Gromov-Hausdorff convergence of state spaces for spectral truncations
GGROMOV–HAUSDORFF CONVERGENCE OF STATE SPACESFOR SPECTRAL TRUNCATIONS
WALTER D. VAN SUIJLEKOM
Abstract.
We study the convergence aspects of the metric on spectral trunca-tions of geometry. We find general conditions on sequences of operator systemspectral triples that allows one to prove a result on Gromov–Hausdorff conver-gence of the corresponding state spaces when equipped with Connes’ distanceformula. We exemplify this result for spectral truncations of the circle, Fourierseries on the circle with a finite number of Fourier modes, and matrix algebrasthat converge to the sphere. Introduction
We continue our study of spectral truncations of (noncommutative) geometrythat we started in [8] and here focus on the metric convergence aspect of so-calledoperator system spectral triples. This is part of a program that tries to extend thespectral approach to geometry to cases where (possibly) only part of the spectraldata is available, very much in line with [9]. And even though the mathematicalmotivation 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 arelimited by the power and resolution of our detectors: we typically study physicalphenomena up to a certain energy scale and with finite resolution.The usual spectral approach to geometry [7] in terms of a ∗ -algebra A of operatorson H and a self-adjoint operator D on H has been adapted in [9, 8] to deal with suchspectral truncations. The ∗ -algebra is replaced by an operator system E (datingback to [6]), which is by definition a ∗ -closed subspace of B ( H ). More precisely, wehave the following definition. Definition 1. An operator system spectral triple is a triple ( E , H , D ) where E is adense subspace of an operator system E in B ( H ) , H is a Hilbert space and D is aself-adjoint operator in H with compact resolvent and such that [ D, T ] is a boundedoperator for all T ∈ E . An operator system comes with an ordering , namely, one can speak of positiveoperators in E ⊆ B ( H ). As a consequence states on E can be defined as positivelinear functionals of norm 1. The above triple then induces a (generalized) distancefunction on the state space S ( E ) by setting(1) d ( ϕ, ψ ) = sup x ∈E {| ϕ ( x ) − ψ ( x ) | : (cid:107) x (cid:107) ≤ } where (cid:107) · (cid:107) denote the Lipschitz semi-norm : (cid:107) x (cid:107) = (cid:107) [ D, x ] (cid:107) ; ( x ∈ E ) . Date : May 18, 2020. a r X i v : . [ m a t h . QA ] M a y WALTER D. VAN SUIJLEKOM If E = A is a ∗ -algebra then this reduces to the usual distance function [7] on thestate space of the C ∗ -algebra A = A . It also agrees with the definition of quantummetric spaces based on order-unit spaces given in [20, 14, 17].Below we will study the properties of this metric distance function and thenotions of Gromov–Hausdorff convergence it gives rise to. We consider sequencesof spectral triples on operator systems and formulate general conditions under whichwe prove the state spaces equipped with the above distance functions to converge toa limiting state space. The latter is also described by an operator system spectraltriple.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 truncationshave been reported in [9, 12, 11]. However, in these works the distance function onstates of the truncated system was only computed after pulling back these statesto the original metric geometry. Extensions of the results contained in the presentpaper 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 approachbelow we re-establish this convergence result.We note that other convergence results on the distance function on quantumspaces are obtained for quantum tori in [16], for coherent states on the Moyal planein [10]. More generally, in [11] certain sets of states have been identified for whichthe Connes’ distance formula has good convergence properties with respect to agiven metric on a Riemannian manifold.2. Gromov–Hausdorff convergence for operator systems
Given a sequence of operator system spectral triples ( E n , H n , D n ) we want tounderstand 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 triplesand let ( E , H , D ) be an operator system spectral triple. An approximate order iso-morphism 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 (cid:48) n both converging to zero such that (cid:107) S n ◦ R n ( a ) − a (cid:107) ≤ γ n (cid:107) a (cid:107) , (cid:107) R n ◦ S n ( h ) − h (cid:107) ≤ γ (cid:48) n (cid:107) h (cid:107) . In other words, we use the Lipschitz semi-norms to quantify how close the pos-itive maps R n and S n are to being each other inverse ( i.e. form an order isomor-phism) as n → ∞ . H CONVERGENCE OF STATE SPACES FOR SPECTRAL TRUNCATIONS 3
We will call a map between operator systems C -contractive if it is contractivewith respect to both the operator norm and the Lipschitz semi-norm (thus assumingthat we are given two operator system spectral triples for them). Finally, we saythat the pair of maps ( R n , S n ) is a C -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 -contractive.Note that the positivity condition on R n , S n in particular implies that we maypull-back states as follows: R ∗ n : S ( E n ) → S ( E ); ϕ n (cid:55)→ ϕ n ◦ R n ,S ∗ n : S ( E ) → S ( E n ); ϕ (cid:55)→ ϕ ◦ S n . Remark 3.
Even though it would be more natural to consider completely positivemaps R n , S n between the operators systems E n and E , this turns out not to benecessary for the proof of our main result. However, in all examples discussedbelow we find that E is a commutative C ∗ -algebra so that these maps are in factcompletely positive ( cf. [18, Theorems 3.9 and 3.11] ). Let us denote the distance functions (1) for ( E n , H n , D n ) and ( E , H , D ) by d E n and d E , respectively. Proposition 4. If ( R n , S n ) is a C -approximate order isomorphism for ( E n , H n , D n ) and ( E , H , D ) , then(1) For all ϕ n , ψ n ∈ S ( E n ) we have d E ( ϕ n ◦ R n , ψ n ◦ R n ) ≤ d E n ( ϕ n , ψ n ) ≤ d E ( ϕ n ◦ R n , ψ n ◦ R n ) + 2 γ (cid:48) n . (2) For all ϕ, ψ ∈ S ( E ) we have d E n ( ϕ ◦ S n , ψ ◦ S n ) ≤ d E ( ϕ, ψ ) ≤ d E n ( ϕ ◦ S n , ψ ◦ S n ) + 2 γ n . Proof.
Since R n is Lipschitz contractive it follows that if (cid:107) a (cid:107) ≤ (cid:107) R n ( a ) (cid:107) ≤
1. Hencesup a ∈E {| ϕ ◦ R n ( a ) − ψ ◦ R n ( a ) | : (cid:107) a (cid:107) ≤ } ≤ sup h ∈E n {| ϕ ( h ) − ψ ( h ) | : (cid:107) h (cid:107) ≤ } . This establishes the first inequality (also proven in [9, Proposition 3.6]).For the second, note that for all h ∈ E n with (cid:107) h (cid:107) ≤ | ϕ n ( h ) − ψ n ( h ) | ≤ | ϕ n ( R n ( S n ( h ))) − ψ n ( R n ( S n ( h ))) | + | ϕ n ( h ) − ϕ n ( R n ( S n ( h ))) | + | ψ n ( h ) − ψ n ( R n ( S n ( h ))) |≤ d E ( ϕ n ◦ R n , ψ n ◦ R n ) + 2 γ (cid:48) n . since (cid:107) ϕ n (cid:107) = (cid:107) ψ n (cid:107) = 1 and (cid:107) S n ( h ) (cid:107) ≤ (cid:107) h (cid:107) ≤
1. The second claim followssimilarly. (cid:3)
The final justification for the above definition of C -approximate order isomor-phism is our following, main result. Theorem 5. If ( R n , S n ) is a C -approximate order isomorphism for ( E n , H n , D n ) and ( E , H , D ) , then the state spaces ( S ( E n ) , d E n ) converge to ( S ( E ) , d E ) in Gromov–Hausdorff distance. WALTER D. VAN SUIJLEKOM
Proof.
Using the idea of ‘bridges’ introduced in [20] we equip the state space S ( E n ⊕ E ) ∼ = S ( E n ) (cid:113) S ( E ) with a distance function (cid:101) d that restricts to the distancefunctions d E n on S ( E n ) and d E on S ( E ), respectively. Explicitly, this distancefunction is given in [20, Theorem 5.2] by (cid:101) d (Φ , Ψ) = sup ( h,a ) ∈E n ⊕E (cid:8) | Φ( h, a ) − Ψ( h, a ) | : max {(cid:107) h (cid:107) , (cid:107) a (cid:107) , γ − n (cid:107) a − S n ( h ) (cid:107)} ≤ (cid:9) for Φ , Ψ ∈ S ( E n ⊕ E ). In order for the last term under the maximum γ − n (cid:107) a − S n ( h ) (cid:107) to be a bridge ( cf. [20, Defn 5.1]) we should check that for any a ∈ E and any δ > h ∈ E n such thatmax {(cid:107) h (cid:107) , γ − n (cid:107) a − S n ( h ) (cid:107)} ≤ (cid:107) a (cid:107) + δ (2)and similarly for E n and E exchanged, that is to say, for any h ∈ E n and any δ > a ∈ E such thatmax {(cid:107) a (cid:107) , γ − n (cid:107) a − S n ( h ) (cid:107)} ≤ (cid:107) h (cid:107) + δ (3)For the first case (2), this follows directly from the assumptions stated in Definition2 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 (cid:107) a (cid:107) ≤ (cid:107) h (cid:107) because R n is contractive, while then (cid:107) a − S n ( h ) (cid:107) =0. We will first show that S ( E ) is in an ε -neighborhood of S ( E n ) with respect tothe distance function (cid:101) d . Indeed, let ϕ ∈ S ( E ) and set ϕ n = ϕ ◦ S n ∈ S ( E n ). Then(4) | (0 ⊕ ϕ )( h, a ) − ( ϕ n ⊕ h, a ) | = | ϕ ( a ) − ϕ ( S n ( h ) | ≤ (cid:107) a − S n ( h ) (cid:107) ≤ γ n which goes to zero as n → ∞ .Next, we claim that with respect to (cid:101) 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 constraintthat max {(cid:107) h (cid:107) , (cid:107) a (cid:107) , γ − n (cid:107) a − S n ( h ) (cid:107)} ≤ | ( ψ n ⊕ h, a ) − (0 , ψ )( h, a ) | = | ψ n ( h ) − ψ n ( R n ( a )) |≤ (cid:107) R n ( a ) − h (cid:107)≤ (cid:107) R n ( a ) − R n ( S n ( h )) (cid:107) + (cid:107) R n ( S n ( h )) − h (cid:107)≤ (cid:107) a − S n ( h ) (cid:107) + γ (cid:48) n (cid:107) h (cid:107) ≤ γ n + γ (cid:48) n , using that R n is a contraction and the convergence of R n ◦ S n ( h ) → h . (cid:3) Examples of Gromov–Hausdorff convergence
Spectral truncations of the circle converge.
We will analyze a spectraltruncation of the distance function on the circle, the latter being described by thespectral triple(5) (cid:18) A = C ∞ ( S ) , H = L ( S ) , D = − i ddx (cid:19) . We will consider a spectral truncation defined by the orthogonal projection P = P n of rank n onto span C { e , e , . . . , e n } for some fixed n ≥
1. An arbitrary element
H CONVERGENCE OF STATE SPACES FOR SPECTRAL TRUNCATIONS 5
Figure 1.
The Fej´er kernel F N = N sin ( Nx/ ( x/ for N = 10. T = P f P in P C ( S ) P can be written as the following n × n Toeplitz matrix withrespect to the orthonormal basis { e k } nk =1 :(6) T = a a − · · · a − n +2 a − n +1 a a a − a − n +2 ... a a . . . ... a n − . . . . . . a − a n − a n − · · · a a . The corresponding operator system
P C ( S ) P = P C ∞ ( S ) P is called the Toeplitzoperator system and is denoted by C ( S ) ( n ) ; it has been analyzed at length in [8].An operator system spectral triple for the Toeplitz operator system is given by( C ( S ) ( n ) , P L ( S ) , P DP ).3.1.1. Fej´er kernel.
Clearly, the compression f (cid:55)→ P f P by P = P n defines a posi-tive map R n : C ( S ) → C ( S ) ( n ) . As in [21, Section 2] we define S n : C ( S ) ( n ) → C ( S ) to be its (formal) adjoint when we equip C ( S ) with the L -norm and C ( S ) ( n ) with the (normalized) Hilbert–Schmidt norm. Let α x denote the nat-ural action of S on C ( S ) ( n ) , and define a norm 1 vector | ψ (cid:105) in P L ( S ) by | ψ (cid:105) = 1 √ n ( e + · · · + e n ) . Proposition 6.
The map S n : C ( S ) ( n ) → C ( S ) defined for any T ∈ C ( S ) ( n ) by S n ( T )( x ) = Tr ( | ψ (cid:105)(cid:104) ψ | α x ( T )) satisfies (cid:104) f, S n ( T ) (cid:105) L ( S ) = 1 n Tr (( R n ( f )) ∗ T ) . Moreover, we may write S n ( R n ( f ))( x ) = n − (cid:88) k = − n +1 (cid:18) − | k | n (cid:19) a k e ikx = ( F n ∗ f )( x ) in terms of the Fej´er kernel F n and the Fourier coefficients a k of f . WALTER D. VAN SUIJLEKOM
Proof.
Let us first check the formula for S n ( T ) by computing thatTr ( | ψ (cid:105)(cid:104) ψ | α x ( T )) = 1 n (cid:88) k,l T kl e i ( k − l ) x = 1 n n − (cid:88) k = − n +1 ( n − | k | ) a k e ikx Thus, S n ( T ) = F n ∗ f when T = P f P and we may use elementary Fourier theoryto derive (cid:104) g, S n ( T ) (cid:105) = (cid:104) g, F n ∗ f (cid:105) = n − (cid:88) k = − n +1 b k a k (cid:18) − | k | n (cid:19) . On the other hand, we have1 n Tr (( R n ( f )) ∗ T ) = 1 n n − (cid:88) k,l = − n +1 b k − l a k − l = 1 n n − (cid:88) k = − n +1 b k a k ( n − | k | ) . (cid:3) The circle as a limit of its spectral truncations.
Let us now show in a seriesof Lemma’s that the conditions of Definition 2 are satisfied.
Lemma 7.
For any f ∈ C ∞ ( S ) we have (cid:107) R n ( f ) (cid:107) ≤ (cid:107) f (cid:107) and (cid:107) [ D, R n ( f )] (cid:107) ≤(cid:107) [ D, f ] (cid:107) .Proof. Since R n ( f ) = P f P and P commutes with D this follows directly since P is a projection. (cid:3) Lemma 8.
There exists a sequence { γ n } converging to 0 such that (cid:107) f − S n ( R n ( f )) (cid:107) ≤ γ n (cid:107) [ D, f ] (cid:107) for all f ∈ C ∞ ( S ) .Proof. Again basic Fourier theory implies that | f ( x ) − S n ( R n ( f ))( x ) | = 12 π (cid:90) π − π F n ( y ) | f ( x ) − f ( y − x ) | dy ≤ π (cid:90) π − π F n ( y ) | y | dy · (cid:107) [ D, f ] (cid:107) =: γ n (cid:107) [ D, f ] (cid:107) . The good kernel properties of the Fej´er kernel imply that γ n → (cid:3) Lemma 9.
For any T ∈ C ( S ) ( n ) we have (cid:107) S n ( T ) (cid:107) ≤ (cid:107) T (cid:107) and (cid:107) [ D, S n ( T )] (cid:107) ≤(cid:107) [ D, T ] (cid:107) .Proof. We have | [ D, S n ( T )( x )] | = | Tr ( | ψ (cid:105)(cid:104) ψ | α x ([ D, T ])) | ≤ (cid:107)| ψ (cid:105)(cid:104) ψ |(cid:107) (cid:107) α x ([ D, T ]) (cid:107) ≤ (cid:107) [ D, T ] (cid:107) . Since this holds for any x , we may take the supremum to arrive at the desiredinequality. The other inequality is even easier. (cid:3) Lemma 10.
There exists a sequence { γ (cid:48) n } converging to 0 such that (cid:107) T − R n ( S n ( T )) (cid:107) ≤ γ (cid:48) n (cid:107) [ D, T ] (cid:107) for all T ∈ C ( S ) ( n ) . H CONVERGENCE OF STATE SPACES FOR SPECTRAL TRUNCATIONS 7
Proof.
Write T = P gP for g = (cid:80) k b k e ikx . Then T − R n ( S n ( T )) = (cid:0) b k − l (cid:1) − (cid:16) − | k − l | n b k − l (cid:17) = (cid:16) | k − l | n b k − l (cid:17) = ( T n − T ∗ n ) (cid:12) (cid:0) k − ln b k − l (cid:1) = 1 n ( T n − T ∗ n ) (cid:12) (cid:0) [ D, T ] (cid:1) in terms of the Schur product (cid:12) with T n and T ∗ n where T n = · · ·
01 1 · · · · · · . Now the norm of the map A (cid:55)→ T n (cid:12) A for A ∈ M n ( C ) coincides with (cid:107) T n (cid:107) cb (cf.[18, Chapter 8]). In [1, Theorem 1] the following estimate for this norm was derived: (cid:107) T n (cid:107) cb ≤ (cid:18) π (1 + log( n )) (cid:19) . Hence we have (cid:107) T − R n ( S n ( T )) (cid:107) ≤ n (cid:107) T n (cid:107) cb (cid:107) [ D, T ] (cid:107) ≤ γ (cid:48) n (cid:107) [ D, T ] (cid:107) where γ (cid:48) n := n (cid:0) π (1 + log( n )) (cid:1) . It is clear that γ (cid:48) n → n → ∞ . (cid:3) Thus we find that the pair of maps ( R n , S n ) for { ( C ( S ) ( n ) , P n L ( S ) , P n DP n ) } n and ( C ∞ ( S ) , L ( S ) , D ) forms a C -approximate order isomorphism. We mayconclude from Theorem 5 that Proposition 11.
The sequence of state spaces { ( S ( C ( S ) ( n ) ) , d n ) } n converges to ( S ( C ( S )) , d ) in Gromov–Hausdorff distance. Using a simple Python script we have computed the distance function for stateson C ( S ) ( n ) of the form S ∗ n (ev x ) for n = 3 , ,
9, where ev x is the pure state on C ( S )given by evaluation at x . The optimization problem for computing the distance hasbeen solved numerically using the standard sequential least squares programming (SLSQP) method and we claim absolutely no originality or proficiency here. Wehave illustrated the numerical results in Figure 2.3.2. Fej´er–Riesz operator systems converge to the circle.
We consider func-tions on S with only a finite number of non-zero Fourier coefficients, analyzed infull detail and in relation with the above spectral truncations of the circle in [8].Therein we have defined the so-called Fej´er–Riesz operator system :(7) C ∗ ( Z ) ( n ) = { a = ( a k ) k ∈ Z : supp( a ) ⊂ ( − n, n ) } . The elements in C ∗ ( Z ) ( n ) are thus given by sequences with finite support of theform a = ( . . . , , a − n +1 , a − n +2 , . . . , a − , a , a , . . . , a n − , a n − , , . . . )and this allows to view C ∗ ( Z ) ( n ) as an operator subsystem of C ∗ ( Z ) ∼ = C ( S ).The adjoint a (cid:55)→ a ∗ is given by a ∗ k = a − k and an element a ∈ C ∗ ( Z ) ( n ) is positiveiff (cid:80) k a k e ikx defines a positive function on S .Since this naturally is an operator subsystem of C ( S ) it is natural to considerthe following spectral triple: WALTER D. VAN SUIJLEKOM
Figure 2.
The distance function d n (0 , x ) ≡ d n (0 , S ∗ n (ev x )) on theToeplitz operator system (Proposition 11) for n = 3 , ,
9. Theblue band corresponds to the lower bounds d (0 , x ) − γ n given inProposition 4 with the constants γ n given in Lemma 8.(8) (cid:18) C ∗ ( Z ) ( n ) , H = L ( S ) , D = − i ddx (cid:19) . We will be looking for positive and contractive maps K n : C ( S ) → C ∗ ( Z ) ( n ) and L n : C ∗ ( Z ) ( n ) → C ( S ) satisfying the conditions of Definition 2 so that we canapply Theorem 5 to conclude Gromov–Hausdorff convergence of the correspondingstate spaces.We introduce K n : C ( S ) → C ∗ ( Z ) ( n ) f (cid:55)→ F n ∗ f where we recall that F n = (cid:80) | k |≤ n − (1 − | k | /n ) e ikx is the Fej´er kernel so that K n indeed maps to C ∗ ( Z ) ( n ) considered as an operator subsystem of C ( S ). Themap L n is simply the linear embedding of C ∗ ( Z ) ( n ) as an operator subsystem of C ∗ ( Z ) ∼ = C ( S ): L n : C ∗ ( Z ) ( n ) → C ( S )( a k ) (cid:55)→ (cid:32) x (cid:55)→ (cid:88) k a k e ikx (cid:33) . Positivity and contractiveness of K n for the norm and Lipschitz norm is an easyconsequence of the good kernel properties of F n while for L n they are triviallysatisfied. H CONVERGENCE OF STATE SPACES FOR SPECTRAL TRUNCATIONS 9
Lemma 12.
There exists a sequence γ n converging to 0 such that (cid:107) L n ◦ K n ( f ) − f (cid:107) ≤ γ n (cid:107) [ D, f ] (cid:107) for all f ∈ C ∞ ( S ) .Proof. Since L n ◦ K n ( f ) = F n ∗ f the proof is analogous to that of Lemma 8. (cid:3) Lemma 13.
There exists a sequence γ (cid:48) n converging to 0 such that (cid:107) K n ◦ L n ( a ) − a (cid:107) ≤ γ (cid:48) n (cid:107) [ D, a ] (cid:107) for all a ∈ C ∗ ( Z ) ( n ) .Proof. From the Fourier coefficients of the Fej´er kernel we find that K n ◦ L n ( a ) − a = (cid:18) − | k | n a k (cid:19) k . We will estimate the sup-norm of the function f ( x ) = n (cid:80) k | k | a k e ikx by the Lip-schitz norm of a . First of all, we may write f as a convolution product f = g ∗ h where g = (cid:80) n − k = − n +1 sgn( k ) e ikx and h = n (cid:80) n − k = − n +1 ka k e ikx = n [ D, a ]. Then (cid:107) f (cid:107) ∞ ≤ (cid:107) g (cid:107) (cid:107) h (cid:107) ∞ where (cid:107) g (cid:107) ≤ (cid:107) g (cid:107) = √ n − . We conclude that (cid:107) g ∗ h (cid:107) ∞ ≤ γ (cid:48) n (cid:107) [ D, a ] (cid:107) ∞ with γ (cid:48) n = √ n − n → n → ∞ . (cid:3) We conclude that the pair of maps ( K n , L n ) for { ( C ∗ ( Z ) ( n ) , L ( S ) , D ) } n and( C ∞ ( S ) , L ( S ) , D ) forms a C -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 )) , d ) in Gromov–Hausdorff distance. We again illustrate the numerical results for the first few cases in Figure 3. Ascompared to the Toeplitz operator system (Figure 2) the optimization is much morecumbersome. This is essentially due to the fact that it involves the computation ofa supremum norm of a trigonometric polynomial.
Remark 15.
If we recall the duality between C ( S ) ( n ) and C ∗ ( Z ) ( n ) as operatorsystems from [8] it is quite surprising that both operator system spectral triplesconverge to the circle as n → ∞ . Matrix algebras converge to the sphere.
In [20, 21] Rieffel analyzedGromov–Hausdorff convergence for so-called quantum metric spaces. Such a spaceis 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 Rieffelshows in [20, Appendix 2] Dirac operators are universal in the sense that the Lip-schitz norms can always be realized as norms of commutators with a self-adjointoperator D .We will here confirm the main results of [21] which is that the matrix algebrasthat describe the fuzzy two-sphere converges in Gromov–Hausdorff distance to theround two-sphere. Even though for much of the analysis we may refer to [20, 21] wedo formulate the main results in our framework of operator system spectral triples. Figure 3.
The distance function d n (0 , x ) ≡ d n (0 , L ∗ n ev x ) on theFej´er–Riesz operator system (Proposition 14) for n = 2 ,
3. Theblue band corresponds to the lower bounds d (0 , x ) − γ n given inProposition 4 with the constants γ N given in Lemma 12.We will describe the round two-sphere by the following spectral triple:(9) ( C ∞ ( S ) , C ⊗ L ( S ) , D S )We write S = { ( x , x , x ) ∈ R : x + x + x = 1 } so that the following vectorfields X jk = x j ∂ k − x k ∂ j ; ( j < k ) . are tangent to S . Of course, these vector fields are fundamental vector fields andgenerate the Lie algebra su (2). Note that the normal vector field is given by (cid:126)x itself.In terms of the three Pauli matrices we may then write the Dirac operator as[23](10) D S = ( (cid:126)x · (cid:126)σ ) (cid:88) j Following [21] we start by definingmaps σ : L ( V n ) → C ( S ) and ˘ σ : C ( S ) → L ( V n ). Given a projection P ∈ L ( V n ),say, on the highest-weight vector of V n , we define the Berezin symbol σ : L ( V n ) → C ( S ) by [5](13) σ ( T )( g ) ≡ σ T ( g ) := Tr( T α g ( P ))where α g is the action of g ∈ SU (2) induced by conjugation on L ( V ). Since α u ( P ) = P for all u ∈ U (1), it follows that σ ( T ) is U (1)-invariant and thus descendsto a function on SU (2) /U (1) = S . Moreover, we readily see that σ is an SU (2)-equivariant map which will turn out to be useful later.We let ˘ σ : C ( S ) → L ( V n ) be the adjoint of the map σ when C ( S ) comesequipped with the L -inner product and L ( V n ) with the Hilbert–Schmidt innerproduct. There is also the following explicit expression ( cf. [21, Sect.2]). Proposition 16. The map ˘ σ defined by ˘ σ ( f ) ≡ ˘ σ f = n (cid:82) f ( g ) α g ( P ) dg satisfies (cid:104) f, σ T (cid:105) = 1 n Tr(˘ σ f T ) . Moreover, we may write the so-called Berezin transform as a convolution product σ (˘ σ f )( g ) = ( f ∗ H P )( g ) ≡ (cid:90) f ( gh − ) H P ( h ) dh where H P is a probability measure defined by H P ( g ) = n Tr( P α g ( P )) . Proof. As in [21, Sect.2] we check the formula for ˘ σ ( f ) by computing that (cid:104) f, σ T (cid:105) = (cid:90) f ( g ) Tr( T α g ( P )) dg = Tr (cid:18)(cid:90) f ( g ) T α g ( P ) dg (cid:19) so that the result follows.For the Berezin transform we then indeed have that σ (˘ σ f )( g ) = Tr (˘ σ f α x ( P )) = Tr (cid:18) n (cid:90) f ( h ) α h ( P ) dhα g ( P ) (cid:19) = n (cid:90) f ( h ) Tr( P α h − g ( P )) dh = n (cid:90) f ( gh − ) H P ( h ) dh using also that H P ( h − ) = H P ( h ). (cid:3) Again, one readily observes that ˘ σ is an SU (2)-equivariant map. The sphere as a limit of matrix algebras. We now show in a series of Lemma’sthat the conditions of Definition 2 hold for R n = ˘ σ and S n = σ . Lemma 17. For any f ∈ C ∞ ( S ) we have (cid:107) ˘ σ f (cid:107) ≤ (cid:107) f (cid:107) and (cid:107) [ D n , ˘ σ f ] (cid:107) ≤ (cid:107) [ D S , f ] (cid:107) .Proof. The contractive property of ˘ σ is proved for instance in [15, Theorem 1.3.5]where ˘ σ is the Berezin quantization map. Then, by SU (2)-equivariance of ˘ σ wehave[ D n , ˘ σ f ] = (cid:88) j There exists a sequence { γ n } converging to 0 such that (cid:107) f − σ (˘ σ f ) (cid:107) ≤ γ n (cid:107) [ D S , f ] (cid:107) for all f ∈ C ∞ ( S ) .Proof. We exploit the expression for σ (˘ σ f ) as a convolution product from Proposi-tion 16. Indeed, | f ( g ) − σ (˘ σ f )( g ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( f ( g ) − f ( h )) H P ( h − g ) dh (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) f (cid:107) Lip (cid:90) d ( g, h ) H P ( h − g ) dh = (cid:107) f (cid:107) Lip (cid:90) d ( e, h ) H P ( h ) dh where d is the SU (2)-invariant (round) distance on SU (2) /U (1) and (cid:107) f (cid:107) Lip is thecorresponding Lipschitz seminorm of f . Since (cid:107) f (cid:107) Lip = (cid:107) [ D S , f ] (cid:107) by standardarguments [7, Sect. VI.1] and (cid:82) d ( e, h ) H P ( h ) dh → n → ∞ , the result follows. (cid:3) Lemma 19. For any T ∈ L ( V n ) we have (cid:107) σ T (cid:107) ≤ (cid:107) T (cid:107) and (cid:107) [ D, S T ] (cid:107) ≤ (cid:107) [ D, T ] (cid:107) .Proof. The map σ is a contraction: (cid:107) σ T (cid:107) = sup g | Tr T α g ( P ) | ≤ (cid:107) T (cid:107) sup g Tr | α g ( P ) | = (cid:107) T (cid:107) . Since σ is also SU (2)-equivariant we again find that[ D S , σ T ] = (cid:88) j Lemma 20. There exists a sequence { γ (cid:48) n } converging to 0 such that (cid:107) T − ˘ σ ( σ T ) (cid:107) ≤ γ (cid:48) n (cid:107) [ D n , T ] (cid:107) for all T ∈ L ( V n ) .Proof. This is based on a highly non-trivial result [21, Theorem 6.1] which statesthat there exists a sequence { γ (cid:48) n } converging to 0 such that (cid:107) T − ˘ σ ( σ T ) (cid:107) ≤ γ (cid:48) 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 g (cid:54) = e (cid:107) α g ( T ) − T (cid:107) l ( g )for a length function g on SU (2) that induces the round metric on S . However,as in the proof of [19, Theorem 3.1] we may estimate L n ( T ) ≤ sup X ∈ su (2) {(cid:107) [ X, T ] (cid:107) : (cid:107) X (cid:107) ≤ } while the right-hand side can be bounded from above by k (cid:107) [ D, T ] (cid:107) for some constant k independent of n (as in the display preceding [19, Theorem 4.2]. (cid:3) We have thus verified that the maps (˘ σ, σ ) between { L ( V n ) , C ⊗ L ( V n ) , D n ) and( C ∞ ( S ) , C ⊗ L ( S ) , D S ) forms a C -approximate order isomorphism and wemay conclude from Theorem 5 that Proposition 21. The sequence of state spaces { ( S ( L ( V n )) , d n ) } n converges inGromov–Hausdorff distance to ( S ( C ( S )) , d ) . References [1] J. R. Angelos, C. C. Cowen, and S. K. Narayan. Triangular truncation and finding the normof a Hadamard multiplier. Linear Algebra Appl. 170 (1992) 117–135.[2] J. W. Barrett. Matrix geometries and fuzzy spaces as finite spectral triples. J. Math. Phys. 56 (2015) 082301, 25.[3] J. W. Barrett and L. Glaser. 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Amer. Math. Soc. J. Math.Phys. 33 (1992) 4011–4019. Institute for Mathematics, Astrophysics and Particle Physics, Radboud UniversityNijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. E-mail address ::