) and (B, LB) be two F –quasi-Leibniz quantum compact metric spaces. An F -tunnel τ = (D, LD, πA, πB) is given by a F –quasi- Leibniz quantum compact metric space (D, LD) as well as two quantum isometries πA :(D, LD) (A, LA) and πB :(D, LD) (B, LB). The domain of τ is (A, LA) and the codomain of τ is (B, LB). We then associate a numerical value to every tunnel. Notation 1.15. If π : A → B is a positive unital linear map from a unital C*- algebra A to a unital C*-algebra B, then the map ϕ ∈ S (B) → ϕ ◦ π ∈ S (A) is denoted by π∗. Notation 1.16. The Hausdorﬀ distance [7, p. 293] induced on the hyperspace of compact subsets of a metric space (X, d) is denoted by Hausd; if X is in fact a vector space and d is induced by some norm k · kX , then Hausd is denoted by Hausk·kX .

Definition 1.17 ([18]). The extent of a tunnel τ = (D, LD, π1, π2) is: n o max Haus (D), π∗( (A )) : j = 1, 2 . mkLD S j S j Now, the dual propinquity can in fact be “specialized”, by which we mean that we may consider only certain tunnels with additional properties. For instance, one might want to work with tunnels involving only strongly Leibniz L-seminorms [28]. The following definition summarizes the properties needed on a class of tunnels to allow for the definition of a version of the dual propinquity. The reader may skip this matter, except for the definition of the inverse of a tunnel in item (2), and simply assume that we will work with all F -tunnels for a given choice of an admissible function F in this paper; however, it is worth to point out that our main result applies equally well to any specialization of the propinquity. Definition 1.18 ([18]). Let F be an admissible function. A class T of tunnels is appropriate for a nonempty class C of F –quasi-Leibniz quantum compact metric spaces when the following properties hold. (1) If τ ∈ T then dom (τ), codom (τ) ∈ C, (2) If (A, LA) and (B, LB) are in C, then there exists τ ∈ T such that dom (τ) = (A, LA) and codom (τ) = (B, LB). −1 (3) If τ = (D, LB, π, ρ) ∈ T then τ = (D, LB, ρ, π) ∈ T . (4) If there exists a full quantum isometry h :(A, LA) → (B, LB) for any −1 (A, LA), (B, LB) ∈ C, then the tunnels (A, LA, idA, h) and (B, LB, h , idB) are in T , where idE is the identity automorphism of any C*-algebra E. (5) If τ, γ ∈ T with codom (τ) = dom (γ) them, for all ε > 0, there exists τ ◦ε γ ∈ T such that dom (τ) = dom (τ ◦ε γ), codom (γ) = codom (τ ◦ε γ) and:

χ (τ ◦ε γ) < χ (τ) + χ (γ) + ε. The most important example of an appropriate class of tunnels is given by: Proposition 1.19 ([18]). If F is an admissible function, then the class of all F - tunnels is appropriate for the class of all F –quasi-Leibniz quantum compact metric spaces. We now have all the ingredients to deﬁne our propinquity. 8 FRÉDÉRIC LATRÉMOLIÈRE

Notation 1.20. Let T be a class of tunnels appropriate for a nonempty class C of F –quasi-Leibniz quantum compact metric spaces for some admissible function F . The class of all tunnels in T from (A, LB) ∈ C to (B, LB) ∈ C is denoted by: h T i Tunnels (A, LA) −→ (B, LB) . Definition 1.21. Let T be a class of tunnels appropriate for a nonempty class C of F –quasi-Leibniz quantum compact metric spaces for some admissible function F . Let (A, LA) and (B, LB) in C. ∗ The dual T -propinquity ΛT ((A, LA), (B, LB)) between (A, LA) and (B, LB) is the real number: n h T io inf χ (τ): τ ∈ Tunnels (A, LA) −→ (B, LB) . Notation 1.22. Let F be some admissible function. If C is the class of all F – quasi-Leibniz quantum compact metric spaces and T is the class of all F tunnels. ∗ The dual T -propinquity is simply denoted by ΛF . If, moreover, F (x, y, z, w) = xw + yz for all x, y, z, w > 0, i.e. if we work within ∗ the class of Leibniz quantum compact metric spaces, then we denote ΛF simply as Λ∗. The main properties of our metric are summarized in the following: Theorem 1.23 ([10, 18]). Let F be an admissible function. If T is a class of tunnels admissible for a nonempty class C of F –quasi-Leibniz quantum compact ∗ metric spaces for some admissible function F , then the dual T -propinquity ΛT is a metric up to full quantum isometry on C. ∗ Moreover the dual propinquity ΛF is complete. Last, when restricted to the class of classical metric spaces, through the identification given in Example (1.4), the propinquity ΛF induces the same topology as the Gromov-Hausdorff distance [5,6] . Tunnels behave, in some ways, like morphisms, albeit their composition is only defined up to an arbitrary choice of a positive constant. We will use the morphism- like properties of tunnels in a fundamental way in this paper and so we now recall them. Lipschitz elements of a quasi-Leibniz quantum compact metric space have images by tunnels, although these images are themselves subsets of the codomain.

Deﬁnition 1.24. Let F be an admissible function, and let (A, LA) and (B, LB) be two F –quasi-Leibniz quantum compact metric spaces. Let τ = (D, LD, πA, πB) be an F -tunnel from (A, LA) to (B, LB). The l-target set of a ∈ dom (LA) for l > LA(a) is:   d ∈ sa (D)   tτ (a|l) = πB(d) ∈ sa (B) LD(d) l . 6  πA(d) = a  Target sets are a form of set-valued morphisms:

Proposition 1.25 ([12, 16]). Let F be an admissible function, and let (A, LA) and (B, LB) be F –quasi-Leibniz quantum compact metric spaces. Let τ be an F -tunnel 0 0 from (A, LA) to (B, LB). For all a, a ∈ dom (LA) and l > max{LA(a), LA(a )}, if 0 0 b ∈ tτ (a|l) and b ∈ tτ (a |l) then: (1) for all t ∈ R: 0 0 b + tb ∈ tτ (a + ta |(1 + |t|)l), SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 9

0 0 (2) if K(a, a , l, τ) = F (kakA + 2lχ (τ), ka kA + 2lχ (τ), l, l) then: 0 0 0 0 0 0 b ◦ b ∈ tτ (a ◦ a |K(a, a , l, τ)) and {b, b } ∈ tτ ({a, a }|K(a, a , l, τ)). The observation on which many proofs, including proofs in this paper, rely, is that target sets size is controlled by the extend of the tunnel.

Proposition 1.26 ([12]). Let F be an admissible function and let (A, LA) and (B, LB) be F –quasi-Leibniz quantum compact metric spaces. Let τ be an F -tunnel from (A, LA) to (B, LB). For all a ∈ dom (LA) and l > LA(a), the target set tτ (a|l) is a nonempty compact subset of sa (B), and moreover if d ∈ sa (D) is chosen so that πA(d) = a and LD(d) 6 l then: kdkD 6 kakA + lχ (τ), so in particular, if b ∈ tτ (a|l) then:

kbkB 6 kakA + 2lχ (τ). 0 0 As a corollary, if a, a ∈ dom (LA) and l > max{LA(a), LA(a )} then: 0 0 0 0 ∀b ∈ tτ (a|l)∀b ∈ tτ (a |l) kb − b kB 6 ka − a kA + 2lχ (τ) and therefore: diam (tτ (a|l), k · kB) 6 4lχ (τ). We conclude this section with a new lemma. A tunnel enables us to pick cor- responding ε-dense ﬁnite subsets of its domain and codomain. This lemma will prove helpful in checking that the morphisms which our main theorem builds will be non-trivial, under appropriate conditions.

Lemma 1.27. Let F be an admissible function. Let (A, LA) and (B, LB) be two F –quasi-Leibniz quantum compact metric spaces. Let ε > 0 and let τ be an F - tunnel from (A, LA) to (B, LB) such that χ (τ) < ε. Let E be a ﬁnite subset of {a ∈ sa (A): LA(a) 6 1} such that for all a ∈ sa (A) with LA(a) 6 1, there exists a0 ∈ E and t ∈ R such that: 0 ka − (a + t1A)kA < ε.

There exists a ﬁnite subset G ⊆ {a ∈ sa (B): LB(a) 6 1} such that: R 0 (1) for all a ∈ sa (B) with LB(a) 6 1, there exists t ∈ and a ∈ G such that 0 ka − (a + t1A)kB 6 3ε, 0 0 (2) for all a ∈ G there exists a ∈ E such that a ∈ tτ (a |1), and conversely, for 0 0 all a ∈ E there exists a ∈ G such that a ∈ tτ −1 (a |1).

Proof. We write τ = (D, LD, πA, πB). For all a ∈ E, choose f(a) ∈ tτ (a|1). Note that LB(f(a)) 6 1 by construction. Moreover, note that a ∈ tτ −1 (f(a)|1). Let G = {f(a): a ∈ E}. Now, let b ∈ sa (B) with LB(b) 6 1. Let a ∈ tτ −1 (b|1). 0 R 0 Since LA(a) 6 1, there exists a ∈ E and t ∈ such that ka − (a + t1A)kA < ε. 0 0 By Deﬁnition (1.24), there exists d ∈ dom (LD) with πA(d) = a , πB(d) = f(a ), 0 0 and LD(d) 6 1. Now, πA(d + t1D) = a + t1A, while πB(d + t1B) = f(a ) + t1B, and moreover: LD(d + t1D) = LD(d) 6 1. 0 0 Therefore f(a ) + t1B ∈ tτ (a + t1A|1), again by Deﬁnition (1.24). It follows by Proposition (1.26) that: 0 0 kb − (f(a ) + t1B)kB 6 ka − (a + t1A)kA + 2χ (τ) 6 3ε. 10 FRÉDÉRIC LATRÉMOLIÈRE

This concludes our proof. We now turn to our main theorem in the next section.

2. Actions of Semigroupoids on Limits We expand on the idea that tunnels are sort of set-valued morphisms by constructing a structure similar to a commutative diagram at the core of our main theorem. Thus, for the next few results, we will work with the following ingredients.

Hypothesis 2.1. Let F be an admissible function. Let (A, LA), (B, LB), (E, LE), and (F, LF) be F –quasi-Leibniz quantum compact metric spaces. Let τ be an F - tunnel from (A, LA) to (E, LE), and let γ be an F -tunnel from (F, LF) to (B, LB). Let ϕ : E → F be a Lipschitz linear map and let D > 0 such that LF ◦ ϕ 6 DLE E and |||ϕ|||F 6 D. As we look at tunnels as a form of morphism and target sets as the mean to get an image for elements by such morphisms, it is natural to deﬁne the image of a set by a tunnel by simply setting, for a tunnel τ from a quasi-Leibniz quantum compact metric space (A, L), and for any nonempty subset A ⊆ dom (L) and for any real l > 0: [ tτ (A|l) = tτ (a|l). a∈A This image of a set is not empty as long as there exists an element a ∈ A such that L(a) 6 l. For our purpose, the following speciﬁc application of this idea will be central.

Deﬁnition 2.2. Given Hypothesis (2.1), for a ∈ dom (LA) and l > LA(a), the l-image-target set of a is deﬁned by:

iϕ,τ (a|l) = ϕ (tτ (a|l)) and the l-forward-target set of a is deﬁned by:

fτ,ϕ,γ (a|l, D) = tγ (iϕ,τ (a|l)|Dl).

Remark 2.3. Using Hypothesis (2.1), we thus have, for a ∈ dom (LA) and l > LA(a): [ [ fτ,ϕ,γ (a|l, D) = tγ (c|Dl) = tγ (ϕ(b)|Dl).

c∈iϕ,τ (a|l) b∈tτ (a|l) We thus form a sort of commutative diagram, though using set-valued functions:

a ∈ sa (A) / fτ,ϕ,γ (a|l, D) ⊆ sa (B) O

tτ (·|l) tγ (·|Dl) tτ (a|l) ⊆ sa (E) ϕ / iτ,ϕ (a|l) ⊆ sa (F) 0 0 By Deﬁnition (2.2), if l > l and D > D, then with the notations of Hypothesis (2.1), we have: 0 0 fτ (a|l, D) ⊆ fτ (a|l ,D ). The forward target sets enjoy properties akin to target sets, which we now establish. We begin with the linearity property. SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 11

0 0 Lemma 2.4. Assume Hypothesis (2.1). If a, a ∈ dom (LA) and l > max{LA(a), LA(a )}, 0 0 then for all t ∈ R, f ∈ fτ,ϕ,γ (a|l, D) and f ∈ fτ,ϕ,γ (a |l, D): 0 0 f + tf ∈ fτ,ϕ,γ (a + ta |(1 + |t|)l, D). 0 0 Proof. Let f ∈ fτ,ϕ,γ (a|l, D) and f ∈ fτ,ϕ,γ (a |l, D). By Deﬁnition (2.2), there 0 0 0 0 exists b ∈ tτ (a|l) and b ∈ tτ (a |l) such that, if c = ϕ(b) and c = ϕ(b ), then 0 0 f ∈ tγ (c|Dl) and f ∈ tγ (c |Dl). 0 0 Now, b + tb ∈ tτ (a + ta |(1 + |t|)l) by Proposition (1.25). So: 0 0 0 c + tc = ϕ(b + tb ) ∈ ϕ (tτ (a + ta |(1 + |t|)l)) . 0 0 In particular, LF(c + tc ) 6 DLE(b + tb ) 6 Dl(1 + |t|). On the other hand, again by Proposition (1.25): 0 0 f + tf ∈ tγ (c + tc |(1 + |t|)Dl). 0 0 Thus f + tf ∈ fτ,ϕ,γ (a + ta |(1 + |t|)l, D). We now establish the main geometric property of forward-target sets — namely, their diameters in norm is controlled by the extend of the tunnels from which they are deﬁned.

0 0 Lemma 2.5. Assume Hypothesis (2.1). Let a, a ∈ dom (LA) and l > max{LA(a), LA(a )}. If c ∈ iτ,ϕ (a|l) then:

kckF 6 D (kakA + 2lχ (τ)) 0 0 and therefore for all c ∈ iτ,ϕ (a |l): 0 0 kc − c kF 6 D (ka − a kA + 4lχ (τ)) .

If moreover f ∈ fτ,ϕ,γ (a|l, D), then:

kfkB 6 D (kakA + 2l (χ (τ) + χ (γ))) 0 0 an therefore for all f ∈ fτ,ϕ,γ (a |l, D): 0 0 kf − f kB 6 D (ka − a kA + 4l (χ (τ) + χ (γ))) so that, in particular:

diam (fτ,ϕ,γ (a|l, D), k · kB) 6 4Dl (χ (τ) + χ (γ)) .

Proof. Let c ∈ iτ,ϕ (a|l). There exists, by Deﬁnition (2.2), an element b ∈ tτ (a|l) such that c = ϕ(b). Now by Proposition (1.26), we have:

(2.1) kckF 6 DkbkE 6 D(kakA + 2lχ (τ)), 0 0 0 0 0 0 as desired. If c ∈ iτ,ϕ (a |l) then c = ϕ(b ) with b ∈ tτ (a|l), and thus c − c = 0 0 0 ϕ(b − b ) with b − b ∈ tτ (a − a |2l) by Proposition (1.25). We conclude: 0 0 kc − c kF 6 D (ka − a kA + 4lχ (τ)) using Expression (2.1). Let now f ∈ fτ,ϕ,γ (a|l, D). By Deﬁnition (2.2), there exists c ∈ iτ,ϕ (a|l) such that f ∈ tγ (c|Dl). By Proposition (1.26) and Expression (2.1), we thus have:

(2.2) kfkB 6 kckF + 2Dlχ (γ) 6 D(kakA + 2l(χ (τ) + χ (γ))). 0 0 Let f ∈ fτ,ϕ,γ (a |l, D). By Lemma (2.4), we have: 0 0 f − f ∈ fτ,ϕ,γ (a − a |2l, D) 12 FRÉDÉRIC LATRÉMOLIÈRE and therefore, by Expression (2.2): 0 0 (2.3) kf − f kB 6 D (ka − a kA + 4l(χ (τ) + χ (γ))) as needed. Setting a = a0 in Expression (2.3), we then get:

diam (fτ,ϕ,γ (a|l, D), k · kB) 6 4Dl(χ (τ) + χ (γ)), thus concluding our lemma. We now complete the algebraic properties of the forward target sets by proving the following lemma.

0 0 Lemma 2.6. Assume Hypothesis (2.1). If a, a ∈ dom (LA) and l > max{LA(a), LA(a )}, 0 0 if f ∈ fτ,ϕ,γ (a|l, D) and f ∈ fτ,ϕ,γ (a |l, D), and if moreover ϕ is a Jordan-Lie morphism, then, setting: 0 0 Mτ,γ (kakA, ka kA, l, D) = max F (kakA + 2lχ (τ), ka kA + 2lχ (τ), l, l) ,

1 F (D (kak + 2lχ (τ) + 2lχ (γ)) ,D (ka0k + 2lχ (τ) + 2lχ (γ)) , Dl, Dl) D A A we conclude: 0 0 0 f ◦ f ∈ fτ,ϕ,γ (a ◦ a |Mτ,γ (kakA, ka kA, l, D),D) and 0 0 0 {f, f } ∈ fτ,ϕ,γ ({a, a }|Mτ,γ (kakA, ka kA, l, D),D). 0 0 Proof. By Deﬁnition (2.2), there exists b ∈ tτ (a|l) and b ∈ tτ (a |l) such that, if 0 0 0 0 c = ϕ(b) and c = ϕ(b ), then f ∈ tγ (c|Dl) and f ∈ fγ (a |Dl). Assume ϕ is a Lie-Jordan morphism. We then have by Proposition (1.25): 0 0 0 b ◦ b ∈ tτ (a ◦ a |F (kakA + 2lχ (τ), ka kA + 2lχ (τ), l, l)) and c ◦ c0 = ϕ(b) ◦ ϕ(b0) = ϕ(b ◦ b0) so: 0 0 0 LF (c ◦ c ) 6 DF (kakA + 2lχ (τ), ka kA + 2lχ (τ), l, l) 6 Mτ,γ (kakA, ka kA, l, D). We also have, again by Proposition (1.25): 0 0 0 f ◦ f ∈ tγ (c ◦ c |F (kckF + 2Dlχ (γ), kc kF + 2Dlχ (γ), Dl, Dl)) with: kckF 6 DkbkE 6 D(kakA + 2lχ (τ)) and 0 0 0 kc kF 6 Dkb kE 6 D(ka kA + 2lχ (τ)), by Lemma (2.5), so:

0 F (kckF + 2Dlχ (τ), kc kF + 2Dlχ (τ), Dl, Dl) 6 0 F (D(kakA + 2lχ (τ)) + 2Dlχ (τ),D(ka kA + 2lχ (τ)) + 2Dlχ (τ), Dl, Dl) 0 6 DMτ,γ (kakA, ka kA, l, D). 0 0 0 By Deﬁnition, we thus have: f ◦ f ∈ fτ,ϕ,γ (a ◦ a |Mτ,γ (kakA, ka kA, l, D),D). The same proof holds for the Lie product. Last, we identify the topological nature of forward target sets. SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 13

Lemma 2.7. Assume Hypothesis (2.1). For all a ∈ dom (LA) and l > LA(a), the sets iτ,ϕ (a|l) and fτ,ϕ,γ (a|l, D) are nonempty and compact, respectively, in sa (F) and sa (B).

Proof. By [12, Lemma 4.2], the set tτ (a|l) is a nonempty compact subset of sa (E). Since ϕ is continuous, iτ,ϕ (a|l) is not empty and compact in sa (F) as well. Let now b ∈ iτ (a|l). By construction, LD ◦ ϕ(b) 6 DLB(b) 6 Dl. Hence again by [12, Lemma 4.2], the set tγ (b|Dl) is a nonempty compact set. In particular, fτ,ϕ,γ (a|l, D) is not empty as well. Moreover, by Lemma (2.5), the following inclusion holds:

fτ,ϕ,γ (a|l, D) ⊆ {b ∈ sa (B): LB(b) 6 Dl, kbkA 6 D (kakA + 2l (χ (τ) + χ (γ)))} , and the set on the right hand-side is compact since L is an L-seminorm. So fτ (a|l, D) is totally bounded in sa (B). To prove that fτ,ϕ,γ (a|l, D) is compact, it is thus suﬃcient to show that is is closed, since it is totally bounded and sa (B) is complete. To this end, we write the tunnel γ as (D, LD, π, ρ). Let (fk)k∈N be a sequence in fτ,ϕ,γ (a|l, D), converging in B to some f. By Deﬁnition (2.2), for each k ∈ N, there exists ck ∈ ϕ (tτ (a|l)) such that fk ∈ tγ (ck|Dl). There exists bk ∈ tτ (a|l) such that ck = ϕ(bk). Now, tτ (a|l) is compact, so there exists a convergent subsequence (bj(k))k∈N converging in norm to some b ∈ tτ (a|l). Therefore, (ϕ(bj(k)))k∈N converges to ϕ(b), which we denote by c. Thus c ∈ iτ,ϕ (a|l). Now, for each k ∈ N, since fj(k) ∈ tγ cj(k) Dl , there exists dk ∈ sa (D) such that LD(dk) 6 Dl while π(dk) = cj(k) and ρ(dk) = fj(k). Once more, (dk)k∈N lies in the compact set:

{w ∈ sa (D): LD(w) 6 Dl, kwkD 6 D(kakA + 2l(χ (τ) + χ (γ)))} and thus we extract a convergent subsequence (dm(k))k∈N with limit d with LD(d) 6 Dl. Moreover by continuity:

lim ρ(dm(k)) = lim fj◦m(k) = f and lim π(dm(k)) = lim cj◦m(k) = c. k→∞ k→∞ k→∞ k→∞

By Definition, we thus conclude that f ∈ tγ (c|Dl) and therefore f ∈ fτ,ϕ,γ (a|l, D). Thus fτ,ϕ,γ (a|l, D) is closed. This concludes our lemma. We now provide the setting for our main theorem. We will prove not only that families of morphisms pass to the limit, informally speaking, for the propinquity, but also that composition of morphisms also passes to the limit. To encode composition, we will use the structure of semigroupoid. A semigroupoid is, informally, a category without the requirement that is contains identities: it is a set endowed with a partially defined associative operation. Formally: Definition 2.8. A semigroupoid (S, S(0), d, c, ◦) is a pair of classes S and S(0), two maps c, d : S → S(0) and a map ◦ : S(2) → S where: S(2) = (s, s0) ∈ S2 : d(s) = c(s0) , such that: (1) for all (s, s0) ∈ S(2), we have d(s ◦ s0) = d(s0) and c(s ◦ s0) = c(s), (2) if s, s0, s00 ∈ S with (s, s0), (s0, s00) ∈ S(2), then (s ◦ s0) ◦ s00 = s ◦ (s0 ◦ s00). 14 FRÉDÉRIC LATRÉMOLIÈRE

Categories are examples of semigroupoids, and a semigroupoid can be turned into a category by adding an element for each object which behaves as an identity: Deﬁnition 2.9. Let (S, S(0), d, c, ◦) be a semigroupoid. An element e ∈ S is a unit when d(e) = c(e) and s ◦ e = s, e ◦ t = t for all s, t ∈ S with d(s) = c(t) = d(e). Given these similarities, a morphism of semigroupoid will be called a functoid if it is not required to preserve units, and a functor if it must. Deﬁnition 2.10. A functoid F from a semigroupoid (S, O, d, c, ◦) to a semigroupoid (C, U, s, t, ) is a pair of maps (which by abuse of notations, we still write F ), one from O to U, and one from S to C, such that:

∀ϕ ∈ S F (d(ϕ)) = s(F (ϕ)) F (c(ϕ)) = t(F (ϕ)) and 0 (2) 0 0 ∀(ϕ, ϕ ) ∈ S F (ϕ ◦ ϕ ) = F (ϕ)F (ϕ ). If moreover, for any unit u ∈ S, the element F (u) is a unit of C, then F is called a functor. We note that Definition (2.10) is meaningful, as we easily check, using the notations of this definition, that if (s, s0) ∈ S(2) then (F (s),F (s0)) ∈ C(2). An important example of semigroupoid is a groupoid, which is a semigroupoid for which all elements are invertible: Definition 2.11. Let (S, S(0), d, c, ◦) be a semigroupoid. An element u ∈ S is invertible when there exists u0 ∈ S, such that: (1) d(u) = c(u0) and d(u0) = c(u), (2) u ◦ u0 is a unit, (3) u0 ◦ u is a unit. The element u0 is called an inverse of u. We note that a trivial argument shows that units in a semigroupoid with a given domain are unique, and that the inverse of an element of a semigroupoid is unique as well. Functors preserve inverse while functoids do not in general. We have now set the framework for our main theorem. We will use repeatedly in its proof the following simple result, which we record as a lemma to help clarify our main argument.

Lemma 2.12. If (An)n∈N and (Bn)n∈N are two sequences of compact subsets of a complete metric space (X, d) such that:

(1) limn→∞ diam (Bn, d) = 0, (2) for all n ∈ N, we have An ⊆ Bn, then the sequence (An)n∈N converges for Hausd if and only if (Bn)n∈N does, in which case they have the same limit, a singleton of X.

Proof. We begin by proving that if a sequence (Cn)n∈N of closed subsets of X with limn→∞ diam (Cn, d) = 0 converges for Hausd then its limit C is a singleton. Since diam (·, d) is a continuous function for Hausd, it is immediate that diam (C, d) = 0. On the other hand, for each n ∈ N, let cn ∈ Cn. Let ε > 0. There exists N ∈ N ε ε such that for all p, q > N, we have Hausd(Cp,Cq) < 3 and diam (Cn, d) < 3 , so: d(cp, cq) 6 diam (Cp, d) + Hausd(Cp,Cq) + diam (Cq, d) < ε, SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 15 and thus (cn)n∈N is a Cauchy sequence; since (X, d) is complete, (cn)n∈N converges to some c ∈ X. Moreover, d(c, Cn) 6 ε for n > N by continuity. On the other 4ε 4ε hand, if x ∈ Cn then d(x, c) < 3 for n > N. Hence Hausd(Cn, {c}) < 3 for n > N and thus C = {c}. We now prove the rest of our lemma. Let ε > 0. There exists N ∈ N such that for all n > N: ε diam (B , d) < . n 2 N ε Since An ⊆ Bn for all n ∈ , we conclude that Hausd(An,Bn) < 2 for all n > N. Now, if (An)n∈N converges for Hausd, and {a} is its limit, then there exists M ∈ N such that for all n > M, we have: ε Haus (A , {a}) < . d n 2 Therefore, for all n > max{N,M}: Hausd(Bn, {a}) 6 Hausd(Bn,An) + Hausd(An, {a}) < ε, and thus (Bn)n∈N converges for Hausd to the same limit {a}. If instead, (Bn)n∈N converges to {b} for Hausd, then similarly, there exists M ∈ N ε such that Hausd(Bn, {b}) < 2 and thus for all n > max{N,M}, we conclude:

Hausd(An, {b}) < ε thus concluding our lemma. We now state and prove our main theorem. Theorem 2.13. Let F be an admissible function. Let T be a class of tunnels appropriate for a nonempty class C of F –quasi-Leibniz quantum compact metric spaces. Let S, S(0), d, c, ◦ be a semigroupoid, where S and S(0) are both countable sets. (0) s s s s For all s ∈ S , let (A , L ) ∈ C, and let (An, Ln)n∈N be a sequence in C such that: ∗ s s s s lim ΛT ((An, Ln), (A , L )) = 0. n→∞ Let D : S → [0, ∞). If, for all s ∈ S and for all n ∈ N, there exists a Lipschitz linear map: s d(s) d(s) c(s) c(s) ϕn :(An , Ln ) → (An , Ln ) with the property: d(s) s An c(s) s d(s) |||ϕn||| c(s) 6 D(s) and Ln ◦ ϕn 6 D(s)Ln , An while for all (s, t) ∈ S(2):

d(t) Ln s t s◦t lim mkD c(s) ϕn ◦ ϕn, ϕn = 0, n→∞ A then there exists a functoid Ψ from S to the category whose objects are the elements of C, and whose morphisms are the Lipschitz linear maps, such that: ∀s ∈ S(0) Ψ(s) = (As, Ls) and d(s) √ A c(s) d(s) ∀s ∈ S |||Ψ(s)|||Ac(s) 6 2D(s) and L ◦ Ψ(s) 6 D(s)L . Furthermore: 16 FRÉDÉRIC LATRÉMOLIÈRE

N s (1) If, for some s ∈ S and for all n ∈ , the map ϕn is positive, and {a ∈ d(s) d(s) dom L : a > 0} is dense in {a ∈ sa A : a > 0}, then Ψ(s) is positive as well. s s (2) If, for some s ∈ S, we can choose (ϕn)n∈N such that ϕn is a Lipschitz morphism for all n ∈ N, then Ψ(s) is a Lipschitz morphism — in particular, Ad(s) |||Ψ(s)|||Ac(s) = 1. 0 0 0 Ld(s) 0 (3) If, for some s, s ∈ S, we have c(s) = c(s ) and d(s) = d(s ), then mkDAc(s) (Ψ(s), Ψ(s )) d(s) 0 Ln s s is a limit point of the sequence mkD c(s) ϕn, ϕn , and so in par- An n∈N ticular:

d(s) 0 d(s) 0 d(s) 0 Ln s s L s s Ln s s lim inf mkD c(s) ϕn, ϕn 6 mkDAc(s) ϕ , ϕ 6 lim sup mkD c(s) ϕn, ϕn . n→∞ An n→∞ An Moreover, if for some s ∈ S, we have d(s) = c(s), then:

s s lim inf mk`Ld(s) (ϕn) 6 mk`Ld(s) (Ψ(s)) 6 lim sup mk`Ld(s) (ϕn). n→∞ n n→∞ n u In particular, if u ∈ S with c(u) = d(u) and if lim mk` u (ϕ ) = 0, n→∞ Ln then Ψ(u) is the identity of Ad(u). (4) If for any unit u ∈ S we have:

u lim mk`Lu (ϕ ) = 0 n→∞ n then Ψ is a functor. (5) If some s ∈ S is invertible, with inverse s0, and if for all n ∈ N, we have:

s◦s s0◦s lim mk` c(s) ϕn = 0 and lim mk` d(s) ϕn = 0, n→∞ Ln n→∞ Ln then Ψ(s) is a bijection from Ad(s) to Ac(s) with inverse Ψ(s0).

Proof. We will employ all the notations given in the assumption of our theorem. Up to extracting subsequences, a standard diagonal argument shows that since S(0) is countable, we may as well assume that: 1 ∀s ∈ S(0) ∀n ∈ N Λ∗ ((As , Ls ), (As, Ls)) . T n n 6 n + 2 (0) N s s s s s For each s ∈ S , n ∈ , let τn ∈ T be a tunnel from (A , L ) to (An, Ln) with s 1 s −1 χ (τn) 6 n+1 . Let γn ∈ T be the inverse tunnel (τn) . For any s ∈ S(0), let Ss be a countable, dense subset of sa (As) with Ss ⊆ dom (Ls). For each s ∈ S, n ∈ N, we denote the forward-image set (see Deﬁnition (2.2)):

f d(s) s c(s) (·|·) τn ,ϕn,γn simply as fs,n (·|·). We now prove our theorem in several steps. The ﬁrst few steps prove the existence of a linear map which will eventually be denoted by Ψ(s) for each s ∈ S. The construction is of intrinsic interest, as it may be possible to use it to extract additional information on these maps; some of the later steps will be examples of this idea. SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 17

Step 1. Let s ∈ S. Let j0 : N → N be a strictly increasing function. Let a ∈ d(s) c(s) dom L . There exists j1 : N → N, strictly increasing, and f ∈ sa A , such that: c(s) d(s) L (f) 6 D(s)L (a) and kfkAc(s) 6 D(s)kakAd(s) , d(s) N and for all l > L (a), the sequence (fs,j0◦j1(n) (a|l, D(s)))n∈ converges, for Hausk·k , to the singleton {f}. Ac(s) Write D = D(s) for this step. Let a ∈ dom Ld(s). By Lemmas (2.5) and (2.7), the sequence: d(s) fs,j0(n) a L (a),D n∈N is a sequence of nonempty compact subsets of:

n c(s) c(s) d(s) d(s) o b ∈ sa A : L (b) 6 DL (a) and kbkAc(s) 6 D(kakAd(s) + 2L (a)) which is itself compact in sa Ac(s) since Lc(s) is an L-seminorm. Thus, there exists a strictly increasing function j such that f a Ld(s)(a),D 1 s,j0◦j1(n) n∈N converges for the Hausdorﬀ distance Hausk·k , since the Hausdorﬀ distance Ac(s) induces a compact topology on the hyperspace of closed subsets of a compact metric spaces (Blaschke’s theorem [2, Theorem 7.2.8]). Let f(a) be the limit of d(s) fs,j ◦j (n) a L (a),D for Hausk·k . By Lemma (2.5), we have: 0 1 n∈N Ac(s) d(s) lim diam fs,j ◦j (n) a L (a),D , k · kAc(s) = 0 n→∞ 0 1 and thus f(a) is a singleton, as desired. We write f(a) = {f}. In particular, we thus note that f is the limit of any sequence (bn)n∈N with bn ∈ d(s) f a L (a),D for all n ∈ N. Thus kfk c(s) = lim kb k c(s) s,j0◦j1(n) A n→∞ n A 6 j0◦j1(n) d(s) DkakAd(s) ; moreover since L is lower semicontinuous, we also have: c(s) c(s) d(s) L (f) lim inf L (bn) DL (a). 6 n→∞ j0◦j1(n) 6 d(s) To conclude this step, let l > L (a). Since for all n ∈ N, we have: d(s) fs,n a L (a),D ⊆ fs,n (a|l, D), so by Lemma (2.12), since by Lemma (2.5) we have: lim diam fs,n (a|l, D), k · k c(s) = 0, n→∞ An we conclude that: f (a|l, D) s,j0◦j1(n) n∈N converges to {f} as well. Step 2. Let s ∈ S and J : N → N be a strictly increasing function. There exists a strictly increasing function j : N → N such that, for all a ∈ Sd(s), there exists ϕs(a) ∈ sa Ac(s), with: s c(s) s d(s) kϕ (a)kAc(s) 6 D(s)kakAd(s) and L ◦ ϕ (a) 6 D(s)L (a), d(s) such that for all l > L (a), the sequence: f s,J◦j(n) (a|l, D(s)) n∈N s converges for Hausk·k to {ϕ (a)}. Ac(s) 18 FRÉDÉRIC LATRÉMOLIÈRE

We now apply Step (1) repeatedly, using a diagonal argument. Fix s ∈ S and d(s) again, just write D = D(s). Write S = {an : n ∈ N}. By Step (1), there s c(s) exists j0 : N → N strictly increasing and ϕ (a0) ∈ sa A , such that for all l Ld(s)(a ), the sequence f (a |l, D) converges to {ϕs(a )} for > 0 s,J◦j0(n) 0 n∈N 0 s c(s) s d(s) Hausk·k , and kϕ (a0)k c(s) Dka0k d(s) while L ◦ ϕ (a0) DL (a0). Ac(s) A 6 A 6 Assume now that for some k ∈ N, there exist strictly increasing functions s s c(s) j0, . . . , jk from N to N and ϕ (a0), . . . , ϕ (ak) ∈ sa A such that for m ∈ d(s) {0, . . . , k} and for all l > L (aj), the sequence: f (a |l, D) s,J◦j0◦···◦jm(n) j n∈N s converges to the singleton {ϕ (aj)} for Hausk·k , with kϕ(aj)k c(s) Dkajk d(s) Ac(s) A 6 A c(s) s d(s) and L ◦ ϕ (aj) 6 DL (aj). By Step (1), there exists a strictly increasing function jk+1 : N → N such that, for all l Ld(s)(a ), the sequence f (a |l, D) converges, > k+1 s,J◦j0◦···◦jk+1(n) k+1 n∈N s s for Hausk·k , to some singleton denoted by {ϕ (ak+1)} with kϕ (ak+1)k c(s) Ac(s) A 6 c(s) s d(s) Dkak+1kAd(s) and L ◦ ϕ (ak+1) 6 DL (ak+1). This completes our induction. Now, let j : m ∈ N 7−→ J ◦ j0 ◦ · · · ◦ jm(m). By construction, for all k ∈ N and d(s) f s for all l > L (ak), the sequence s,j(n) (ak|l, D) n∈N converges to {ϕ (ak)} for Hausk·k . Ac(s) Step 3. There exists a strictly increasing function j : N → N such that, for all s ∈ S and for all a ∈ Sd(s), there exists ϕs(a) ∈ sa Ac(s), with:

s c(s) s d(s) kϕ (a)kAc(s) 6 D(s)kakAd(s) and L ◦ ϕ (a) 6 D(s)L (a), d(s) such that for all l > L (a), the sequence: f s,j(n) (a|l, D(s)) n∈N s converges for Hausk·k to {ϕ (a)}. Ac(s)

Since S is countable, we write it as S = {sm : m ∈ N} for this step. We now s0 apply Step (2) repeatedly. First, let j0 : N → N and ϕ be given by Step (2) for s0 and J : n ∈ N 7→ n. Assume we have constructed, for some k ∈ N, strictly increasing maps j0,. . . ,jk from N to N, and functions ϕs0 : Sd(s0) → sa Ac(s0),. . . ,ϕsk : Sd(sk) → sa Ac(sk) d(s ) d(s ) such that for all m ∈ {0, . . . , k}, a ∈ S m , and l > L m (a), we have:

c(sm) sm d(s) sm L ◦ ϕ (a) 6 D(sm)L (a) and kϕ (a)kAc(m) 6 D(sm)kakAd(sm) ,

sm and the sequence fs ,j ◦...◦j (n) (a|l, D(sm)) converges to {ϕ (a)} for Hausk·k . m 0 m n∈N Ac(s) We apply Step (2) with J : n ∈ N → j0 ◦ ... ◦ jk and sk+1: thus there exists s d(k+1) c(k+1) jk+1 : N → N strictly increasing and ϕ k+1 : S → sa A satisfying the conclusions of Step (2). This completes our induction. We conclude this step by setting j : m ∈ N 7−→ j0 ◦ · · · ◦ jm(m). Step 4. There exists a strictly increasing function j : N → N and, for all s ∈ S, a function ϕs : dom Ld(s) → sa Ac(s) such that for all a ∈ dom Ld(s) and for all d(s) l > L (a), and D > D(s), the sequence: f s,j(n) (a|l, D) n∈N SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 19

s converges to {ϕ (a)} for Hausk·k ; moreover: Ac(s) s c(s) s d(s) kϕ (a)kAc(s) 6 D(s)kakAd(s) and L (ϕ (a)) 6 D(s)L (a). Let j and, for all s ∈ S, let ϕs : Sd(s) → sa Ac(s) be constructed by Step (3). Fix s ∈ S and write D > D(s). d(s) 0 d(s) 0 Let a ∈ dom L , and ε > 0. There exists a ∈ S such that ka−a kAd(s) < ε d(s) d(s) 0 4D . Let l = max{L (a), L (a )}. Since by Lemma (2.5): 0 lim diam (fs,n (a |l, D), k · kAc(s) ) = 0 n→∞ 0 and since fs,j(n) (a |l, D(s)) converges to a singleton for Hausk·k , by Lemma n∈N Ac(s) f 0 (2.12), the sequence s,j(n) (a |l, D) n∈N also converges to the same singleton. 0 Since fs,j(n) (a |l, D) is convergent for the Hausdorﬀ distance Hausk·k n∈N Ac(s) by Step (3), it is Cauchy for Hausk·k . Let N ∈ N such that, for all p, q N, Ac(s) > we have: 0 0 ε Hausk·k fs,j(p) (a |l, D), fs,j(q) (a |l, D) < . Ac(s) 3 d(s) d(s) 0 By Lemma (2.5), we have, for all l > max{L (a), L (a )}, for all n ∈ N, for 0 0 all b ∈ fs,j(n) (a|l, D) and for all b ∈ fs,j(n) (a |l, D):

0 0 1 1 ε 4l kb − b k D ka − a k d(s) + 2l + + . Ac(s) 6 A j(n) + 1 j(n) + 1 6 4 j(n) + 1 N 4l ε Let M ∈ such that for all n > M, we have j(n)+1 6 12 , so that: 0 ε Hausk·k c(s) fs,j(n) (a|l, D), fs,j(n) (a |l, D) 6 . A 3 Hence for all p, q > max{N,M}: Hausk·k fs,j(p) (a|l, D), fs,j(q) (a|l, D) Ac(s) 0 Hausk·k fs,j(p) (a|l, D), fs,j(p) (a |l, D) 6 Ac(s) 0 0 + Hausk·k fs,j(p) (a |l, D), fs,j(q) (a |l, D) Ac(s) 0 + Hausk·k fs,j(q) (a |l, D), fj(q) (a|l, D) Ac(s) < ε. Hence f (a|l, D) is a Cauchy sequence for Haus . Since (sa Ac(s), k· s,j(n) n∈N k·k c(s) A k c(s) ) is complete, so is Haus and thus f (a|l, D) converges for A k·k c(s) s,j(n) n∈N A Hausk·k to some set f(a); again since limn→∞ diam fs,j(n) (a|l, D), k · k c(s) = Ac(s) A 0, the set f(a) is a singleton which we denote by {ϕs(a)}. We observe that if in fact a ∈ Sd(s), then we introduce no confusion with this notation since by construction, f s s,j(n) (a|l, D) n∈N converges to {ϕ (a)} as deﬁned by Step (3). 00 0 Using Lemma (2.12) and the observation that for all n ∈ N, and for all l > l > Ld(s)(a), we have: 0 00 fs,j(n) (a|l ,D) ⊆ fs,j(n) (a|l ,D) d(s) we conclude that for all l > L (a), the sequence: f s,j(n) (a|l, D) n∈N s converges for Hausk·k to {ϕ (a)}. Ac(s) 20 FRÉDÉRIC LATRÉMOLIÈRE

d(s) f d(s) In particular, if a ∈ dom L , then s,j(n) a L (a),D(s) n∈N converges to {ϕs(a)}, and thus as before, since Lc(s) is lower semi-continuous, we conclude c(s) s d(s) d(s) that L ◦ ϕ (a) 6 D(s)L (a) for all a ∈ dom L ; Similarly kϕ(a)kAc(s) 6 D(s)kakAd(s) . Step 5. For all s ∈ S, the map ϕs : dom Ld(s) → dom Lc(s) is linear and s ϕ (1Ad(s) ) = 1Ac(s) . 0 d(s) Fix s ∈ S and write D = D(s). Let a, a ∈ dom L , t ∈ R, and l > d(s) d(s) 0 N 0 0 max{L (a), L (a )}. For each n ∈ , let bn ∈ fs,j(n) (a|l, D) and bn ∈ fs,j(n) (a |l, D). s 0 s 0 By construction, (bn)n∈N converges to ϕ (a) while (bn)n∈N converges to ϕ (a ) in sa Ac(s). On the other hand, by Lemma (2.4): N 0 0 ∀n ∈ bn + tbn ∈ fs,j(n) (a + ta |(1 + |t|)l, D(s)), 0 s 0 c(s) so (bn + tbn)n∈N converges to ϕ (a + ta ) in sa A . By uniqueness of the limit, we conclude: ϕs(a + ta0) = ϕs(a) + tϕs(a0). s We also note that since ϕn maps unit to unit, we have 1Ac(s) ∈ fs,n (1Ad(s) |0,D) N s for all n ∈ . Thus ϕ (1d(s)) = 1Ac(s) . Step 6. For all s ∈ S, the map ϕs has a unique extension (still denoted by ϕs) as a continuous linear map from Ad(s) to Ac(s) with: sa(Ad(s)) • |||ϕs||| D(s), sa(Ac(s)) 6 d(s) √ s A • |||ϕ |||Ac(s) 6 2D(s), c(s) s d(s) • L ◦ ϕ 6 D(s)L . •∀ a ∈ Ad(s) ϕs(a∗) = ϕs(a)∗.

s d(s) c(s) c(s) s Fix s ∈ S. The map ϕ : dom L → dom L satisﬁes L ◦ ϕ 6 D(s)Ld(s) by Step (4). d(s) s For all a ∈ dom L , we have kϕ (a)kAc(s) 6 D(s)kakAd(s) by construction, so ϕs : dom Ld(s) → sa Ac(s) is a continuous linear map. Hence it is uniformly continuous on the dense subspace dom Ld(s) of sa Ad(s) and therefore, it has a unique uniformly continuous extension as a linear map from sa Ad(s) to sa Ac(s) with norm at most D(s). As a side note, if Ld(s)(a) = ∞ (equivalently if a ∈ d(s) d(s) c(s) s d(s) sa A \ dom L ) then the inequality L ◦ ϕ (a) 6 D(s)L (a) = ∞ is trivial. For all a ∈ Ad(s), we then set: ϕs(a) = ϕs (

2 2 6 2D(s) kakAc(s) . Last, by construction, ϕs(a∗) = ϕs(

0 0 Step 7. For all s, s0 ∈ S, if d(s) = c(s0), then ϕs ◦ ϕs = ϕs◦s .

0 0 Let s, s0 ∈ S such that d(s) = c(s0). By assumption, (Ad(s), Ld(s)) = (Ac(s ), Lc(s )), so the statement of this step is at least meaningful. We also note that it is obviously 0 0 suﬃcient to prove that ϕs ◦ ϕs (a) = ϕs◦s (a) for all a ∈ sa Ad(s). Moreover, by assumption:

0 0 0 N c(s ) c(s ) s s d(s) d(s) ∀n ∈ (An , Ln ) = codom ϕn = dom (ϕn) = (An , Ln ).

d(s0) d(s0) Let a ∈ sa A and write l = L (a). For all n ∈ N, let an ∈ t d(s0) (a|l), τj(n) 0 s0 c(s ) d(s) and write bn = ϕj(n)(an). Note that bn ∈ Aj(n) = Aj(n). 0 0 c(s0) Let cn ∈ t c(s0) (bn|D(s )l) — in particular, cn ∈ fs0,j(n) (a|l, D(s )) ⊆ A . γj(n) 0 s◦s 0 In addition, let en ∈ t c(s) ϕj(n)(an) D(s ◦ s )l , so that by Deﬁnition (2.2), we γj(n) 0 c(s) have en ∈ fs◦s0,j(n) (a|l, D(s ◦ s )) (so in particular, en ∈ A ). By construction, s◦s0 c(s) s0 d(s) (en)n∈N converges to ϕ (a) in A , while (cn)n∈N converges to ϕ (a) in A . s 0 c(s) Let dn ∈ t c(s) ϕj(n)(bn) D(s)D(s )l — so dn ∈ A . Now, since we chose γj(n) −1 0 d(s) d(s) cn ∈ t d(s) (bn|D(s )l), and since γj(n) = τj(n) , we also have by symmetry bn ∈ γj(n) 0 0 t d(s) (cn|D(s )l). So by Deﬁnition (2.2), we conclude that dn ∈ fs,j(n) (cn|D(s )l, D(s)). τj(n) 0 s 0 Let hn ∈ fs,j(n) ϕ (a) D(s )l, D(s) . By construction, (hn)n∈N converges to 0 ϕs ◦ ϕs (a) in Ac(s). On the other hand, by Lemma (2.5):

s0 0 s kdn − hnkAc(s) 6 D(s) kcn − ϕ (a)kAd(s) + 4lD(s )χ τj(n) so

0 6 lim sup kdn − hnkAc(s) n→∞ s0 0 s D(s) lim kcn − ϕ (a)k c(s0) + 4lD(s ) lim χ τ = 0 6 n→∞ A n→∞ j(n)

s s0 so (dn)n∈N converges to ϕ ◦ ϕ (a) as well. Let ε > 0 and let D = max{D(s)D(s0),D(s ◦ s0)}. N s s0 ε Let N1 ∈ such that for all n > N1, we have kdn − ϕ ◦ ϕ (a)kAc(s) < 4 . N s◦s0 ε Let N2 ∈ such that for all n > N2, we have ken − ϕ (a)kAc(s) < . 4 N s ε Let N3 ∈ such that for all n > N3, we have χ τj(n) < 4Dl . N Last, by assumption, there exists N4 ∈ such that for all n > N4 and for all 0 d(s0) d(s0) 0 a ∈ sa A with L (a ) 6 l, we have:

0 0 ε ϕs◦s (a0) − ϕs ◦ ϕs (a0) < . j(n) j(n) j(n) c(s) Aj(n) 4 22 FRÉDÉRIC LATRÉMOLIÈRE

We record that by assumption, we have: s 0 s dn ∈ t c(s) ϕj(n)(bn) D(s)D(s )l ⊆ t c(s) ϕj(n)(bn) Dl γj(n) γj(n) and 0 0 s◦s 0 s◦s en ∈ t c(s) ϕj(n)(a) D(s ◦ s )l ⊆ t c(s) ϕj(n)(a) Dl . γj(n) γj(n) For all n > max{N1,N2,N3,N4}, we thus have: s◦s0 s s0 0 6 kϕ (a) − ϕ ◦ ϕ (a)kAc(s) s◦s0 s s0 6 kϕ (a) − enkAc(s) + ken − dnkAc(s) + kdn − ϕ ◦ ϕ (a)kAc(s) 2ε + ke − d k c(s) 6 4 n n A 2ε s◦s0 s c(s) 6 + kϕj(n)(an) − ϕj(n)(bn)kAc(s) + 2lDχ τj(n) by Prop (1.26) 4 j(n) 3ε s◦s0 s s0 6 + kϕj(n)(an) − ϕj(n)(ϕj(n)(an))kAc(s) 4 j(n) 6 ε. Hence, as ε > 0 is arbitrary, we conclude:

0 0 0 0 ϕs◦s (a) − ϕs ◦ ϕs (a) = 0 so ϕs◦s (a) = ϕs ◦ ϕs (a). Ac(s) This concludes this step. Summary 2.14. By setting Ψ(s) = (As, Ls) for all s ∈ S(0), and Ψ(s) = ϕs for all s ∈ S, we have proven that Ψ is a functoid from (S, S(0), d, c, ◦) to the category of F –quasi-Leibniz quantum compact metric spaces whose arrows are Lipschitz linear maps. We now turn to various additional properties for our maps ϕs (s ∈ S) under more stringent conditions. We begin with positivity. N s Step 8. If, for some s ∈ S and for all n ∈ , the map ϕn is positive, and d(s) d(s) s {a ∈ dom L : a > 0} is dense in {a ∈ sa A : a > 0}, then ϕ is positive as well. d(s) d(s) Let a ∈ dom L with a > 0 and write l = L (a); we also write D = D(s). c(s) N Let µ ∈ S A . Let ε > 0. There exists N1 ∈ such that, for all n > N, we c(s) s s can ﬁnd fn ∈ A such that kfn − ϕ (a)kAc(s) < ε, with fn ∈ t c(s) (ϕn(bn)|Dl), γj(n) where bn ∈ t d(s) (a|l). τj(n) N There exists N2 ∈ such that for all n > N2 we have: ε ε χ τ d(s) < and χ γc(s) < . j(n) 3(l + 1) j(n) 3(Dl + 1)

Let N = max{N1,N2} and n > N. We will need, for this step only, a notation for the tunnels involved in the d(s) d(s) d(s) d(s) d(s) c(s) following computation. We write τj(n) = (Dn , Rn , πn , ρn ) and γj(n) = c(s) c(s) c(s) c(s) (Dn , Rn , ρn , πn ). c(s) By Deﬁnition (1.17) and [18, Proposition 2.12], there exists ηn ∈ S Aj(n) ε s such that mk c(s) (µ, ηn) < . Since ϕ is positive and unital, the map Rn 3(Dl+1) j(n) SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 23

s d(s) µn = ηn ◦ ϕn is a state of Aj(n). Again by Deﬁnition (1.17), there exists νn ∈ d(s) ε S (A ) with mk d(s) (νn, µn) < . Rn 3(l+1) d(s) Moreover, by Deﬁnition (1.24), since bn ∈ t d(s) (a|l), there exists dn ∈ Dn τj(n) d(s) d(s) d(s) with Rn (dn) 6 l such that πn (dn) = a and ρn (dn) = bn. There also exists 0 c(s) c(s) 0 c(s) 0 d(s) 0 s dn ∈ Dn such that Rn (dn) 6 Dl, πn (dn) = fn and ρn (dn) = ϕn(bn). We then compute: (2.4) s s |µ(ϕ (a)) − νn(a)| 6 |µ(ϕ (a)) − µ(fn)| + |µ(fn) − νn(a)| s 6 kϕ (a) − fnkA + |µ(fn) − νn(a)| ε + |µ(f ) − ν (a)| 6 3 n n ε + |µ(f ) − µ (b )| + |µ (b ) − ν (a)| 6 3 n n n n n n ε + |µ(f ) − η (ϕs (b ))| + |µ ◦ ρd(s)(d ) − ν ◦ πd(s)(d )| 6 3 n n n n n n n n n n ε c(s) 0 c(s) 0 6 + |µ ◦ πn (dn) − ηn ◦ ρn (dn)| + lmk d(s) (νn, µn) 3 Rn ε 6 + Dlmk c(s) (µ, ηn) + lmk d(s) (νn, µn) 3 Rn Rn ε Dlε lε + + = ε. 6 3 3Dl 3l Now, since a > 0, for any n > N, we have νn(a) > 0. As the limit of the sequence of nonnegative numbers (νn(a))n∈N by Expression (2.4), we conclude that s c(s) s µ(ϕ (a)) > 0. As µ ∈ S (A ) was an arbitrary state, we conclude that ϕ (a) > 0. By continuity of ϕs, since the space of positive elements in Ac(s) is closed in norm, and by assumption that any positive element of Ad(s) is the limit of positive elements in dom Ld(s), we conclude that ϕs is a positive map, as claimed. N s Step 9. If, for some s ∈ S and for all n ∈ , the map ϕn is a unital *-morphism, A then ϕ is a unital *-morphism; in particular |||ϕ|||B = 1. N s Fix s ∈ S such that for all n ∈ , the map ϕn is a unital *-morphism. For this step, set D = D(s). s We begin by proving that ϕ , restricted to dom (LA), is a Jordan-Lie morphism. 0 d(s) d(s) d(s) 0 Let a, a ∈ dom L and l > max{L (a), L (a )}. For each n ∈ N, let 0 0 0 bn ∈ fj(n) (a |l, D) and bn ∈ fj(n) (a |l, D). By construction, (bn)n∈N converges to s 0 s 0 ϕ (a) while (bn)n∈N converges to ϕ (a ). On the other hand, by Lemma (2.6) (in particular, using the notations therein), for all n ∈ N, we have: 0 0 0 bn ◦ bn ∈ fj(n) (a ◦ a |M(kakA, ka kA, l, l),D) 0 s 0 so (bn ◦ bn)n∈N converges to ϕ (a ◦ a ). By uniqueness of the limit, we conclude: ϕs (a ◦ a0) = ϕs(a) ◦ ϕs(a0). The same results holds for the Lie product. Thus ϕs restricted to dom Ld(s) is a Jordan-Lie morphism. By continuity, ϕs is a Jordan-Lie morphism on from sa Ad(s) to sa Ac(s). So ϕs is a unital Jordan-Lie morphism on from sa Ad(s) to sa Ac(s). We note that by construction, ϕs(a∗) = ϕs(a)∗. 24 FRÉDÉRIC LATRÉMOLIÈRE

Let now a, b ∈ Ad(s). We ﬁst note that: ϕs(a ◦ b) = ϕs(

d(s) 0 d(s) 0 d(s) 0 Ln s s L s s Ln s s lim inf mkD c(s) ϕn, ϕn 6 mkDAc(s) ϕ , ϕ 6 lim sup mkD c(s) ϕn, ϕn . n→∞ An n→∞ An Moreover, if for some s ∈ S, we have d(s) = c(s), then:

s s s lim inf mk`Ln (ϕn) 6 mk`L(ϕ ) 6 lim sup mk`Ln (ϕn). n→∞ n→∞ Fix s, s0 ∈ S with the desired properties and write D = max{1,D(s),D(s0)}. d(s) d(s) Let ε > 0. Let E be a ﬁnite subset of {a ∈ sa A : L (a) 6 1} such that for d(s) d(s) 0 all a ∈ sa A with L (a) 6 1 there exists a ∈ E and t ∈ R such that:

0 ε ka − (a + t1 )k d(s) < . A A 12D 0 Let N ∈ N such that for all n > N, we have for all a ∈ E: 0 s 0 ε Hausk·k fs,j(n) (a |1,D), {ϕ (a )} < Ad(s) 8 and: 0 s0 0 ε Hausk·k fs0,j(n) (a |1,D), {ϕ (a )} < Ad(s) 8 while: ε χ τ s < . j(n) 12D Note that N exists since E is ﬁnite and by construction of ϕs above. Fix n > N. d(s) Let Gn ⊆ Aj(n) be given by Lemma (1.27), so that: d(s) d(s) R 0 • for all a ∈ sa Aj(n) with Ln (a) 6 1, there exists t ∈ and a ∈ Gn 0 ε such that ka − (a + t1A)k d(s) , An 6 4D 0 0 • for all a ∈ Gn there exists a ∈ E such that a ∈ t d(s) (a |1), and conversely, τj(n) 0 0 for all a ∈ E there exists a ∈ Gn such that a ∈ t d(s) (a |1). γj(n) SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 25

d(s) d(s) 0 Let a ∈ sa A with L (a) 6 1. There exists a ∈ E and t ∈ R such that 0 ε ε ka − (a + t1A)kAd(s) < 12D < 4D . We then have:

s0 s s0 0 ϕ (a) − ϕ (a) 6 ϕ (a − (a + t1A)) Ac(s) Ac(s) s0 0 s 0 + ϕ (a + t1A) − ϕ (a + t1A) Ac(s) s 0 + kϕ (a − (a + t1A))kAc(s)

Dε s0 0 s 0 Dε + kϕ (a ) − ϕ (a )k c(s) + 6 4D A 4D ε s0 0 s 0 + kϕ (a ) − ϕ (a )k c(s) . 6 2 A 0 s s0 Let a ∈ G ∩ t s (a |1). Let c = ϕ (a ) and d = ϕ (a ). Let c ∈ n n τj(n) n j(n) n n j(n) n 0 tγj(n) (cn|D) and d ∈ tγj(n) (dn|D). In particular, c ∈ fs,j(n) (a |1,D) and d ∈ 0 s 0 ε fs0,j(n) (a |1,D). Thus by construction, kϕ (a ) − ckAd(s) 6 8 . We then compute:

s0 0 s 0 ε kϕ (a ) − ϕ (a )k c(s) + kd − ck d(s) A 6 4 A ε ε s0 s 6 + 2 + kϕj(n)(an) − ϕj(n)(an)kAd(s) by Proposition (1.26) 4 12D j(n) d(s) L 0 ε j(n) s s 6 + mkD c(s) ϕj(n), ϕj(n) . 2 Aj(n)

d(s) Hence for all a ∈ sa (A) with L (a) 6 1:

d(s) 0 L 0 s s ε ε j(n) s s kϕ (a) − ϕ (a)kAc(s) 6 + + mkD c(s) ϕj(n), ϕj(n) , 2 2 Aj(n)

d(s) d(s) L L s s0 j(n) s s0 i.e. mkDAc(s) ϕ , ϕ 6 mkD c(s) ϕj(n), ϕj(n) + ε. Aj(n) We proceed symmetrically to show the converse inequality. Namely, let an ∈ Ad(s) with Ld(s) (a ) 1. There exists t ∈ R and a0 ∈ G such that ka − (a0 + j(n) Aj(n) n 6 n n n n ε t1A)k d(s) < 4D . As before, we have: Aj(n)

s0 s ε s0 0 s 0 kϕj(n)(an) − ϕj(n)(an)kAd(s) 6 + kϕj(n)(an) − ϕj(n)(an)kAd(s) . j(n) 2 j(n) 0 s 0 Let a ∈ E such that a ∈ t d(s) (an|1). Let c ∈ t d(s) ϕj(n)(an) D . Note that by γj(n) γj(n) symmetry, we have an ∈ t d(s) (a|1), so that by construction c ∈ fs,j(n) (a|1,D) and τj(n) s ε thus kϕ (a) − ckAd(s) < 8 . 0 0 s 0 s Similarly, let d ∈ t d(s) ϕj(n)(an) D , and note d ∈ fs0,j(n) (a|1,D) so kϕ (a) = γj(n) ε dk < 8 . s0 0 s 0 ε From this, we then obtain as before that kϕj(n)(an) − ϕj(n)(an)k d(s) 6 2 + Aj(n) Ld(s) s0 s mkDAc(s) ϕ , ϕ . Thus we get:

d(s) L 0 d(s) 0 j(n) s s L s s mkD c(s) ϕj(n), ϕj(n) 6 mkDAc(s) ϕ , ϕ + ε. Aj(n) 26 FRÉDÉRIC LATRÉMOLIÈRE

Hence we have shown that, for all ε > 0, there exists N ∈ N such that for all n > N, we have: d(s) d(s) 0 L 0 d(s) 0 L s s j(n) s s L s s mkDAc(s) ϕ , ϕ − ε 6 mkD c(s) ϕj(n), ϕj(n) 6 mkDAc(s) ϕ , ϕ + ε. Aj(n) Thus: d(s) L 0 d(s) 0 j(n) s s L s s lim mkD c(s) ϕj(n), ϕj(n) = mkDAc(s) ϕ , ϕ . n→∞ Aj(n) In particular, we have:

d(s) 0 d(s) 0 d(s) 0 Ln s s L s s Ln s s lim inf mkD c(s) ϕn, ϕn 6 mkDAc(s) ϕ , ϕ 6 lim sup mkD c(s) ϕn, ϕn . n→∞ An n→∞ An If c(s) = d(s) then our argument above can be applied as well when we replace s0 s0 d(s) d(s) ϕn and ϕ with the identity of Aj(n) and A , and adjusting the proof accordingly, in which case we conclude: s s s lim inf mk`Ld(s) (ϕn) 6 mk`Ld(s) (ϕ ) 6 lim sup mk`Ld(s) (ϕn). n→∞ n n→∞ n u Now, assume u ∈ S such that d(u) = c(u) and limn→∞ mk` s(u) (ϕ ) = 0. By Ln n u u d(u) this step, we conclude that mk`Ls(u) (ϕ ) = 0 so ϕ is the identity of A . u From this, and by definition, we conclude that if all units u of S satisfy limn→∞ mk` d(u) (ϕ ) = Ln n 0 then Ψ is a functor. Moreover, if s ∈ S is invertible and we set u = s ◦ s0 and v = s0 ◦ s, where s0 is the inverse of s, and if: u s◦s0 lim mk` d(u) (ϕn) = lim mk` d(u) (ϕn ) = 0 n→∞ Ln n→∞ Ln and v s0◦s lim mk` d(v) (ϕn) = lim mk` d(v) (ϕn ) = 0 n→∞ Ln n→∞ Ln 0 then ϕu and ϕv are the identities of Ad(u) and Ad(v). Since moreover ϕu = ϕs◦s = 0 0 0 ϕs ◦ ϕs and ϕv = ϕs ◦s = ϕs ◦ ϕs, we conclude that ϕs is invertible with inverse 0 ϕs . This concludes our step and our theorem. Remark 2.15. We note that the morphisms obtained by Theorem (2.13) are cer- tainly not unique in general. There are many different choices being made in their construction: the choice of the tunnels and the choices of subsequences of target sets from compactness. The proof shows how all these choices can be made in a coherent fashion to preserve composition, and ensure that the limit of identity automorphisms is the identity — thus preserving inverse relationships. Remark 2.16. We need an additional assumption regarding the density of the positive elements with finite L-seminorm within the positive elements of the underlying C*-algebra for Item (1) of Theorem (2.13) which does not appear for the rest of the theorem, and in particular for Item (3) regarding *-morphisms. This assumption is needed for the proof of Item (1) as we have no multiplicative property for this step, and not for Item (3) because *-morphisms are automatically positive. We note that for classical metric space, this special assumption is always satisfied because the truncation of a Lipschitz function is Lipschitz. On the other hand, we also note that when L-seminorms are obtained from ergodic group actions as in [23], which is an often used construction, then again this special property is SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 27 met, using a standard regularization argument akin to the proof of the density of the domain of such L-seminorms in [23]. However, in general, the situation can be more involved and the Leibniz property does not seem to ensure that this special condition will always hold. We proved our main Theorem (2.13) with the construction of the dual propinquity in [18]. We could just as easily make the same argument, with trivial changes, adapted to the original construction in [12] based on journeys — this would techni- cally allow us to use somewhat more general classes of tunnels. Among the possible choices of tunnels, we could then employ the tunnels obtained directly from bridges [15]. The reason one would wish to adapt the argument in such an easy manner is if it is possible to extract more information about the functor constructed by Theorem (2.13) based on a more stringent choice of tunnels — for instance, when spaces converge for the quantum propinquity [15].

3. Applications As a ﬁrst observation, we apply Theorem (2.13) to the construction of a single *-morphism. The result is interesting because it is not necessarily easy to construct morphisms between C*-algebras, yet this ﬁrst corollary shows a new method based on metric approximations. Corollary 3.1. Let F be an admissible function. Let (A, LA) and (B, LB) be two F – A B quasi-Leibniz quantum compact metric spaces, and let (An, Ln )n∈N and (B, Ln )n∈N be two sequences of F –quasi-Leibniz quantum compact metric spaces such that: ∗ A A ∗ B lim ΛF ((An, Ln ), (A, L )) = lim ΛF ((Bn, Ln ), (B, LB)) = 0. n→∞ n→∞ Let D > 0. If for each n ∈ N, there exists a Lipschitz morphism (resp. a Lipschitz linear map) ϕn : An → Bn such that: N B A (1) ∀n ∈ Ln ◦ ϕn 6 DLn , (2) |||ϕ |||An D, n Bn 6 then there exists a Lipschitz morphism (resp. a Lipschitz linear map) ϕ : A → B such that: A (1) |||ϕ|||B 6 D (2) LB ◦ ϕ 6 LA. A B N If moreover (An, Ln ) = (Bn, Ln ) for all n ∈ then:

A lim inf mk`L (ϕn) 6 mk`LA (ϕ) 6 lim sup mk`LA (ϕn). n→∞ n n→∞

If furthermore, ϕn is a bijection for infinitely many n ∈ N, then ϕ may be chosen to be a bijection as well. Proof. All but the last statement result from applying Theorem (2.13) to the semigroupoid ({s}, {a, b}, c, d, ◦) with d(s) = a and c(s) = b (thus ◦ is the empty map). Setting (Aa, La) = (A, LA), (Ab, Lb) = (B, LB) and for all n ∈ N, setting a a A b b B s (An, Ln) = (An, Ln ), (An, Ln) = (Bn, Ln ), and ϕn = ϕn, Theorem (2.13) applies. If ϕn is invertible for infinitely many n ∈ N, then up to extracting a subsequence, we may as well assume that ϕn is invertible for all n ∈ N. Then we employ the groupoid ({s, s0, a, b}, {a, b}, d, c, ◦), with the same definitions as above, with a and b serving as the units for the homonymous domains, and s0 the inverse of s, with s0 −1 N ϕn = ϕn for all n ∈ . Theorem (2.13) applies again. 28 FRÉDÉRIC LATRÉMOLIÈRE

As an obvious remark, we point out that if we are given two convergent sequences A B (An, Ln )n∈N and (Bn, Ln )n∈N of quasi-Leibniz quantum compact metric spaces which are entry-wise fully quantum isometric, then by [12], their respective limits (A, LA) and (B, LB) are also fully quantum isometric: A B A B 0 Λ((A, L ), (B, L )) = lim Λ((An, Ln ), (B, Ln )) = lim 0 = 0. 6 n→∞ n→∞ This coincidence property was a strong motivation behind the construction of the propinquity, and is proven using in part the quasi-Leibniz property. Thus our present Theorem (2.13) provides another illustration of the same principle that using the quasi-Leibniz property and the definition of the propinquity enables us to build *-morphisms between C*-algebras, albeit Theorem (2.13) application in the single morphism case is only new and interesting for Lipschitz morphisms which are not full quantum isometries. We now turn to the main motivation for this paper: the existence of a nontrivial action of a group on the limit of a sequence of quasi-Leibniz quantum compact metric spaces, when the group is the limit of groups acting on each term of the sequence. One way to see the importance of this result is with a sight on mathematical physics applications: a model obtained as a limit for the propinquity of covariant models for some group will bear the same covariance. There are in fact different manners to formalize this problem. We will study the equivariant propinquity in a subsequent paper, to keep the present work of manageable length, and also because we take the perspective that this is a work on the properties of the Gromov-Hausdorff propinquity. For this purpose, we encode covariance in a strong form, in terms of a naturally enhanced Gromov-Hausdorff convergence for monoids. We choose to construct the Gromov-Hausdorff group distance in terms of ε-isometries to keep our presentation brief. We will use the term monoid to mean an associative magma with an identity element (a unit). A metric monoid (G, δG) is a monoid G endowed with a distance δG which induces a topology on G, and for which the multiplication of G is continuous.

Deﬁnition 3.2. Let (G1, δ1) and (G2, δ2) be two compact metric monoids with respective identity elements e1 and e2. An ε-almost isometric isomorphism (ς1, ς2), for ε > 0, is an ordered pair of maps ς1 : G1 → G2 and ς2 : G2 → G1 such that for all {j, k} = {1, 2}: 0 0 0 ∀g, g ∈ Gj ∀h ∈ Gk |δk(ςj(g)ςj(g ), h) − δj(gg , ςk(h))| 6 ε, and ςj(ej) = ek. The set of all ε-almost isometric isomorphism is denoted by:

AIsoε ((G1, δ1) → (G2, δ2)). Almost isometric isomorphisms behave well under composition, which will be the key property to obtain the triangle inequality for our new distance between compact metric monoids.

Lemma 3.3. Let (G1, δ1), (G2, δ2) and (G3, δ3) be three compact metric monoids with respective identity elements e1, e2 and e3.

If (ς1, κ1) ∈ AIsoε1 ((G1, δ1) → (G2, δ2)) and (ς2, κ2) ∈ AIsoε2 ((G2, δ2) → (G3, δ3)) for some ε1 > 0 and ε2 > 0, then:

(ς2 ◦ ς1, κ1 ◦ κ2) ∈ AIsoε1+ε2 ((G1, δ1) → (G3, δ3)). SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 29

Moreover ς(e1) = e3 and κ(e3) = e1. Thus (ς, κ) ∈ AIsoε1+ε2 ((G1, δ1) → (G3, δ3)) as desired. We use almost isometric isomorphisms to deﬁne our Gromov-Hausdorﬀ distance for compact metric monoids.

Deﬁnition 3.4. The Gromov-Hausdorﬀ monoid distance Υ((G1, δ1), (G2, δ2)) between two metric compact monoids (G1, δ1) and (G2, δ2) is given by:

Υ((G1, δ1), (G2, δ2)) = inf {ε > 0|AIsoε ((G1, δ1) → (G2, δ2)) 6= ∅} . In order to prove that our distance Υ is indeed a metric on the class of compact metric monoid up to isometric isomorphism, we will ﬁrst characterize almost isometric isomorphisms in terms of some natural properties, which we encode in the following deﬁnition for convenience.

Deﬁnition 3.5. Let (G1, δ1) and (G2, δ2) be two metric monoids and ε > 0. An ordered pair ς1 : G1 → G2 and ς2 : G2 → G1 of functions is called ε-near isometric 0 isomorphism when, for all {j, k} = {1, 2} and g, g ∈ Gj: 0 0 (1) δk(ςj(g)ςj(g ), ςj(gg )) 6 ε, (2) δj(ςk(ςj(g)), g) 6 ε, 0 0 (3) |δk(ςj(g), ςj(g )) − δj(g, g )| 6 ε, (4) δk(ςj(ej), ek) 6 ε. If ς1 and ς2 map units to units, then the near isometric isomorphism is called unital. We now show that almost isometric isomorphism and near isometric isomorphism only diﬀer in the way they quantify the defect of these maps to be actual isometric isomorphism.

Lemma 3.6. Let (G1, δ1), (G2, δ2) be two metric monoids and ε > 0.

(1) If (ς1, ς2) ∈ AIsoε ((G1, δ1) → (G2, δ2)) then for all {j, k} = {1, 2}: (a) ∀g ∈ Gj ∀h ∈ Gk |δk(ςj(g), h) − δj(g, ςk(h))| 6 ε, (b) ∀g ∈ Gj δj(ςk ◦ ςj(g), g) 6 ε, 0 0 0 (c) ∀g, g ∈ Gj δk(ςj(g)ςj(g ), ςj(gg )) 6 2ε, 0 0 0 (d) ∀g, g ∈ Gj |δk(ςj(g), ςj(g )) − δj(g, g )| 6 2ε; in particular (ς1, ς2) is a unital 2ε-near isometric isomorphism; (2) if (ς1, ς2) is a unital ε-near isometric isomorphism, then:

(ς1, ς2) ∈ AIso3ε ((G1, δ1) → (G2, δ2));

(3) if (ς1, ς2) is a ε-near isometric isomorphism, then:

AIso6ε ((G1, δ1) → (G2, δ2)) 6= ∅. 30 FRÉDÉRIC LATRÉMOLIÈRE

Proof. First, let (ς1, ς2) ∈ AIsoε ((G1, δ1) → (G2, δ2)). Let {j, k} = {1, 2}. If g ∈ Gj and h ∈ Gk then:

|δk(ςj(g), h) − δj(g, ςk(h))| = |δk(ςj(g)ek, h) − δj(gej, ςk(h))|

= |δk(ςj(g)ςj(ej), h) − δj(gej, ςk(h))| 6 ε.

In particular, if g ∈ Gk then:

δk(ςj(ςk(g)), g) = |δk(ςj(ςk(g)), g) − δj(ςk(g), ςk(g))| + δj(ςk(g), ςk(g)) (3.1) 6 ε + 0 = ε. Thus (1a) and (1b) are proven for all {j, k} = {1, 2}. 0 Using Inequality (3.1), if g, g ∈ Gj, then:

0 0 δk(ςj(g)ςj(g ), ςj(gg )) 0 0 0 0 0 0 6 |δk(ςj(g)ςj(g ), ςj(gg )) − δj(gg , ςk(ςj(gg )))| + δj(gg , ςk(ςj(gg ))) 6 ε + ε = 2ε. 0 We can now conclude that for all g, g ∈ Gj: 0 0 0 0 |δk(ςj(g), ςj(g )) − δj(g, g )| = |δk(ςj(ej)ςj(g), ςj(g )) − δj(g, g )| 0 0 6 ε + |δj(ejg, ςk ◦ ςj(g )) − δj(g, g )| 0 0 6 ε + δj(ςk ◦ ςj(g ), g ) 6 2ε. This concludes our proof of (1). Let now (ς1, ς2) be a ε-near isometric isomorphism. We note that for all {j, k} = 0 {1, 2}, g, g ∈ Gj and h ∈ Gk:

0 0 |δk(ςj(g)ςj(g ), h) − δj(gg , ςk(h))| 0 0 0 0 6 |δk(ςj(g)ςj(g ), h) − δk(ςj(gg ), h)| + |δk(ςj(gg ), h) − δj(gg , ςk(h))| 0 0 0 0 6 δk(ςj(g)ςj(g ), ςj(gg )) + |δk(ςj(gg ), h) − δj(gg , ςk(h))| 6 ε + 2ε = 3ε.

Thus if (ς1, ς2) is unital, we have shown that (ς1, ς2) ∈ AIso3ε ((G1, δ1) → (G2, δ2)). Assume now that (ς1, ς2), still a ε-near isometry, is not longer unital. Deﬁne for both j ∈ {1, 2}: ( 0 ek if g = ej, ςj : g ∈ Gj 7→ ςj(g) otherwise. 0 0 Let now {j, k} = {1, 2}. Note that of course, δj(ςk ◦ ςj(ej), ej) = 0. Let now g ∈ Gj. If g 6= ej we check: 0 0 0 δk(ςj(g)ςj(e), ςj(ge)) = δk(ςj(g), ςj(g)) = 0, SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 31

0 0 and similarly, all needed properties for (ς1, ς2) to be a near isometric isomorphism are trivially met, with the following computation the only relevant one here:

0 0 δk(ςj(g), ςj(ej)) − δj(g, ej) 0 6 δk(ςj(g), ej) − δj(ςj(g), ςj(ej)) + |δk(ςj(g), ςj(ej)) − δj(g, ej)| 6 δk(ej, ςj(ej)) + |δk(ςj(g), ςj(ej)) − δj(g, ej)| 6 2ε. 0 0 0 0 Thus, (ς1, ς2) is a unital 2ε-near isometric isomorphism, and therefore (ς1, ς2) ∈ AIso6ε ((G1, δ1) → (G2, δ2)) as desired. We are now ready to prove that Υ is indeed a metric on compact metric monoids up to isometric isomorphisms.

Theorem 3.7. For any metric compact monoids (G1, δ1), (G2, δ2) and (G3, δ3):

(1) Υ((G1, δ1), (G2, δ2)) < ∞, (2) Υ((G1, δ1), (G2, δ2)) = Υ((G2, δ2), (G1, δ1)), (3) Υ((G1, δ1), (G3, δ3)) 6 Υ((G1, δ1), (G2, δ2)) + Υ((G2, δ2), (G3, δ3)), (4) Υ((G1, δ1), (G1, δ1)) = 0, (5) If Υ((G1, δ1), (G2, δ2)) = 0 then there exists a monoid isometric morphism from (G1, δ1) to (G2, δ2). In particular, Υ is a metric up to metric group isometric isomorphism on the class of metric compact groups. Moreover, if GH is the usual Gromov-Hausdorff distance between compact metric spaces then GH 6 4Υ on the class of metric compact monoids. Proof. The symmetry of Υ follows obviously from the symmetry of the definition of almost isometric isometries. We now prove the triangle inequality. Let υ1 = Υ((G1, δ1), (G2, δ2)) and υ2 = Υ((G2, δ2), (G3, δ3)). Let ε > 0. Let (ς , ) ∈ AIso ε ((G , δ ) → (G , δ )). Similarly, let (ς , ) ∈ 1 κ1 υ1+ 2 1 1 2 2 2 κ2 AIso ε ((G , δ ) → (G , δ )). By Lemma (3.3), if ς = ς ◦ ς and = ◦ υ2+ 2 2 2 3 3 2 1 κ κ1 κ2 then (ς, κ) ∈ AIsoυ1+υ2+ε ((G1, δ1) → (G3, δ3)). Therefore by Definition (3.4), we conclude: Υ((G1, δ1), (G3, δ3)) 6 υ1 + υ2 + ε. As ε > 0 is arbitrary, we conclude:

Υ((G1, δ1), (G3, δ3)) 6 υ1 + υ2 = Υ((G1, δ1), (G2, δ2)) + Υ((G2, δ2), (G3, δ3)), as desired. We now prove that Υ((G, δG), (H, δH )) = 0 implies the existence of an isometric monoid isomorphism between (G, δG) and (H, δH ). Let S be a countable dense subset of G. Let G0 be the submonoid generated in G by S, which is also countable as it is the set of all ﬁnite products of elements in S. Similarly, let H0 be a countable dense submonoid of H. Assume ﬁrst that Υ((G, δG), (H, δH )) = 0. For each n ∈ N, let:

(ςn, n) ∈ AIso 1 ((G, δG) → (H, δH )). κ n+1

For each g ∈ G0 and any strictly increasing sequence j : N → N, any subsequence (ςj(n)(g)) admits a convergent subsequence (ςj◦k(n)) in the compact space H. Thus, as G0 is countable, a diagonal argument shows that there exists a strictly increasing 32 FRÉDÉRIC LATRÉMOLIÈRE sequence j of natural numbers such that, for all g ∈ G0, the sequence (ςj(n)(g)) converges to a limit we denote by ς(g) (similar to Steps (2) and (3) in the proof of Theorem (2.13)). 0 0 0 Now, for all g, g ∈ G0, we have δH (ς(g), ς(g )) = limn→∞ δH (ςj(n)(g), ςj(n)(g )) so ς is an isometry (using Assertion (1d) of Lemma (3.6)). Moreover, by Assertion 0 (1c) of Lemma (3.6), we also have that for all g, g ∈ G0: 0 0 0 0 δH (ς(gg ), ς(g)ς(g )) = lim δH (ςj(n)(gg ), ςj(n)(g)ςj(n)(g )) = 0. n→∞

So ς is a monoid morphism from G0 to H. Now, as a uniformly continuous function over the dense subset G0 of G, ς admits a unique uniformly continuous extension to G which we still denote by ς. It is immediate that ς is an isometry. Moreover, as the product of G is continuous as a map from G × G to G for the topology induced by δG on G and its product on G × G, it is immediate that ς is a monoid morphism as well. Now, by the same argument, there exists an isometric map κ : H → G and a N N strictly increasing k : → such that for all h ∈ H0, the sequence κj◦k(n)(h) converges to κ(h). By Assertion (1b) of Lemma (3.6), we have κ ◦ ς is the identity of G0 and ς ◦ κ is the identity of H0. By continuity, this means that ς and κ are inverse of each others. Now, κ is a monoid morphism because it is the inverse of ς, itself a monoid morphism. Alternatively, Assertion (1c) of Lemma (3.6) proves that κ is a morphism. Thus, Υ((G, δG), (H, δH )) = 0 implies that there exists an isometric monoid isomorphism ς : G → H as desired. Conversely, if there exists an isometric monoid isomorphism ς from (G, δG) onto −1 (H, δH ) then ς, ς ∈ AIso0 ((G, δG) → (H, δH )) so Υ((G, δG), (H, δH )) = 0. We also note that we can always pick g0 ∈ G, h0 ∈ H and define ς : g ∈ G 7→ h0, κ : h ∈ H 7→ g0; we easily check that (ς, κ) ∈ AIsoM ((G, δG) → (H, δH )) where M = max{diam (G, δG), diam (H, δH )}, so Υ((G, δG), (H, δH )) 6 M. Now, let (ς, κ) ∈ AIsoε ((G, δG) → (H, δH )) for ε > Υ((G, δG), (H, δH )) for two compact metric monoids (G, δG) and (H, δH ). By Assertion (1b) and (1d) of Lemma (3.6), we easily check that ς is a 2ε-isometry in the sense of [2, Definition 7.3.27]. Hence by [2, Corollary 7.3.28], the Gromov-Hausdorff distance between (G, δG) and (H, δH ) is no more than 4 times ε. We thus conclude:

GH((G, δG), (H, δH )) 6 4Υ((G, δG), (H, δH )). We conclude with the observation that a morphism of monoid between groups is in fact a group morphism. This completes our proof. A special case of obvious interest is given by converging sequences of closed subgroups for the Hausdorﬀ metric induced by a continuous length function on a compact group: the following proposition shows that such a sequence would also converge for our distance Υ even though we do not a priori assume more than convergence in a metric sense, with no algebraic condition. Proposition 3.8. Let (G, δ) be a compact metric group, with δ translation invariant. If H ⊆ G is a closed subgroup of G and (Hn)n∈N is a sequence of closed subgroups of G converging to H for the Hausdorﬀ distance Hausδ induced by δ on the closed subsets of G, then:

lim Υ((Hn, δ), (H, δ)) = 0. n→∞ SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 33

Remark 3.9. If ` is a length function over a group G such that `(g−1hg) = `(h) for all g, h ∈ G, then the distance δ : g, h ∈ G 7→ `(h−1g) is both left and right invariant, i.e. translation invariant.

Proof. For any closed subset F of G and g ∈ G, we choose an element pF (g) ∈ F such that:

δ(g, pF (g)) = min {δ(g, h): h ∈ F } . The existence of such an element is guaranteed by compactness; our choice is arbitrary: in particular, we have no expectation of any regularity for the function pF thus deﬁned. For all n ∈ N, set:

κn : g ∈ H 7→ pHn (g) and $n : g ∈ Hn 7→ pH (g).

We immediately note that κn and ϕn both ﬁx the identity element of G for all n ∈ N. N ε Let N ∈ such that, for all n > N, we have Hausδ(Hn,H) < 3 . Let n > N. ε Note that by deﬁnition of Hausδ, we have max{δ(κn(g), g), δ($n(h), h)} < 3 for all g ∈ H and h ∈ Hn. 0 Let g, g ∈ H and h ∈ Hn.

0 0 |δ(κn(g)κn(g ), h) − δ(gg , $n(h))| 0 0 0 0 = |δ(κn(g)κn(g ), h) − δ(gg , h) + δ(gg , h) − δ(gg , $n(h))| 0 0 0 0 6 |δ(κn(g)κn(g ), h) − δ(gg , h)| + |δ(gg , h) − δ(gg , $n(h))| 0 0 6 δ(κn(g)κn(g ), gg ) + δ(h, $n(h)) ε δ( (g) (g0), gg0) + 6 κn κn 3 ε δ( (g) (g0), (g)g0) + δ( (g)g0, gg0) + 6 κn κn κn κn 3 ε δ( (g0), g0) + δ( (g), g) + 6 κn κn 3 ε ε ε + + = ε. 6 3 3 3 All the above computations are equally valid if we replace κn by $n and vice- and-versa, up to choosing elements in the appropriate domains, of course. Thus (κn, ϕn) is an ε-almost isometric isomorphism, so Υ((Hn, δ), (H, δ)) 6 ε for all n > N. This concludes our proof. 0 As a last observation, if we deﬁned Υ ((G1, δ1), (G2, δ2)) as

inf {ε > 0|∃ a ε-near isometric isomorphism from (G1, δ1) to (G2, δ2)} 0 for two compact metric monoids (G1, δ1) and (G2, δ2), then Υ is a priori an in- frametric rather than a metric, since near isometric isomorphisms do not behave as well as almost isometric isomorphisms for composition. On the other hand, 1 0 0 2 Υ 6 Υ 6 6Υ by Lemma (3.6), so Υ and Υ deﬁne the same topology on the class of compact metric monoids. We are now ready to prove our main application of Theorem (2.13).

Theorem 3.10. Let (G, δG) be a compact metric monoid. Let (Gn, δn)n∈N be a sequence of compact metric monoids, converging to G for the Gromov-Hausdorﬀ 34 FRÉDÉRIC LATRÉMOLIÈRE monoid distance Υ. For each n ∈ N, choose:

(ςn, κn) ∈ AIsoεn ((Gn, δn) → (G, δ)) where (εn)n∈N is some sequence converging to 0 and εn > Υ((G, δ), (Gn, δn)) for all n ∈ N. Let D : G → [0, ∞) be a continuous function. Let K : [0, ∞) → [0, ∞) be continuous with K(0) = 0. Let C be a nonempty class of F –quasi-Leibniz quantum compact metric spaces for some admissible function F , and let T be a class of tunnels compatible with C. Let (An, Ln)n∈N be a sequence in C, such that for each n ∈ N, there exists an action αn by Lipschitz linear endomorphisms of Gn on An, where: ∀n ∈ N, g ∈ G |||αg ||| D(ς (g)) and L ◦ αg D(ς (g))L , n n An 6 n n n 6 n n and 0 N 0 g g 0 ∀n ∈ , g, g ∈ Gn, a ∈ sa (An) αn(a) − αn (a) 6 K (δn(g, g )) Ln(a). An If there exists (A, L) ∈ C such that: ∗ lim ΛT ((An, Ln), (A, L)) = 0 n→∞ then there exists a strongly continuous action α of the monoid G by Lipschitz linear endomorphisms such that for all g ∈ G: g g (3.2) |||α |||A 6 D(g) and L ◦ α 6 D(g)L, and for all g, h ∈ G: (3.3) Ln κn(g) κn(h) L g h Ln κn(g) κn(h) lim inf mkDA αn , αn 6 mkDA α , α 6 lim sup mkDA αn , αn , n→∞ n n→∞ n while for all g, h ∈ G, and a ∈ dom (L), we have: g h α (a) − α (a) A 6 L(a)K(δG(g, h)).

If moreover for all n ∈ N, the actions αn of the monoid Gn is by positive unital linear maps, then α is a strongly continuous action of G by positive unital linear maps.

Remark 3.11. With the notations of Theorem (3.10), if G is a group, and Gn is a N g g group for all n ∈ , then of course |||α |||A = 1 as α is a *-automorphism for all g ∈ G. Remark 3.12. We emphasize that while Theorem (3.10) requires convergence of the quantum metric spaces and of the groups or monoids, there is not requirement of convergence involving the actions themselves. Proof. Since G is a compact metric space, it is separable. Let E be a countable dense subset of G. Let H be the sub-monoid generated by E. Since H consists of all the ﬁnite products of elements of the countable set E, it is itself countable. As a monoid, H is also a semigroupoid in a trivial manner (formally, the semigroupoid is (H, {e}, c, d, ·) with c and d the constant functions equal to e and · the multiplication of H, and e is the unit of H. With these notations, we would set e e e e N (A , L ) = (A, L) and (An, Ln) = (An, Ln) for all n ∈ , though we will not use this heavier notation here). SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 35

N Let g ∈ H, n ∈ and a ∈ dom (Ln) with Ln(a) 6 1. We now compute: 0 0 0 0 κn(g) κn(g ) κn(gg ) κn(g)κn(g ) κn(gg ) αn ◦ αn (a) − αn (a) = αn (a) − αn (a) An An 0 0 6 K (δn (κn(g)κn(g ), κn(gg ))) Ln(a) 0 0 6 K (δn (κn(g)κn(g ), κn(gg ))) . 0 0 By Assertion (1c) of Lemma (3.6), the sequence (δn (κn(g)κn(g ), κn(gg )))n∈N converges to 0. So: 0 0 lim K (δn ( n(g) n(g ), n(gg ))) = 0. n→∞ κ κ κ

en We also record that αn , for en ∈ Gn the unit of Gn, is the identity map. Thus we ﬁt the hypothesis of Theorem (2.13). Therefore, there exists an action α of H on A, with the property that, for all a ∈ sa (A) with L(a) 6 1 and for all g, h ∈ H: g h kα (a) − α (a)kA 6 lim sup K(δn(κn(g), κn(h))) 6 K(δG(g, h)), n→∞ and in addition, Expressions (3.2) and (3.3) hold for g ∈ H. Note that α is a monoid action, i.e. in particular the unit of G acts as the identity. It thus immediately follows by homogeneity that for all a ∈ dom (L) and g ∈ H: g h kα (a) − α (a)kA 6 L(a)K(δG(g, h)). Therefore g ∈ H 7→ αg(a) is uniformly continuous for all a ∈ dom (L) since lim0 K = 0, and thus it can be extended uniquely to a uniformly continuous function g ∈ G 7→ αg(a), with the additional property that for all g, h ∈ G: g h α (a) − α (a) A 6 L(a)diam (A, L)K(δG(g, h)) by continuity of K (and δG). It is straightforward that for all a ∈ dom (L): (1) for all g, h ∈ G, we have by continuity of the multiplication on G that αg ◦ αh(a) = αgh(a), (2) for all g ∈ G, the lower semi-continuity of L and the continuity of D gives g us L ◦ α (a) 6 D(g)L(a), g (3) for all g ∈ G, we also have by continuity that kα (a)kA 6 D(g)kakA. In particular, Expression (3.2) holds for all g ∈ G. Therefore, α is a monoid action of G via Lipschitz linear maps. Let g ∈ G. Let ε > 0, and let R > 0 be some upper bound for the convergent sequence (diam (An, Ln))n∈N. Since K is continuous and K(0) = 0, there exists ε δ > 0 such that for t ∈ [0, δ), we have K(t) < 2 . δ Let h ∈ H with δG(g, h) < 3 . N δ Now, there exists N ∈ such that for all n > N, we have εn < 6 . Let n > N. By Deﬁnition (3.2), we have: δ |δ ( (g), (h)) − δ (g, h)| < . n κn κn G 3 Hence δn(κn(g), κn(h)) < δ. We now compute, for any a ∈ dom (Ln) with Ln(a) 6 1:

κn(g) κn(h) κn(g) κn(h) a − αn (a) − a − αn (a) 6 αn (a) − αn (a) An An An 36 FRÉDÉRIC LATRÉMOLIÈRE ε K(δ ( (g), (h))) . 6 n κn κn 6 2 Therefore: ε κn(g) κn(h) lim sup mk`Ln (α ) 6 lim sup mk`Ln (α ) + , n→∞ n→∞ 2 and ε n(g) n(h) lim inf mk`L (ακ ) lim inf mk`L (ακ ) − . n→∞ n > n→∞ n 2 Now: g h g h ka − α (a)kA − a − α (a) A 6 α (a) − α (a) A ε K(δ (g, h)) < . 6 G 2 Thus: g ε h ka − α (a)k + a − α (a) A 6 2 A ε κn(h) 6 + lim sup mk`Ln (α ) since h ∈ H, 2 n→∞ ε ε κn(g) κn(g) 6 + + lim sup mk`Ln (α ) = ε + lim sup mk`Ln (α ), 2 2 n→∞ n→∞ and therefore:

g κn(g) mk`L(α ) 6 lim sup mk`Ln (α ) + ε. n→∞ Similarly:

g n(h) mk`L(α ) lim inf mk`L (ακ ) − ε. > n→∞ n Since ε > 0 is arbitrary, we conclude that for all g ∈ G:

κn(g) g κn(g) lim inf mk`Ln (αn ) 6 mk`L(α ) 6 lim sup mk`Ln (α ). n→∞ n→∞ Thus Expression (3.3) now holds for all g ∈ G. Let now a ∈ sa (A), ε > 0 and g ∈ G. Since D is continuous, let η > 0 such that if δG(g, h) < η, then |D(g) − D(h)| < ε. 0 0 ε By density, there exists a ∈ dom (L) such that ka − a k < 3(D(g)+ε) . Now, there ε exists η2 > 0 such that if t ∈ [0, η2) then K(t) < 3 max{L(a0),1} . Let now h ∈ G with δG(g, h) < min{η, η2}. We compute:

g h α (a) − α (a) A g 0 g 0 h 0 h 0 6 kα (a − a )kA + α (a ) − α (a ) A + α (a − a ) A 0 0 0 6 D(g) ka − a kA + L(a )K(δG(g, h)) + D(h) ka − a kA 6 ε. Therefore, for all a ∈ sa (A), the function g ∈ G 7→ αg(a) is continuous, as desired. We now turn to an application of our work to certain examples. We can begin to address two related difficult questions using our work in this paper, and we will see that quantum metric geometric ideas might continue to prove helpful in their study. Ergodic actions of compact groups on unital C*-algebras have been of great interests, and it has been challenging to determine, for a given compact group, SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 37 even compact Lie groups, what are all the C*-algebras on which the group acts ergodically, outside of some special cases. On the other hand, another difficult question in noncommutative metric geometry is to determine the closure of certain classes of quasi-Leibniz quantum compact metric spaces for the propinquity. For instance, it remains unclear what the closure of the finite dimensional quasi-Leibniz quantum compact metric spaces is at the time of this writing. When restricting our focus to finite dimensional algebras whose quantum metrics come from an ergodic action of compact groups, our present work will enable us to make significant advance is answering the closure question. As a first simple observation, working with groups enables us to strengthen the conclusions of Theorem (3.10), by obtaining actions by full quantum isometries. The following proposition will be used to this end in our next theorem on group actions. Proposition 3.13. If α is an action of a compact group G on some quasi-Leibniz g quantum compact metric space (A, L) by 1-Lipschitz automorphisms (i.e. L◦α 6 L for all g ∈ G), then for all g ∈ G, the automorphism αg is a full quantum isometry of (A, L).

g Proof. Let g ∈ G. By assumption, for all a ∈ sa (A), we have L ◦ α (a) 6 L(a). Thus for all a ∈ dom (L):

g g−1 g−1 L(a) = L(α (α )) 6 L(α (a)). g Now, since the above holds for all g ∈ G, then we also have L(a) 6 L ◦ α (a) for g g g all a ∈ dom (L). Hence L ◦ α 6 L 6 L ◦ α and thus L ◦ α = L. g Consequently, α is a full quantum isometry for any g ∈ G. As an important corollary of Theorem (3.10), we obtain:

Theorem 3.14. Let (G, δG) be a compact metric group. Let (Gn, δn)n∈N be a sequence of compact metric groups, converging to G for the Gromov-Hausdorﬀ monoid distance Υ. For each n ∈ N, choose:

(ςn, κn) ∈ AIsoεn ((Gn, δn) → (G, δ)) where (εn)n∈N is some sequence of positive numbers converging to 0 with εn > Υ((G, δ), (Gn, δn)) for all n ∈ N. Let D : G → [0, ∞) be a continuous function. Let K : [0, ∞) → [0, ∞) be continuous with K(0) = 0. Let C be a nonempty class of F –quasi-Leibniz quantum compact metric spaces for some admissible function F , and let T be a class of tunnels compatible with C. Let (An, Ln)n∈N be a sequence in C, such that for each n ∈ N, there exists an action αn by Lipschitz automorphisms of Gn on An, where: N g ∀n ∈ , g ∈ Gn Ln ◦ αn 6 D(ςn(g))Ln, and 0 N 0 g g 0 ∀n ∈ , g, g ∈ Gn, a ∈ sa (An) αn(a) − αn (a) 6 K (δn(g, g )) Ln(a). An If there exists (A, L) ∈ C such that: ∗ lim ΛT ((An, Ln), (A, L)) = 0 n→∞ 38 FRÉDÉRIC LATRÉMOLIÈRE then there exists a strongly continuous action α of the group G by Lipschitz automorphisms such that for all g ∈ G: g L ◦ α 6 D(g)L, and for all g ∈ G: κn(g) g κn(g) lim inf mk`Ln αn 6 mk`L (α ) 6 lim sup mk`Ln αn , n→∞ n→∞ while for all g, h ∈ G, and a ∈ dom (L), we have: g h α (a) − α (a) A 6 L(a)K(δG(g, h)). Furthermore: g (1) if D 6 1 then for all g ∈ G, the *-automorphism α is a full quantum isometry, (2) if for all n ∈ N, the action αn of Gn on An is ergodic, then the action α of G on A is also ergodic. Proof. As a corollary of Theorem (3.10), there exists an action of G on A by Lip- schitz *-endomorphisms, only noting that we can apply Theorem (2.13) to obtain an action of G by *-automorphisms. g By Proposition (3.13), if D 6 1 then in fact, α is a full quantum isometry of (A, L). We now turn to the ergodicity property. Step 1. We assume, for the rest of this proof, that for all n ∈ N, the action αn of g C Gn on An is ergodic, namely that {a ∈ A : ∀g ∈ G α (a) = a} = 1An . Our goal is to prove that α is also ergodic. For this and the next few steps, we will need to be more speciﬁc about the construction of the action α provided by Theorem (2.13). We start this proof at Step 4 of the proof of our main theorem (2.13). Thus, we have chosen a tunnel τn in T from (A, L) to (An, Ln) with: 1 χ (τ ) Λ ((A , L ), (A, L)) + , n 6 T n n n + 1 −1 N and we let γn = τn for all n ∈ . To simplify notations, we have extracted a subsequence from (An, Ln)n∈N which we still denote by (An, Ln)n∈N such that for all g ∈ G, a ∈ dom (L) and l > L(a): g lim Hausk·k (fn,g (a|l, 1), {α (a)}) = 0, n→∞ A where fn,g (a|l, 1) is a shorthand for f κn(g) (a|l, 1). τn,αn ,γn Now, for each n ∈ N, and a ∈ An, we set: Z E g n(a) = αn(a) dλn(g) Gn where λn is the Haar probability measure on Gn. Of course, En is the conditional expectation of An onto the ﬁxed point algebra for αn which is, for all n ∈ N, C reduced to 1An by assumption (indeed: Z Z h E hg g −1 E αn( n(a)) = αn (a) dλn(g) = αn(a) dλn(gh ) = n(a) Gn Gn since λn is translation invariant.) SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 39

E R g Similarly, let : a ∈ A 7→ G α (a) dλ(g). Our goal is thus to prove that the range of E is C1A. Our first observation is that: Step 2. It is sufficient to show that if a ∈ dom (L) and αg(a) = a for all g ∈ G, then a ∈ R1A. g Indeed, should this be the case, then, first, for all g ∈ G we have L(α (a)) 6 L(a) g so α (a) ∈ {b ∈ sa (A): L(b) 6 L(a), kbkA 6 kakA}. As L is an L-seminorm, the E R g latter set is compact for k · kA. Thus, (a) = G α (a) dλ(g) lies in the same set, and in particular, it lies in dom (L). g Since α (E(a)) = E(a) for all g ∈ G, we conclude that E(a) ∈ R1A, by the assumption of our step. g Let a ∈ sa (A) such that α (a) = a for all g ∈ G, so ka−E(a)kA = 0. By density of dom (L), there exists (an)n∈N in dom (L) converging for k · kA to a. g h By continuity, the function g ∈ G 7→ kα (an) − α (a)kA is pointwise convergent to 0, and it is bounded. Hence by Lebesgues’ dominated convergence theorem (as λ is a bounded measure, so bounded functions are integrable), we conclude: Z E E g g k (an) − (a)kA 6 kα (an) − α (a)k dλ(g) G −−−−→n→∞ 0. E R E R N R E Hence (a) ∈ 1A since ( (an))n∈N ∈ 1A ans 1A is closed in A. As a = (a), we conclude that a ∈ R1A. Last, if a ∈ A and for all g ∈ G, we have αg(a) = a, then for all g ∈ G we have g 1 ∗ 1 ∗ g 1 ∗ 1 ∗ α 2 (a + a ) = 2 (a + a ) and α ( 2i (a − a )) = 2i (a − a ), so: 1 1 1 1 αg(a) = αg (a + a∗) + (a − a∗) = (a + a∗) + (a − a∗) = a. 2 2i 2 2i This concludes our proof of our first step.

g Step 3. If a ∈ dom (L) and for all g ∈ G, we have α (a) = a, then a ∈ R1A. Let now a ∈ dom (L) such that αg(a) = a for all g ∈ G. Our goal is to prove R a ∈ 1A. Without loss of generality, we shall assume that L(a) 6 1. Let ε > 0. As K is continuous and K(0) = 0, there exists δ > 0 such that if R ε δ t ∈ , |t| < δ then K(t) < 7 . Let F ⊆ G be a ﬁnite, 3 -dense subset of G. N δ Let N0 ∈ such that for all n > N0, we have n < 3 . Note that by assumption, κn(F ) is δ-dense: if g ∈ Gn then $(g) ∈ G, so there exists h ∈ F such that δ δG(g, h) < 3 . Now:

δGn (g, κn(h)) 6 δGn (g, κn ◦ $n(g)) + δGn (κn ◦ $n(g), κn(h)) 6 εn + (δG($n(g), h) + εn) δ 2ε + 6 n 3 < δ. N ε Let N ∈ such that, for all n > N, we have χ (τn) < 7 . N For each g ∈ F , let Ng ∈ such that if n > Ng and c ∈ fn,g (a|1, 1) then g ε N kc − α (a)kA < 7 . Let M = max{N,Ng : g ∈ F } ∈ . Let n > M. 40 FRÉDÉRIC LATRÉMOLIÈRE Let b ∈ t (a|1). For all g ∈ G, let c ∈ t ακn(g)(b) 1 . Note that n τn n,g γn n cn,g ∈ fn,g (a|1, 1). Thus by Proposition (1.26):

κn(g) bn − αn (bn) 6 2χ (τn) · 1 + ka − cn,gkA An ε 2 + ka − αg(a)k + kαg(a) − c k 6 7 A n,g A 3ε < , 7 g since ka − α (a)kA = 0.

Let now g ∈ Gn, and h ∈ κn(F ) ⊆ Gn such that δGn (g, h) < δ. We estimate: g h h g kbn − αn(bn)kAn 6 kbn − αn(bn)kAn + kαn(bn) − αn(bn)kAn 3ε + K(δ (g, h)) 6 7 Gn 4ε . 6 7 Hence (as λn(Gn) = 1): Z g kbn − En(bn)k = bn − α (bn) dλn(g) An n Gn An Z g = (bn − αn(bn)) dλn(g) G (3.4) n An Z g 6 kbn − αn(bn)kAn dλn(g) Gn 4ε . 6 7 E N To ease notations, let n(bn) = tn1An for all n ∈ . The sequence (tn)n∈N is bounded since, for all n ∈ N: E |tn| = k n(bn)kAn 6 kbnkAn 6 χ (τn) + kakA 6 1 + kakA by Proposition (1.26).

Let f : N → N be strictly increasing so that (tf(n))n∈N converges to some t ∈ R. 0 N 0 ε Let N ∈ such that, for n > N , we have |t − tf(n)| < 7 . R Observe that by construction, x1An ∈ tτn (x1A|l) for any x ∈ , l > 0 and n ∈ N. 0 Let n > max{M,N }. We compute: ka − t1AkA 6 ka − tf(n)1AkA + |tf(n) − t| ε < ka − t 1 k + f(n) A A 7 2ε ε + kb − t 1 k + by Proposition (1.26), 6 7 f(n) f(n) Af(n) Af(n) 7 3ε = kb − E (b )k + f(n) f(n) f(n) Af(n) 7 4ε 3ε + = ε by Eq. (3.4). 6 7 7 Hence a ∈ R1A as desired. This concludes our proof. SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 41

We now apply Theorem (3.14) to determining certain closures of quasi-Leibniz quantum compact metric spaces under the propinquity. Corollary 3.15. Let G be a compact group, of unit e ∈ G, endowed with a continuous length function ` such that: ∀g, h ∈ G `(hgh−1) = `(g). Let G be a collection of closed subgroups of G, closed for the Hausdorﬀ distance Haus` induced by `. Let C be the class of all quasi-Leibniz quantum compact metric spaces (A, L) such that: (1) there exists H ∈ G and there exists a strongly continuous ergodic action α of H on A, (2) for all a ∈ sa (A): ka − αg(a)k L(a) sup A : g ∈ G \{e} , > `(g) (3) for all g ∈ G, the automorphism αg is 1-Lipschitz. The class C is not empty and closed for the dual propinquity.

g Proof. Let (A, L) ∈ C. For all g ∈ G and a ∈ dom (L), note that ka − α (a)k 6 g h −1 `(g)L(a) by construction; thus kα (a)−α (a)kA 6 L(a)`(gh ). Denoting δ(g, h) = `(gh−1) for all g, h ∈ G, we thus have: g h α (a) − α (a) A 6 L(a)δ(g, h). By Proposition (3.13), the automorphisms αg are full quantum isometries for L for all g ∈ G. Moreover, we note in passing that by construction: ( ) αg(αh(a)) − αh(a) L(αh(a)) = sup A : g ∈ G \{e} `(g)

−1  h h gh   α α (a) − a  = sup A : g ∈ G \{e}  `(g) 

 −1  αh gh(a) − a  A  sup −1 : g ∈ G \{e}  `(h gh)  = L(a), so the class C is not empty. (There exists an irreducible representation of G on n g g g ∗ some C as G is compact. Therefore, setting α : M ∈ Mn 7→ U M (U ) for all g ∈ G deﬁnes an ergodic action of G on a full matrix algebra; as seen, (Mn, a 7→ n g o kα (a)−akMn sup `(g) : g ∈ G \{e} ) lies in C). Let now (An, Ln)n∈N be any sequence in C converging to some (B, LB) for ΛT . For each n ∈ N, there exists by assumption on C, a group Hn ∈ G , and an ergodic action αn of Hn on An. Now, G is closed for Haus`, so it compact (as G is compact). Thus there exists a subsequence (Hf(n))n∈N converging to some closed subgroup H ∈ G of G. By Proposition (3.8), the sequence (Hf(n))n∈N converges to H for Υ as well. 42 FRÉDÉRIC LATRÉMOLIÈRE

Thus, (Af(n), Lf(n))n∈N meets the hypothesis of Theorem (3.14). Therefore, there exists an ergodic action β of G on (B, LB) by full quantum isometries, such g that for all a ∈ sa (B) and g ∈ G, we have ka − β (a)kB 6 `(g)LB(a). Thus (B, LB) ∈ C, as desired. In particular, we see that: Corollary 3.16. Let ` be a continuous length function on the d-torus Td. Let C be the class quasi-Leibniz quantum compact metric spaces (A, L) such that: (1) A is finite dimensional, (2) there exists an ergodic action α of Td on A, (3) for all a ∈ sa (A), we have: ka − αg(a)k L(a) = sup A : g ∈ Td \{(1,..., 1)} . `(g) All the infinite dimensional quasi-Leibniz quantum compact metric spaces in the closure of C are quantum tori. Proof. Quantum tori are the only (infinite dimensional) C*-algebras carrying an d ergodic action of the torus T [21]. An interesting direction for future research exploits Theorem (3.14) as follows. Let G be a compact Lie group. In general, it is a delicate problem to determine all C*-algebras on which G acts ergodically and classify them. However, we now propose an approach to this question. Indeed, G acts ergodically on many matrix algebras. Theorem (3.14) suggests that limits of such finite dimensional C*-algebras, endowed with the natural L-seminorm from the ergodic action of G and some fixed choice of length function on G [23], will carry ergodic actions of G. We make two observations. First, we conjecture that in fact, all these C*-algebras will lie in the closure of the class of these finite dimensional C*-algebras, for the propinquity. Sec- ond, we believe that as finite dimensional actions are of course tightly related to the representations of G (or projective representations, more generally), the question of finding all C*-algebras on which G acts ergodically will amount to understand- ing under what conditions these actions give rise to converging sequences for the propinquity. A natural group to start exploring the application of our suggested technique is SU(2). Rieffel proved in [30] that all C*-algebras of continuous functions on co-adjoint orbits of SU(2) are obtained as limits, for the propinquity, of full matrix algebras carrying the adjoint action of SU(2) induced by well-chosen irreducible representations of SU(2). More generally, it would be natural to ask whether one can obtain all possible C*-algebras of continuous sections of the sort of matrix- bundles over SU(2)-homogeneous spaces, which are all the examples of C*-algebras with an ergodic action of SU(2) by [32]. As a prelude to this effort, we offer the following result. Note that in general, the closure of finite quasi-Leibniz quantum compact metric spaces contain all nuclear, quasi-diagonal quasi-Leibniz quantum compact metric space, and in particular, it contains all the quantum tori, including the non-type I variety [10]. It is very unclear what conditions in general are needed on a class of quantum metrics spaces built over type I C*-algebras to ensure that the closure of this class for the propinquity consists only of type I C*-algebras. Bringing together our present work with the powerful results in [32], we however obtain: SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 43

Corollary 3.17. Let ` be a continuous length function over SU(2). If C is the class of all ﬁnite dimensional quasi-Leibniz quantum compact metric spaces (A, L) such that: (1) there exists a strongly ergodic action α of SU(2) on A by *-automorphisms, (2) for all a ∈ sa (A): ka − αg(a)k L(a) = sup A : g ∈ SU(2), g 6= I `(g) then the closure of C consists of quasi-Leibniz quantum compact metric spaces of the form (A, L) where A is type I. Proof. This results from Corollary (3.16) and [32], whose corollary is that a C*- algebra admits an ergodic action of SU(2) only if it is type I.

Another application of Theorem (2.13) involves inductive limits of groups. No- tably, the following result does not require that the inductive limit be locally compact. This is a step in a possibly new approach to construct actions of non-locally compact groups on C*-algebras. An inductive sequence of groups, for our purpose, is given by a sequence of groups (Gn)n∈N together with group monomorphisms θn : Gn → Gn+1. We will assume all the groups Gn to be topological groups with a separable, ﬁrst countable topology — hence, sequences will prove suﬃcient to test continuity. The inductive limit group G = lim (G , θ ) is the quotient of Q G by −→n→∞ n n n∈N n the equivalence relation: 0 N 0 (gn)n∈N ≡ (gn)n∈N ⇐⇒ ∃N ∈ ∀n > N θn(gn) = θn(gn).

Let N ∈ N. We define, for g ∈ GN , the element ηN (g) as the equivalence class in G of:   identity of G if n < N,  n    g if n = N,  θ (g) if n > N. n n∈N Thus defined, the maps ηn are group monomorphisms for all n ∈ N. We hence- N S forth identify Gn with ηn(Gn) in G for all n ∈ . Thus G = n∈N Gn with Gn ⊆ Gn+1 for all n ∈ N. The inductive topology on G is by definition, the final topology for the maps θn. With the above identification, it becomes the largest topology which makes, for all n ∈ N, the inclusion of Gn in G continuous. In particular, U ⊆ G is open in the inductive topology if and only if for all N ∈ N, the set U ∩ GN is open in GN . In our next theorem, we work directly with inductive limits seen as increasing unions of groups with the inductive space topology. As a side note, this topology does not turn G to a topological group in general. Theorem 3.18. Let F be an admissible function, C a nonempty class of F –quasi- Leibniz quantum compact metric spaces and T an appropriate class of tunnels for C. Let G be a group with unit e and (Gn)n∈N an increasing sequence, for inclusion, S N of subgroups of G such that G = n∈N Gn. We assume that for all n ∈ , the group Gn is a separable first-countable topological group and that the inclusion map Gn ,→ Gn+1 is continuous. We endow G with the inductive limit topology. 44 FRÉDÉRIC LATRÉMOLIÈRE

Let (An, Ln)n∈N be a sequence in C, and for each n ∈ N, let αn be an action of Gn by *-automorphisms of An. Let D : G → [0, ∞) be a continuous function such that for all n ∈ N and for all g ∈ Gn: g Ln ◦ αn 6 D(g)Ln. Let K : G × G → [0, ∞) be continuous, such that K(g, g) = 0 for all g ∈ G and: h g ∀n ∈ N, g, h ∈ Gn, a ∈ dom (Ln) α (a) − α (a) Ln(a)K(h, g). n n An 6 If (A, L) ∈ C satisﬁes: ∗ lim ΛT ((An, Ln), (A, L)) = 0, n→∞ then there exists an strongly continuous action α of G on A such that: g (1) for all g ∈ G, we have L ◦ α 6 D(g)L, (2) for all g, h ∈ G, and a ∈ dom (L), we have: g h α (a) − α (a) A 6 L(a)K(δG(g, h)), (3) the following inequalities hold for all g ∈ G, and for all N ∈ N such that g ∈ GN : g g g lim inf mk`Ln (αn) mkL(α ) lim sup mk`Ln (αn), n→∞ 6 6 n→∞ n N > n>N

Proof. For any n ∈ N, the group Gn is assume separable, so it admits a dense countable subgroup Hn. If U is nonempty open in G then for some N ∈ N, we have U ∩GN is open and not empty, and thus HN ∩U is not empty as well. Consequently, S S H = n∈N Hn is a dense subgroup n∈N Gn. Notably, H is countable. N g g g Let g ∈ H. For all n ∈ , if g ∈ Hn then write ϕn = αn; otherwise set ϕn to be the identity of An. 0 N 0 If g, g ∈ H, there exists N ∈ such that g, g ∈ GN . For n > N we then have: g g0 gg0 g g0 gg0 ϕn ◦ ϕn − ϕn = αn ◦ αn − αn = 0 An e and thus, we meet Theorem (2.13) hypothesis (noting that mk`Ln (αn) = 0 for all n ∈ N and αn are actions by *-automorphisms). We thus conclude that there exists an action α of H on A by *-automorphisms such that in particular: h g α (a) − α (a) A 6 L(a)K(h, g), reasoning in the same manner as in the proof of Theorem (3.10). Let now g ∈ G, and N ∈ N such that g ∈ GN . Since HN is dense in GN , there exists a sequence (gk)k∈N in HN such that limk→∞ gk = g. Thus in particular, if a ∈ dom (L) then: p,q→∞ gp gq kα (a) − α (a)kA 6 L(a)K(gp, gq) −−−−−→ L(a)K(g, g) = 0

gk and thus (α (a))k∈N is Cauchy in sa (A), which is complete. It thus converges. 0 Note that moreover, if we had chosen N ∈ N for which g ∈ GN 0 and some other sequence (hk)k∈N converging in GN 0 to g, then, without loss of generality, 0 we may assume N > N; the sequence (zk)k∈N obtained by interleaving (gk)k∈N z and (hk)k∈N converges to g in GN 0 and thus as above, (α k (a))k∈N must converge g h for all a ∈ dom (L); thus both subsequences (α k (a))k∈N and (α k (a))k∈N have the same limit. We denote this limit by αg(a). SEMIGROUPOID ACTIONS AND THE GROMOV-HAUSDORFF PROPINQUITY 45

0 0 A simple argument shows that αg is a *-morphism and αg ◦ αg (a) = αgg (a) for all a ∈ A, so α is an algebraic action of G on A by *-automorphisms. By construction, we then note that for all a ∈ dom (L), the following inequality holds:

g h gk hk α (a) − α (a) = lim kα (a) − α (a)kA A k→∞ 6 lim sup L(a)K(gk, hk) = L(a)K(g, h). k→∞ 0 0 ε Let a ∈ sa (A). Let ε > 0. There exists a ∈ dom (L) such that ka − a kA < 3 . Since K is continuous, there exists an open subset U of G such that e ∈ U and ε for all g ∈ U we have K(g, e) < 3(diam(A,L)L(a0)+1) . We then have for all g ∈ U: g g e kα (a) − akA = kα (a) − α (a)kA g 0 g 0 e 0 e 0 6 kα (a − a )kA + kα (a ) − α (a )kA + kα (a − a )kA ε 2ka − a0k + ε. 6 A 3 6 We conclude that we have proven that for all a ∈ sa (A): lim kαg(a) − ak = 0 g→e A and thus it follows immediately that α is strongly continuous. Now, the same general techniques used in the proof of Theorem (3.14) applies to complete the theorem — with K is now a function over G × G.

References 1. K. Aguilar and F. Latrémolière, Quantum ultrametrics on af algebras and the Gromov– Hausdorff propinquity, Studia Mathematica 231 (2015), no. 2, 149–194, ArXiv: 1511.07114. 2. D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Texts in Math- ematics, vol. 33, American Mathematical Society, 2001. 3. M. Christ and M. A. Rieffel, Nilpotent group C∗-algebras-algebras as compact quantum metric spaces, Submitted (2015), 22 pages, ArXiv: 1508.00980. 4. A. Connes, Compact metric spaces, Fredholm modules and hyperfiniteness, Ergodic Theory and Dynamical Systems 9 (1989), no. 2, 207–220. 5. D. Edwards, The structure of superspace, Studies in Topology (1975), 121–133. 6. M. Gromov, Groups of polynomial growth and expanding maps, Publications mathématiques de l’ I. H. E. S. 53 (1981), 53–78. 7. F. Hausdorff, Grundzüge der Mengenlehre, Verlag Von Veit und Comp., 1914. 8. D. Kerr, Matricial quantum Gromov-Hausdorff distance, J. Funct. Anal. 205 (2003), no. 1, 132–167, math.OA/0207282. 9. N. Landsman, Mathematical topics between classical and quantum mechanics, Springer Mono- graphs in Mathematics, Springer-Verlag, 1998. 10. F. Latrémolière, Convergence of fuzzy tori and quantum tori for the quantum Gromov– Hausdorff Propinquity: an explicit approach., Münster Journal of Mathematics 8 (2015), no. 1, 57–98, ArXiv: math/1312.0069. 11. , Curved noncommutative tori as Leibniz compact quantum metric spaces, Journal of Math. Phys. 56 (2015), no. 12, 123503, 16 pages, ArXiv: 1507.08771. 12. , The dual Gromov–Hausdorff Propinquity, Journal de Mathématiques Pures et Ap- pliquées 103 (2015), no. 2, 303–351, ArXiv: 1311.0104. 13. , Equivalence of quantum metrics with a common domain, Journal of Mathematical Analysis and Applications 443 (2016), 1179–1195, ArXiv: 1604.00755. 14. F. Latrémolière, The modular Gromov–Hausdorff propinquity, Submitted (2016), 67 pages, ArXiv: 1608.04881. 46 FRÉDÉRIC LATRÉMOLIÈRE

15. F. Latrémolière, The Quantum Gromov-Hausdorff Propinquity, Trans. Amer. Math. Soc. 368 (2016), no. 1, 365–411, electronically published on May 22, 2015, http://dx.doi.org/10.1090/tran/6334, ArXiv: 1302.4058. 16. , A compactness theorem for the dual Gromov-Hausdorff propinquity, Indiana Univer- sity Journal of Mathematics 66 (2017), no. 5, 1707–1753, ArXiv: 1501.06121. 17. F. Latrémolière, Heisenberg modules over quantum 2-tori are metrized quantum vector bundles, Submitted (2017), 38 pages, ArXiv: 1703.07073. 18. F. Latrémolière, The triangle inequality and the dual Gromov-Hausdorff propinquity, Indiana University Journal of Mathematics 66 (2017), no. 1, 297–313, ArXiv: 1404.6633. 19. F. Latrémolière and J. Packer, Noncommutative solenoids and the gromov-hausdorff propinquity, Proc. Amer 145 (2017), no. 5, 1179–1195, ArXiv: 1601.02707. 20. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 825–920. 21. P. Milnes and S. Walters, C∗-algebras characterized by ergodic actions of the n-torus, Bulletin of the London Mathematical Society 32 (2000), no. 4, 465–470. 22. N. Ozawa and M. A. Rieffel, Hyperbolic group C∗-algebras and free product C∗-algebras as compact quantum metric spaces, Canad. J. Math. 57 (2005), 1056–1079, ArXiv: math/0302310. 23. M. A. Rieffel, Metrics on states from actions of compact groups, Documenta Mathematica 3 (1998), 215–229, math.OA/9807084. 24. , Metrics on state spaces, Documenta Math. 4 (1999), 559–600, math.OA/9906151. 25. , Group C∗-algebras as compact quantum metric spaces, Documenta Mathematica 7 (2002), 605–651, ArXiv: math/0205195. 26. , Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance, Mem. Amer. Math. Soc. 168 (2004), no. 796, 67–91, math.OA/0108005. 27. , Distances between matrix alegbras that converge to coadjoint orbits, Proc. Sympos. Pure Math. 81 (2010), 173–180, ArXiv: 0910.1968. 28. , Leibniz seminorms for "matrix algebras converge to the sphere", Clay Math. Proc. 11 (2010), 543–578, ArXiv: 0707.3229. 29. , Leibniz seminorms and best approximation from C∗-subalgebras, Sci China Math 54 (2011), no. 11, 2259–2274, ArXiv: 1008.3773. 30. , Matricial bridges for "matrix algebras converge to the sphere", Submitted (2015), 31 pages, ArXiv: 1502.00329. 31. , Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (March 2004), no. 796, math.OA/0011063. 32. A. Wassermann, Ergodic actions of compact groups on operator algebras. III. classification for SU(2), Inventiones Math 93 (1988), no. 2, 309–354. 33. N. Weaver, Lipschitz algebras, World Scientific, 1999.

E-mail address: [email protected] URL: http://www.math.du.edu/~frederic

Department of Mathematics, University of Denver, Denver CO 80208