Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero between them with respect to the modular Gromov-Hausdorff propinquity.


Introduction
This paper was inspired by an apparent connection between Jonathan Rosenberg's work on Riemannian metrics on a generically transcendental quantum torus and Levi-Civita connections for these metrics [14], and Frédéric Latrémolière's work on metrized quantum vector bundles and the modular Gromov-Hausdorff propinquity [6].
The subject of the Gromov-Hausdorff propinquity has its origin in Marc Rieffel's observation [12] that in certain papers on theoretical physics, statements can be found regarding the convergence of a sequence of operator algebras to an operator algebra. He deduced that the bookkeeping device used by the authors of these papers to prove convergence can be described as a metric structure on unital C * -algebras. Seeing that the Gromov-Hausdorff distance for compact metric spaces enables us to discuss the convergence of a sequence of compact metric spaces to a compact metric space, he defined for the class of order-unit spaces endowed with a special metric structure (called the compact quantum metric spaces) an analogous distance called the quantum Gromov-Hausdorff distance. The relation between order-unit spaces and C * -algebras is made clear when one knows that the space of self-adjoint elements of a unital C * -algebra is an order-unit space.
The quantum Gromov-Hausdorff distance suffers from some deficiencies. Designed for orderunit spaces, it does not incorporate the multiplicative structure of a C * -algebra. Also, it was unknown if distance zero between two C * -algebras necessarily means that they are * -isomorphic. These problems were settled when Latrémolière defined, in [4], the quantum Gromov-Hausdorff propinquity for the class of unital C * -algebras endowed with a special metric structure (called the quantum compact metric spaces).
Recently, Latrémolière was able to generalize the Gromov-Hausdorff propinquity to Hilbert C * -modules over a quantum compact metric space endowed with a special metric structure [6]. He calls these objects metrized quantum vector bundles, regarding them to be a noncommutative generalization of vector bundles over a compact Riemannian manifold endowed with a metric. The modular Gromov-Hausdorff propinquity then allows us to formalize the concept of convergence for metrized quantum vector bundles. Distance zero between metrized quantum vector bundles is equivalent to the existence of an isomorphism between them, in terms of their Hilbert-C * -module structures and their metric structures.
In Section 3, we will define Riemannian metrics on generically transcendental quantum tori and provide a brief overview of Rosenberg's work on Levi-Civita connections for these metrics.
In Section 4, we will define Latrémolière's quantum compact metric spaces and metrized quantum vector bundles. We will then show how to build a metrized quantum vector bundle from a Riemannian metric on a generically transcendental quantum torus and the Levi-Civita connection for the metric.
In Section 5, we will prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero between them with respect to the modular Gromov-Hausdorff propinquity.
In Section 6, we will pose some open questions that serve as the basis for future work in this area.

Preliminaries
This section serves to standardize notation and conventions.
Let N denote the set of positive integers, and for each m ∈ N, let [m] df = N ≤m . Throughout this paper, fix n ∈ N as well as a generically transcendental 1 skew-adjoint (n × n)-matrix Θ having entries in C.
Fix also an arbitrary norm N on R n . Of particular physical importance is the Euclidean norm on R n .
Let A Θ denote the n-dimensional quantum torus corresponding to Θ, which is the universal C * -algebra generated by n unitary elements u 1 , . . . , u n satisfying the relation u k u j = e 2πiΘ jk u j u k for all j, k ∈ [n].
Let α denote the canonical action of T n on A Θ given by α t (u j ) = t j · u j for all t ∈ T n and j ∈ [n], where t = (t j ) j∈ [n] . Then let ∂ 1 , . . . , ∂ n denote the coordinate directional-derivative operators associated to α.
Let A ∞ Θ denote the * -subalgebra of smooth elements of A Θ for α, which we refer to as the n-dimensional smooth quantum torus corresponding to Θ. Then let D Θ denote the R-vector space of * -derivations on A ∞ Θ . Given a vector space V , a seminorm L on V , and r ∈ R >0 , let Given a * -algebra A, let its set of self-adjoint elements and its set of skew-adjoint elements be denoted by A sa and A ska respectively, and the * -algebra of (n × n)-matrices with entries in A be denoted by M n (A).
Given a C * -algebra A, let its state space be denoted by S (A).

Generically transcendental quantum tori as noncommutative Riemannian manifolds
This section gives a brief overview of Riemannian metrics on a generically transcendental quantum torus and Rosenberg's Levi-Civita connections for these metrics, as defined in [14]. Throughout this section, A denotes a unital C * -algebra.
Definition 3.1. Let χ(A) denote the free left A-module with rank n whose underlying C-vector space is A n , where the left action • of a ∈ A on X ∈ χ(A) is given by left multiplication of each component of X by a. For every j ∈ [n], let e j denote the element of χ(A) that has 1 A in the j-th component and 0 A 's elsewhere. Define the standard A-valued inner product ·| · st and its associated (standard) norm · st on χ(A) by ∀ a 1 , . . . , a n , b 1 , . . . , b n ∈ A : We can define a C * -algebraic norm · Mn(A) on M n (A) by g Mn(A) df = T g L(χ(A), ·|· st ) for all g ∈ M n (A). Henceforth, we will view M n (A) as a C * -algebra.
Then the following statements hold: 1) χ(A), ·| · g is a left Hilbert A-module (we denote the associated norm on χ(A) by · g ), 3) e j | e k g = g jk for all j, k ∈ [n].
Proof . By hypothesis, g has an invertible positive square root in M n (A), so We can thus see that ·| · g is a sesquilinear form. Furthermore, As T √ g is A-linear, we find that Next, we have for all X ∈ χ(A) that and if X| X g = 0 A , then T √ g (X) | T √ g (X) st = 0 A , so T √ g (X) = 0 χ(A) and thus X = 0 χ(A) . Continuing, so if (X k ) k∈N is a Cauchy sequence in χ(A) with respect to · g , then T √ g (X k ) k∈N is a Cauchy sequence in χ(A) with respect to · st that has a · st -limit X , as (χ(A), · st ) is a complete metric space. Hence, Therefore, χ(A), · g is a complete metric space, which establishes (1) and (2). Finally, (3) follows from the definition of ·| · g .
The following lemma will be needed in Section 4.
so · g and · h are equivalent norms on χ(A). In particular, Proof . Observe that Interchanging g and h in the relations above, we get as required. Finally, as · st = · I , where I is the identity of M n (A), the second part comes from letting g = I.
It was established in [2] that r j · ∂ j + ad a r 1 , . . . , r n ∈ R and a ∈ ( where the hypothesis that Θ is generically transcendental plays a role. For any a, b ∈ A Θ , we have ad a = ad b if and only if a − b ∈ C · 1 A Θ , so if τ denotes the faithful tracial state on A Θ , then (3.1) can be rewritten as Now, for every a ∈ (A Θ ) ska , the following observations can be made: • as a is skew-adjoint, we have a = ib for some self-adjoint b ∈ A Θ , so Hence, a − τ (a) · 1 A Θ is skew-adjoint as well.
By [14,Theorem 1.4], the decomposition of every element of D Θ into an R-linear combination of the ∂ j 's and an inner * -derivation is unique, so the observations above tell us that D Θ can be linearly parametrized by the normed R-vector space R n × (A ∞ Θ ) ska ∩ ker(τ ) , whose norm · we have chosen to be defined by ∀ r 1 , . . . , r n ∈ R, ∀ a ∈ (A ∞ Θ ) ska ∩ ker(τ ) : (r 1 , . . . , r n ; a) df = N (r 1 , . . . , r n ) + a A Θ .
We may then transfer · to an R-vector-space norm · D Θ on D Θ in an obvious way.
For the rest of this paper, we will let χ Θ df = χ(A Θ ) and satisfying the following four properties: 1) ∇ is R-linear in the first variable and C-linear in the second, If ∇ further satisfies the following two properties, then we call it a Levi-Civita connection for g: for all j, k ∈ [n] (this property is known as the torsion-freeness of ∇ (we will not need this property in our proofs), for all δ ∈ D Θ and X, Y ∈ χ ∞ Θ (this property is known as the compatibility of ∇ with g).
We now state, for the class of all generically transcendental quantum tori, a noncommutative version of the fundamental theorem of Riemannian geometry.
Theorem 3.6 ( [14]). For every Riemannian metric g, there exists a unique Levi-Civita connection ∇ g for g, which necessarily satisfies the following identity 2 :

Metrized quantum vector bundles over generically transcendental quantum tori
We will first define quantum compact metric spaces and metrized quantum vector bundles. We will then show how to build metrized quantum vector bundles over a generically transcendental quantum torus from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. • F is non-decreasing in each argument, If F (w, x, y, z) = wz + xy for all w, x, y, z ∈ [0, ∞), then we say that it is Leibniz.
metrizes the weak- * topology on S (A), If F is implicitly understood, then we simply call (A, L) a quasi-Leibniz quantum compact metric space, and if F is Leibniz, then we simply call (A, L) a Leibniz quantum compact metric space. The seminorm L is called a Lipschitz seminorm, and it is what endows A with its special metric structure.
Example 4.3. The following classes of C * -algebras can be endowed with Lipschitz seminorms that turn them into quasi-Leibniz quantum compact metric spaces: AF C * -algebras [1], curved noncommutative tori [3], noncommutative solenoids [8], and C * -algebras equipped with an ergodic action of a compact group [10].

Definition 4.4 ([6]
). An admissible triple is an ordered triple (F, G, H) with the following properties: • F is an admissible function, • G and H are non-decreasing in each of their arguments, • (x + y)z ≤ G(x, y, z) and 2xy ≤ H(x, y) for all x, y, z ∈ [0, ∞).
If (F, G, H) is implicitly understood, then we simply call (X, ·| · , D, A, L) a metrized quantum vector bundle.
Example 4.6. In [6], Latrémolière constructed metrized quantum vector bundles from actual vector bundles over compact Riemannian manifolds that provide motivating examples for Definition 4.5. In the same paper, he also showed that free Hilbert modules over the underlying unital C * -algebra of a quasi-Leibniz quantum compact metric space can be turned into metrized quantum vector bundles. The tools developed in [6] are then applied in [7] to prove the convergence of Heisenberg modules over quantum 2-tori with respect to the modular Gromov-Hausdorff propinquity.
Viewing T n as R n /Z n , we may define an N -dependent continuous length function on T n by Using , we may then define an N -dependent seminorm L on a dense Jordan-Lie subalgebra of (A Θ ) sa by

L. Huang
The action α of T n on A Θ is ergodic, so (A Θ , L) is a Leibniz quantum compact metric space by [10,Section 2]. Furthermore, according to [12,Proposition 8.6], we have This alternative expression for L on (A ∞ Θ ) sa will play an important role in what is to follow. As This completes the proof.
Note: For the rest of this section only, g denotes a Riemannian metric. Proof . It is clear from the definition of χ ∞ Θ that a • X ∈ χ ∞ Θ . Next, by the Leibniz rule, Hence, we have for all δ ∈ D Θ satisfying δ D Θ ≤ 1 that At the same time, it is straightforward to see that as required.
Choose r 1 , . . . , r n ∈ R satisfying N (r 1 , . . . , r n ) ≤ 1, and let δ = n j=1 r j · ∂ j . Then by the Leibniz rule, Rearranging the terms from both ends of (4.2) and then taking the norm of both sides, we obtain Hence, a j A Θ ≤ g It then follows from another application of Lemma 3.3 that As (δ(a j )) = δ( (a j )) and (δ(a j )) = δ( (a j )) for all j ∈ [n], we have Recalling that r 1 , . . . , r n ∈ R are arbitrary subject to N (r 1 , . . . , r n ) ≤ 1, an application of (4.1) yields Consequently, where the object on the right-hand side is a · A Θ -compact subset of A Θ (see [5,Remark 2.46]). Hence, as X ∈ D 1 g is arbitrary, we have shown that there exists a single · A Θ -compact subset of A Θ that contains the A Θ -coefficients of all elements of D 1 g , which implies that D 1 g is · st -precompact in χ Θ . However, · st and · g are equivalent by Lemma 3.3, so D 1 g is · g -pre-compact in χ Θ .

Definition 4.12 ([9]
). Let A be a normed C-vector space and C a circled convex subset of A. The Minkowski gauge functional associated to C is then the function p that satisfies Dom(p) = { a ∈ A| there exists an r ∈ R >0 such that a ∈ r · C}; ∀ a ∈ Dom(p) : p(a) df = inf({ r ∈ R >0 | a ∈ r · C}). Now, let D g,M denote the Minkowski gauge functional associated to D 1 g · g . Our objective is to show that χ Θ , ·| · g , D g,M , A Θ , L is an (F, G, H)-metrized quantum vector bundle, where F is Leibniz, and G and H are defined as in Propositions 4.9 and 4.10 respectively. Before proceeding further, let us first collect some facts about Minkowski gauge functionals. 3) if a ∈ Dom(L M ), then there is a sequence (b k ) k∈N in B such that -semicontinuous on (B, · B ), where · B denotes the restriction of · A to B.
We must first establish that D g,M is a norm on its domain. Let X ∈ Dom(D g,M ) satisfy D g,M (X) = 0. Then (3) of Lemma 4.13 tells us that there is a sequence (X k ) k∈N in χ ∞ Θ such that D g (X k ) → D g,M (X) = 0 and X k → X with respect to · g , but X k g ≤ D g (X k ) by construction for every k ∈ N, so X k g → 0, or equivalently, X k → 0 χ Θ with respect to · g . Therefore, X = 0 χ Θ , but as Minkowski gauge functionals are already absolutely homogeneous, subadditive and non-negative, D g,M is indeed a norm on its domain.
Let X ∈ Dom(D g,M ). Then (3) of Lemma 4.13 tells us that there is a sequence (X k ) k∈N in χ ∞ Θ such that D g (X k ) → D g,M (X) and X k → X with respect to · g , but, as before, X k g ≤ D g (X k ) by construction for every k ∈ N, so X g ≤ D g,M (X).
It remains to verify properties (6) and (7) of Definition 4.5, which we now turn our attention to.
Proposition 4.14. Let a ∈ Dom(L) and X ∈ Dom(D g,M ). Then a • X ∈ Dom(D g,M ) and Proof . By (3) of Lemma 4.13, there is a sequence (X k ) k∈N in χ ∞ Θ such that • X k → X with respect to · g .
Let µ denote the normalized Haar measure on T n . It is a well-known result of the theory of smooth vectors for a strongly continuous Lie-group action on a C * -algebra that a net (f ν ) ν∈N in C ∞ c (T n ) exists such that (a) f ν is non-negative and In fact, as f ν is non-negative for all ν ∈ N and as self-adjoint elements of a C * -algebra are preserved under * -homomorphisms, (b) can be strengthened to say that a ν ∈ (A ∞ Θ ) sa for all ν ∈ N . Proceeding, we have for all ν ∈ N and t ∈ T n \ {0} that Consequently, a ν ∈ Dom(L) and L(a ν ) ≤ L(a) for all ν ∈ N . Invoking Proposition 4.9, we get where the last inequality is due to the fact that G is non-decreasing in each argument. By (2) of Lemma 4.13, Y ∈ Dom(D g,M ) and Let > 0. As the right-hand side of this inequality converges to G a A Θ , L(a), D g,M (X) by the continuity of G, we find for all ν ∈ N and k ∈ N sufficiently large that As a ν • X k → a • X with respect to · g , it follows from the closure clause implicit in (1) of Lemma 4.13 that a • X ∈ Dom(D g,M ) and   4.15. Let X, Y ∈ Dom(D g,M ). Then X| Y g , X| Y g ∈ Dom(L) and where H : [0, ∞) 2 → [0, ∞) is defined as in Proposition 4.10.
Proof . By (3) of Lemma 4.13, there are sequences (X k ) k∈N and (Y k ) k∈N in χ ∞ Θ such that We already have from Proposition 4.10 that Let > 0. As the right-hand side of this inequality converges to H(D g,M (X), D g,M (Y )) by the continuity of H, we find for all k ∈ N sufficiently large that However, > 0 is arbitrary, so as required. Remark 4.17. We could have worked with more general C * -algebras in this paper, but here are reasons why we focus only on generically transcendental quantum tori: 1) quantum tori are a convenient source of compact quantum metric spaces, 2) the norm · D Θ on the space of * -derivations on A ∞ Θ enables us to conveniently prove the inequalities necessary to achieve the objective of this paper, 3) by Theorem 3.6, we have a single parameter for the D-norms on our metrized quantum vector bundles.

A distance-zero result for the modular Gromov-Hausdorff propinquity
In [6], Latrémolière introduced the modular Gromov-Hausdorff propinquity as a means of measuring, for an admissible triple (F, G, H), how close two (F, G, H)-metrized quantum vector bundles are to each other, in the sense of how close they are to being isomorphic, in terms of their Hilbert-C * -module structures and their metric structures. Our aim in this section is to prove the following zero-distance result.  The domain of γ, denoted by Dom(γ), is defined to be Ω 1 , and the co-domain of γ, denoted by Co-Dom(γ), is defined to be Ω 2 . We say that (Ω 1 , Ω 2 , B, b, π 1 , π 2 , N , α, β) is a modular bridge from Ω 1 to Ω 2 .
A modular bridge between two metrized quantum vector bundles yields useful numerical quantities that indicate how close the metrized quantum vector bundles are to each other. Before defining these quantities, we require a preliminary definition. 1. The bridge seminorm of γ is the seminorm bn γ on A 1 ⊕ A 2 defined by 2. The basic reach ρ (γ) of γ is the Hausdorff distance, in the seminormed space (A 1 ⊕A 2 , bn γ ), between the embedded images of Dom(L 1 ) and Dom(L 2 ), i.e., bn γ (a 1 , a 2 ), sup 3. The height of γ is the non-negative quantity ς(γ) defined by where • Haus mk L 1 denotes the mk L 1 -induced Hausdorff distance between subsets of S (A 1 ), and • Haus mk L 2 denotes the mk L 2 -induced Hausdorff distance between subsets of S (A 2 ). 4. The deck seminorm of γ is the seminorm dn γ on X 1 ⊕ X 2 defined by 5. The modular reach of γ is the non-negative quantity ρ (γ) defined by 6. The imprint of γ is the non-negative quantity (γ) defined by where • Haus k Ω 1 denotes the k Ω 1 -induced Hausdorff distance between subsets of X 1 , and • Haus k Ω 2 denotes the k Ω 2 -induced Hausdorff distance between subsets of X 2 .
Remark 5.6. Lemma 4.19 of [6] establishes that the numerical quantities defined in Definition 5.5 are finite. We now define a full quantum isometry between metrized quantum vector bundles. Definition 5.8 ([6]). Let (F, G, H) be an admissible triple. Let Ω 1 and Ω 2 be (F, G, H)-metrized quantum vector bundles. A full quantum isometry from Ω 1 to Ω 2 is then an ordered pair (L , D) with the following properties: • L is a unital * -isomorphism from A 1 to A 2 , • D is a continuous linear isomorphism (not necessarily unitary) from X 1 to X 2 , • D(a • ζ) = L (a) • D(ζ) for all a ∈ A 1 and ζ ∈ X 1 , The following theorem says that the modular Gromov-Hausdorff propinquity has the required properties to qualify as a (class) pseudometric.

L. Huang
We thus obtain~·~r ·g = √ r~·~g, so r·g . Now, we have by Definition 4.12 that as · g and · r·g are equivalent = sup X ∈ χ Θ there exists a t ∈ R >0 such that X ∈ t · D 1 r·g · r·g = Dom(D r·g,M ).
Next, observe for all X ∈ Dom(D g,M ) = Dom(D r·g,M ) that as · g and · r·g are equivalent We can now verify without difficulty that (L , D) is a full quantum isometry from Ω r to Ω s , where However, in order to give the reader an appreciation for the sophistication involved when trying to resolve more difficult propinquity problems (see the conclusion below), we will prove Theorem 5.1 directly.
Proof of Theorem 5.1. By Lemma 5.10, we may define a bijection β r,s : D 1 r·g,M → D 1 s·g,M by It follows that is an (F, G, H)-modular bridge, whose associated numerical quantities we now seek to compute. To perform these computations, first note that bn γr,s is a seminorm on A Θ ⊕ A Θ such that ∀ a 1 , a 2 ∈ A Θ : bn γr,s (a 1 , We can then make the following observations regarding the basic reach, height and imprint of γ r,s : • The basic reach of γ r,s is 0: Hence, λ(γ r,s ) = ρ (γ r,s ), i.e., the length of γ r,s equals its modular reach, which we will now prove is also 0. Observe for all X, Y ∈ D 1 r·g,M that bn γr,s X| Y r·g , β r,s (X)| β r,s (Y ) s·g = X| Y r·g − β r,s (X)| β r,s (Y ) s·g A As the argument is symmetric in X and Y , we also have for all X, Y ∈ D 1 r·g,M that bn γr,s Y | X r·g , β r,s (Y )| β r,s (X) s·g = 0, so dn γr,s Id This completes the proof.

Conclusion
The only property of Levi-Civita connections used in this paper is metric compatibility. We could have allowed our D norms to depend not only on the choice of a Riemannian metric, but also on the choice of a connection that is not necessarily torsion-free. However, this would introduce an extra degree of variability that could unnecessarily complicate the study of convergence questions in modular propinquity. Many unanswered questions remain. The four most important ones (in our opinion) are the following: • Is D g lower-semicontinuous on χ ∞ Θ for any Riemannian metric g? If this is the case, then (4) of Lemma 4.13 says that D g,M extends D g , which would render D g,M more manageable to deal with.
• Can we find a topology on M n (A Θ ) so that if h → g in the space of Riemannian metrics with respect to this topology, then Λ mod This question can be posed so straightforwardly because we consider only Levi-Civita connections.
• Can we incorporate the Riemannian curvature operator [14, Definition 3.1] into the current work?
• Although Rosenberg considers only free modules in [14], can we extend our results to nonfree modules? In [7], Latrémolière establishes a technical propinquity result for metrized quantum vector bundles whose underlying modules are non-free Heisenberg modules over quantum 2-tori, but it is not clear at this time how his result may be extended to non-free Hilbert modules over quantum tori of arbitrary dimension.