Gromov-Hausdorff convergence of state spaces for spectral truncations

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov-Hausdorff convergence of the corresponding state spaces when equipped with Connes' distance formula. We exemplify this result for spectral truncations of the circle, Fourier series on the circle with a finite number of Fourier modes, and matrix algebras that converge to the sphere.


Introduction
We continue our study of spectral truncations of (noncommutative) geometry that we started in [8] and here focus on the metric convergence aspect of so-called operator system spectral triples. This is part of a program that tries to extend the spectral approach to geometry to cases where (possibly) only part of the spectral data is available, very much in line with [9]. And even though the mathematical motivation should be sufficient, there is a clear physical motivation for this. Indeed, from experiments we will only have access to part of the spectrum since we are limited by the power and resolution of our detectors: we typically study physical phenomena up to a certain energy scale and with finite resolution.
The usual spectral approach to geometry [7] in terms of a * -algebra A of operators on H and a self-adjoint operator D on H has been adapted in [9,8] to deal with such spectral truncations. The * -algebra is replaced by an operator system E (dating back to [6]), which is by definition a * -closed subspace of B(H). More precisely, we have the following definition. Definition 1. An operator system spectral triple is a triple (E, H, D) where E is a dense subspace of an operator system E in B(H), H is a Hilbert space and D is a self-adjoint operator in H with compact resolvent and such that [D, T ] is a bounded operator for all T ∈ E.
An operator system comes with an ordering, namely, one can speak of positive operators in E ⊆ B(H). As a consequence states on E can be defined as positive linear functionals of norm 1. The above triple then induces a (generalized) distance function on the state space S(E) by setting (1) d(ϕ, ψ) = sup where · 1 denote the Lipschitz semi-norm: If E = A is a * -algebra then this reduces to the usual distance function [7] on the state space of the C * -algebra A = A . It also agrees with the definition of quantum metric spaces based on order-unit spaces given in [20,14,17].
Below we will study the properties of this metric distance function and the notions of Gromov-Hausdorff convergence it gives rise to. We consider sequences of spectral triples on operator systems and formulate general conditions under which we prove the state spaces equipped with the above distance functions to converge to a limiting state space. The latter is also described by an operator system spectral triple.
We exemplify our main result on Gromov-Hausdorff convergence by considering: • spectral truncations on the circle; • Fourier series with only a finite number of non-zero Fourier coefficients; • matrix algebras converging to the sphere.
Previous results in the literature on the distance function for spectral truncations have been reported in [9,12,11]. However, in these works the distance function on states of the truncated system was only computed after pulling back these states to the original metric geometry. Extensions of the results contained in the present paper to tori are contained in the master's thesis [4]. The convergence of matrix algebras to the sphere was studied by Rieffel in [21] while computer simulations were performed in [3]. Using the general approach below we re-establish this convergence result.
We note that other convergence results on the distance function on quantum spaces are obtained for quantum tori in [16], for coherent states on the Moyal plane in [10]. More generally, in [11] certain sets of states have been identified for which the Connes' distance formula has good convergence properties with respect to a given metric on a Riemannian manifold.

Gromov-Hausdorff convergence for operator systems
Given a sequence of operator system spectral triples (E n , H n , D n ) we want to understand when and how this approximates an operator system spectral triple (E, H, D). We will adopt the point of view of [20] and consider the convergence (in Gromov-Hausdorff distance) of the corresponding state spaces S(E n ) → S(E) equipped with the distance formula (1).

Definition 2.
Let {(E n , H n , D n )} n be a sequence of operator system spectral triples and let (E, H, D) be an operator system spectral triple. An approximate order isomorphism for this set of data is given by linear maps R n : E → E n and S n : E n → E for any n such that the following three condition hold: (1) the maps R n , S n are positive maps (2) there exist sequences γ n , γ n both converging to zero such that In other words, we use the Lipschitz semi-norms to quantify how close the positive maps R n and S n are to being each other inverse (i.e. form an order isomorphism) as n → ∞.
We will call a map between operator systems C 1 -contractive if it is contractive with respect to both the operator norm and the Lipschitz semi-norm (thus assuming that we are given two operator system spectral triples for them). Finally, we say that the pair of maps (R n , S n ) is a C 1 -approximate order isomorphism if (R n , S n ) is an approximate order isomorphism in the above sense and for which all maps R n and S n are C 1 -contractive.
Note that the positivity condition on R n , S n in particular implies that we may pull-back states as follows: Remark 3. Even though it would be more natural to consider completely positive maps R n , S n between the operators systems E n and E, this turns out not to be necessary for the proof of our main result. However, in all examples discussed below we find that E is a commutative C * -algebra so that these maps are in fact completely positive (cf.  (1) For all ϕ n , ψ n ∈ S(E n ) we have (2) For all ϕ, ψ ∈ S(E) we have Proof. Since R n is Lipschitz contractive it follows that if a 1 ≤ 1 then also R n (a) 1 ≤ 1. Hence This establishes the first inequality (also proven in [9, Proposition 3.6]).
For the second, note that for all h ∈ E n with h 1 ≤ 1 we have since ϕ n = ψ n = 1 and S n (h) 1 ≤ h 1 ≤ 1. The second claim follows similarly.
The final justification for the above definition of C 1 -approximate order isomorphism is our following, main result.
Proof. Using the idea of 'bridges' introduced in [20] we equip the state space S(E n ⊕ E) ∼ = S(E n ) S(E) with a distance function d that restricts to the distance functions d En on S(E n ) and d E on S(E), respectively. Explicitly, this distance function is given in [20,Theorem 5 In order for the last term under the maximum γ −1 n a−S n (h) to be a bridge (cf. [20,Defn 5.1]) we should check that for any a ∈ E and any δ > 0 there is an h ∈ E n such that and similarly for E n and E exchanged, that is to say, for any h ∈ E n and any δ > 0 there is an a ∈ E such that For the first case (2), this follows directly from the assumptions stated in Definition 2 as we may take h = R n (a). For the second case (3), for any h ∈ E n we may take a = S n (h) so that a 1 ≤ h 1 because R n is contractive, while then a − S n (h) = 0.
We will first show that S(E) is in an ε-neighborhood of S(E n ) with respect to the distance function d. Indeed, let ϕ ∈ S(E) and set ϕ n = ϕ • S n ∈ S(E n ). Then which goes to zero as n → ∞.
Next, we claim that with respect to d also S(E n ) is in an ε-neighborhood of S(E). Thus, take ψ n ∈ S(E n ) and define ψ = ψ n • R n . Then under the constraint that max{ h 1 , a 1 , γ −1 using that R n is a contraction and the convergence of R n • S n (h) → h.

Examples of Gromov-Hausdorff convergence
3.1. Spectral truncations of the circle converge. We will analyze a spectral truncation of the distance function on the circle, the latter being described by the spectral triple We will consider a spectral truncation defined by the orthogonal projection P = P n of rank n onto span C {e 1 , e 2 , . . . , e n } for some fixed n ≥ 1. An arbitrary element T = P f P in P C(S 1 )P can be written as the following n × n Toeplitz matrix with respect to the orthonormal basis {e k } n k=1 : The corresponding operator system P C(S 1 )P = P C ∞ (S 1 )P is called the Toeplitz operator system and is denoted by C(S 1 ) (n) ; it has been analyzed at length in [8].
An operator system spectral triple for the Toeplitz operator system is given by (C(S 1 ) (n) , P L 2 (S 1 ), P DP ).
3.1.1. Fejér kernel. Clearly, the compression f → P f P by P = P n defines a positive map R n : C(S 1 ) → C(S 1 ) (n) . As in [21, Section 2] we define S n : C(S 1 ) (n) → C(S 1 ) to be its (formal) adjoint when we equip C(S 1 ) with the L 2 -norm and C(S 1 ) (n) with the (normalized) Hilbert-Schmidt norm. Let α x denote the natural action of S 1 on C(S 1 ) (n) , and define a norm 1 vector |ψ in P L 2 (S 1 ) by |ψ = 1 √ n (e 1 + · · · + e n ) .
Moreover, we may write in terms of the Fejér kernel F n and the Fourier coefficients a k of f .
Proof. Let us first check the formula for S n (T ) by computing that Thus, S n (T ) = F n * f when T = P f P and we may use elementary Fourier theory to derive On the other hand, we have 3.1.2. The circle as a limit of its spectral truncations. Let us now show in a series of Lemma's that the conditions of Definition 2 are satisfied.
Proof. Since R n (f ) = P f P and P commutes with D this follows directly since P is a projection.
Lemma 8. There exists a sequence {γ n } converging to 0 such that Proof. Again basic Fourier theory implies that The good kernel properties of the Fejér kernel imply that γ n → 0. Since this holds for any x, we may take the supremum to arrive at the desired inequality. The other inequality is even easier. for all T ∈ C(S 1 ) (n) .
Proof. Write T = P gP for g = k b k e ikx . Then in terms of the Schur product with T n and T * n where Now the norm of the map A → T n A for A ∈ M n (C) coincides with T n cb (cf. [18,Chapter 8]). In [1, Theorem 1] the following estimate for this norm was derived: T n cb ≤ 1 + 1 π (1 + log(n)) .
Using a simple Python script we have computed the distance function for states on C(S 1 ) (n) of the form S * n (ev x ) for n = 3, 5, 9, where ev x is the pure state on C(S 1 ) given by evaluation at x. The optimization problem for computing the distance has been solved numerically using the standard sequential least squares programming (SLSQP) method and we claim absolutely no originality or proficiency here. We have illustrated the numerical results in Figure 2. 3.2. Fejér-Riesz operator systems converge to the circle. We consider functions on S 1 with only a finite number of non-zero Fourier coefficients, analyzed in full detail and in relation with the above spectral truncations of the circle in [8]. Therein we have defined the so-called Fejér-Riesz operator system: The elements in C * (Z) (n) are thus given by sequences with finite support of the form a = (. . . , 0, a −n+1 , a −n+2 , . . . , a −1 , a 0 , a 1 , . . . , a n−2 , a n−1 , 0, . . .) and this allows to view C * (Z) (n) as an operator subsystem of C * (Z) ∼ = C(S 1 ).
The adjoint a → a * is given by a * k = a −k and an element a ∈ C * (Z) (n) is positive iff k a k e ikx defines a positive function on S 1 .
Since this naturally is an operator subsystem of C(S 1 ) it is natural to consider the following spectral triple: Figure 2. The distance function d n (0, x) ≡ d n (0, S * n (ev x )) on the Toeplitz operator system (Proposition 11) for n = 3, 5, 9. The blue band corresponds to the lower bounds d(0, x) − 2γ n given in Proposition 4 with the constants γ n given in Lemma 8.
We will be looking for positive and contractive maps K n : C(S 1 ) → C * (Z) (n) and L n : C * (Z) (n) → C(S 1 ) satisfying the conditions of Definition 2 so that we can apply Theorem 5 to conclude Gromov-Hausdorff convergence of the corresponding state spaces.
We introduce where we recall that F n = |k|≤n−1 (1 − |k|/n)e ikx is the Fejér kernel so that K n indeed maps to C * (Z) (n) considered as an operator subsystem of C(S 1 ). The map L n is simply the linear embedding of C * (Z) (n) as an operator subsystem of C * (Z) ∼ = C(S 1 ): Positivity and contractiveness of K n for the norm and Lipschitz norm is an easy consequence of the good kernel properties of F n while for L n they are trivially satisfied.

Lemma 12.
There exists a sequence γ n converging to 0 such that Proof. Since L n • K n (f ) = F n * f the proof is analogous to that of Lemma 8.
Lemma 13. There exists a sequence γ n converging to 0 such that for all a ∈ C * (Z) (n) .
Proof. From the Fourier coefficients of the Fejér kernel we find that We will estimate the sup-norm of the function f (x) = 1 n k |k|a k e ikx by the Lipschitz norm of a. First of all, we may write f as a convolution product f = g * h where g = n−1 k=−n+1 sgn(k)e ikx and h = 1 We conclude that the pair of maps (K n , L n ) for {(C * (Z) (n) , L 2 (S 1 ), D)} n and (C ∞ (S 1 ), L 2 (S 1 ), D) forms a C 1 -approximate order isomorphism and we have Proposition 14. The sequence of state spaces {(S(C * (Z) (n) ), d n )} n converges to (S(C(S 1 )), d) in Gromov-Hausdorff distance.
We again illustrate the numerical results for the first few cases in Figure 3. As compared to the Toeplitz operator system ( Figure 2) the optimization is much more cumbersome. This is essentially due to the fact that it involves the computation of a supremum norm of a trigonometric polynomial.
Remark 15. If we recall the duality between C(S 1 ) (n) and C * (Z) (n) as operator systems from [8] it is quite surprising that both operator system spectral triples converge to the circle as n → ∞.

3.3.
Matrix algebras converge to the sphere. In [20,21] Rieffel analyzed Gromov-Hausdorff convergence for so-called quantum metric spaces. Such a space is given by a pair (A, L) of an order-unit space A and a so-called Lipschitz norm L on A. At first sight, such spaces appear to be more general than (operator system) spectral triples and the distance function they give rise to. However, as Rieffel shows in [20, Appendix 2] Dirac operators are universal in the sense that the Lipschitz norms can always be realized as norms of commutators with a self-adjoint operator D.
We will here confirm the main results of [21] which is that the matrix algebras that describe the fuzzy two-sphere converges in Gromov-Hausdorff distance to the round two-sphere. Even though for much of the analysis we may refer to [20,21] we do formulate the main results in our framework of operator system spectral triples. We will describe the round two-sphere by the following spectral triple: We write S 2 = {(x 1 , x 2 , x 3 ) ∈ R 3 : x 2 1 + x 2 2 + x 2 3 = 1} so that the following vector fields (j < k). are tangent to S 2 . Of course, these vector fields are fundamental vector fields and generate the Lie algebra su (2). Note that the normal vector field is given by x itself.
In terms of the three Pauli matrices we may then write the Dirac operator as [23] (10) Note that ( x· σ) acts as the chirality operator and makes sure that the spinor bundle on S 2 is actually non-trivial (as it should). A Dirac operator on the fuzzy sphere was introduced in [13] (see also [2]). It is based on the following spectral triple (11) (L(V n ), where V n is the n-dimensional irreducible representation of SU (2) and where L jk are generators of su(2) in the n-dimensional representation. The comparison with (10) is convincing, except for the absence of the chirality operator in the case of the fuzzy sphere. However, as shown in [2] this can be repaired for by a doubling of the representation space and a corresponding doubling constructing for the Dirac operator. For our purposes, both of these Dirac operators on the fuzzy sphere give rise to the same Lipschitz norm so we may just as well work with the Dirac operator defined in (12).
Let us now proceed to show that there is a C 1 -approximate order isomorphism (σ, σ) for the sequence of spectral triples defined in (11) and the spectral triple of (9). As a consequence, we thus rederive the main conclusion of [21,Theorem 3.2] that the fuzzy sphere converges to the two-sphere in Gromov-Hausdorff distance as n → ∞, though this time formulated in terms of the above spectral triples.
We letσ : C(S 2 ) → L(V n ) be the adjoint of the map σ when C(S 2 ) comes equipped with the L 2 -inner product and L(V n ) with the Hilbert-Schmidt inner product. There is also the following explicit expression (cf. [21,Sect.2]). Moreover, we may write the so-called Berezin transform as a convolution product where H P is a probability measure defined by H P (g) = n Tr(P α g (P )).
3.3.2. The sphere as a limit of matrix algebras. We now show in a series of Lemma's that the conditions of Definition 2 hold for R n =σ and S n = σ.
Proof. This is based on a highly non-trivial result [21, Theorem 6.1] which states that there exists a sequence {γ n } converging to 0 such that T −σ(σ T ) ≤ γ n L n (T ) for all T ∈ L(V n ), where L n is the Lipschitz norm on L(V n ) defined by L n (T ) = sup