Towards a Monge-Kantorovich metric in noncommutative geometry
TTowards a Monge-Kantorovich metricin noncommutative geometry
Pierre Martinetti ∗ CMTP & Dipartimento di Matematica, Universit`a di Roma Tor Vergata, I-00133Universit`a di Napoli Federico II, I-00185
Proceeding of the international conference for the Centenary of Kantorovich,
Monge-Kantorovich optimal transportation problem, transport metricand their applications , St-Petersburg, June 2012.
Abstract
We investigate whether the identification between Connes’ spectral distance in noncommutativegeometry and the Monge-Kantorovich distance of order 1 in the theory of optimal transport - thathas been pointed out by Rieffel in the commutative case - still makes sense in a noncommutativeframework. To this aim, given a spectral triple ( A , H , D ) with noncommutative A , we introduce a“Monge-Kantorovich”-like distance W D on the space of states of A , taking as a cost function thespectral distance d D between pure states. We show in full generality that d D ≤ W D , and exhibitseveral examples where the equality actually holds true, in particular on the unit two-ball viewedas the state space of M ( C ). We also discuss W D in a two-sheet model (product of a manifold by C ), pointing towards a possible interpretation of the Higgs field as a cost function that does notvanish on the diagonal. I Introduction
In [6] Connes noticed that the geodesic distance on a compact Riemannian manifold M (con-nected and without boundary) can be retrieved in purely algebraic terms, from the knowledge ofboth the algebra C ∞ ( M ) of smooth functions on M and the signature (or Hodge-Dirac operator) d + d † , where d is the exterior derivative. Explicitly, one has d geo ( x, y ) = sup f ∈ C ∞ ( M ) (cid:8) | δ x ( f ) − δ y ( f ) | , (cid:13)(cid:13) [ d + d † , π ( f )] (cid:13)(cid:13) ≤ (cid:9) (1.1)where- π denotes the representation of the commutative algebra C ∞ ( M ) by multiplication on theHilbert space Ω • ( M ) of square integrable differential forms on M ;- the norm of the commutator is the operator norm on B (Ω • ( M )) (bounded operators onΩ • ( M )): (cid:107) A (cid:107) = sup (cid:107) ψ (cid:107) Ω =1 (cid:107) Aψ (cid:107) Ω (1.2)for any A ∈ B (Ω • ( M )), with (cid:107)·(cid:107) Ω the Hilbert space norm of Ω • ( M );- δ x : f → f ( x ) is the evaluation at x ∈ M .Evaluations are nothing but the pure states of the C ∗ -closure C ( M ) of C ∞ ( M ). Recall thata state of a C ∗ -algebra A is a positive linear form on A with norm 1. The space of states,denoted S ( A ), is convex and its extremal points are called pure states . By Gelfand theorem, anycommutative C ∗ -algebra A is isomorphic to the algebra of functions vanishing at infinity on itspure state space P ( A ) and - conversely - any locally compact topological space X is homeomorphicto the pure state space of C ( X ), A (cid:39) C ( P ( A )) X (cid:39) P ( C ( X ) . (1.3)In modern terms, the category of commutative C ∗ -algebras is (anti)-isomorphic to the category oflocally compact topological spaces. The compact case corresponds to unital algebras. ∗ [email protected] a r X i v : . [ m a t h - ph ] O c t ith Gelfand theorem in minds, it is natural to extend (1.1) to non-pure states ϕ, ˜ ϕ ∈ S ( C ( M )),defining d d + d † ( ϕ, ˜ ϕ ) . = sup f ∈ C ∞ ( M ) (cid:8) | ϕ ( f ) − ˜ ϕ ( f ) | , (cid:13)(cid:13) [ d + d † , π ( f )] (cid:13)(cid:13) ≤ (cid:9) . (1.4)Since the commutativity of the algebra C ∞ ( M ) does not enter (1.1), another natural extensionis to the noncommutative setting. Namely, given a noncommutative algebra A acting by π on someHilbert space H , together with an operator D on H such that [ D, π ( a )] is bounded for any a ∈ A ,one defines [7] for any ϕ, ˜ ϕ ∈ S ( A ) d D ( ϕ, ˜ ϕ ) . = sup a ∈B D ( A ) | ϕ ( a ) − ˜ ϕ ( a ) | , (1.5)where B D ( A ) . = { a ∈ A , (cid:107) [ D, π ( a )] (cid:107) ≤ } (1.6)denotes the D -Lipschitz ball of A . It is easy to check that d D satisfies all the properties of adistance on S ( A ) (see e.g. [15, p.35]), except it may be infinite. Following the terminology of [16],we call it the spectral distance a , but depending on the authors it may be called Connes or thenoncommutative distance. Notice that - as in the commutative case - when A is not a C ∗ -algebrawe consider the states of its C ∗ -closure in the operator norm coming from the representation π .Therefore the spectral distance (1.5) appears as a generalization to the noncommutative settingof the Riemannian geodesic distance. The latter is retrieved between pure states in the commutativecase d d + d † ( δ x , δ y ) = d geo ( x, y ) . (1.7)For non-pure states (still in the commutative case), Rieffel seems to have been the first to notice in[18] that (1.4) was nothing but Kantorovich’s dual formulation of the minimal transport betweenprobability measures, with cost function the geodesic distance. Indeed, the set of states of C ( M ) (cid:51) ϕ is in 1-to-1 correspondence with the set of probability measures Prob( M ) (cid:51) µ on M , ϕ ( f ) = (cid:90) M f d µ, (1.8)and it is not difficult to check (as recalled in section II) that d d + d † ( ϕ, ˜ ϕ ) = W ( µ, ˜ µ ) (1.9)where W denotes the Monge-Kantorovich (or Wasserstein) distance of order one with cost d geo . b The same result holds on a locally compact manifold, as soon as it is complete [11].In this contribution, we investigate how the identification of the spectral distance with theMonge-Kantorovich metric could still make sense in a noncommutative context. Namely, given a non-commutative algebra A acting on some Hilbert H together with an operator D such that [ D, π ( a )] isbounded for any a ∈ A , is there some “optimal transport in noncommutative geometry” such thatthe associated Monge-Kantorovich distance coincides with the spectral distance (1.5) ? We providea tentative answer, introducing on the state space S ( A ) a new distance W D , obtained by takingas a cost function the spectral distance d D on the pure state space P ( A ). The main propertiesof this “Monge-Kantorovich”-like distance are worked out in proposition III.1: it is shown in fullgenerality that d D ≤ W D on S ( A ), with equality on P ( A ) as well as between any non-pure statesobtained as convex linear combinations of the same two pure states. In particular, this allows toshow that d D = W D on the two-ball, viewed as the space of states of A = M ( C ).We begin with the commutative case (section II), recalling in proposition II.1 why d d + d † coin-cides with the Monge-Kantorovich distance W . Then we investigate the noncommutative case andintroduce the new distance W D in section III. Section IV deals with examples. a Because it is a distance associated with a spectral triple, cf section III.1 b In this contribution we are only interested in the Monge-Kantorovich distance of order 1 with cost the geodesicdistance. From now on we simply call it the Monge-Kantorovich distance.. I The commutative case
II.1 Monge-Kantorovich and spectral distance
Recall that given two probability measures µ , ˜ µ on a metric space ( X , d ) (non-necessarily com-pact), the Monge-Kantorovich distance is W ( µ, ˜ µ ) = inf π (cid:90) X ×X d π d ( x, y ) (2.1)where the infimum is on all the measures on X × X whose marginals are µ and ˜ µ . In his seminalwork [13, 14], Kantorovich showed there exists a dual formulation, W ( µ, ˜ µ ) = sup (cid:107) f (cid:107) Lip ≤ , f ∈ L ( µ ) ∩ L ( µ ) (cid:18)(cid:90) X f d µ − (cid:90) X f d˜ µ (cid:19) (2.2)for any pair of probability measures on X such that the right-hand side in the above expression isfinite. The supremum is on all real µ, ˜ µ -integrable real functions f that are 1-Lipschitz, that is tosay | f ( x ) − f ( y ) | ≤ d ( x, y ) ∀ x, y ∈ X . (2.3)Take now ( X , d ) a locally compact Riemannian manifold ( M , d geo ) and consider the spectraldistance (1.4). The formula is the same as in the compact case, except that we want the algebrato be represented by bounded operators. So instead of C ∞ ( M ) we look for the supremum on thealgebra C ∞ ( M ) of smooth functions vanishing at infinity. Let ϕ, ˜ ϕ be two states of C ( M ) definedby probability measures µ, ˜ µ via formula (1.8). That the Monge-Kantorovich distance W ( µ, ˜ µ )equals the spectral distance d d + d † ( ϕ, ˜ ϕ ) follows from the three well known points:- the supremum in (1.5) can be equivalently searched on selfadjoint elements [12]. In thecommutative case, this means we assume f ∈ C ∞ ( M ) is real.- for real functions, the norm of the commutator [ d + d † , π ( f )], as an operator on Ω • ( M ), isprecisely the Lipschitz norm of f (see [10] and also section II.3): (cid:107) f (cid:107) Lip = (cid:13)(cid:13) [ d + d † , π ( f )] (cid:13)(cid:13) . (2.4)- the supremum on 1-Lipschitz smooth functions vanishing at infinity in the spectral distanceformula is the same as the supremum on 1-Lipschitz continuous functions non-necessarilyvanishing at infinity in Monge-Kantorovich formula (for details cf e.g. [11, § M be complete. Under this conditionone obtains Proposition II.1 [18, 11] On a (connected, without boundary) complete Riemannian manifold M , for any state ϕ, ˜ ϕ ∈ S ( C ( M )) one has d d + d † ( ϕ, ˜ ϕ ) = W ( µ, ˜ µ ) . (2.5) II.2 On the importance of being complete
It is not known to the author whether Kantorovich duality holds for non-complete manifolds(in the literature the completeness condition seems to be always assumed). In any case, (2.2) stillmakes sense as a definition of the Monge-Kantorovich distance for non-complete manifolds. Theimportance of the completeness condition is illustrated by simple examples, taken from [11].Let M be a compact manifold and N = M (cid:114) { x } . For example M = S = [0 ,
1] and N = (0 , S is W M ( x, y ) = min {| x − y | , − | x − y |} , (2.6)which differs from W N ( x, y ) = | x − y | . On the contrary, for M = S and N = S (cid:114) { x } , onehas W M = W N . Removing a point from a complete compact manifold may change or not theMonge-Kantorovich distance. n the contrary, removing a point does not modify the spectral distance, in the sense that d M d + d † ( ϕ , ϕ ) = sup f ∈ C ∞ ( M ) (cid:8) | ( ϕ − ϕ )( f ) | ; || f || Lip ≤ (cid:9) = sup f ∈ C ∞ ( M ) ,f ( x )=0 (cid:8) | ( ϕ − ϕ )( f ) | ; || f || Lip ≤ (cid:9) = sup f ∈ C ∞ ( N ) (cid:8) | ( ϕ − ϕ )( f ) | ; || f || Lip ≤ (cid:9) = d N d + d † ( ϕ , ϕ ) . Here we noticed that because C ∞ ( M ) has a unit , if f attains the supremum then so does f − f ( x ) (the argument is still valid if the supremum is not attained, by considering a sequenceof element in the Lipschitz ball tending to the infimum).To summarize, the spectral and the Monge-Kantorovich distances are equal on the incompletemanifold S (cid:114) { x } , but are not equal on (0 , II.3 Spin, Laplacian and the Lipschitz ball
There exist alternative definitions of the Lipschitz ball (1.6). Instead of the signature operator d + d † , one can use as well the Dirac (or Atiyah) operator ∂/ = − i dim M (cid:88) µ =1 γ µ ∂ µ . (2.7)Recall that the γ µ ’s are selfadjoint matrices of dimension M . = 2 E ( m ) , m . = dim M , spanning anirreducible representation of the Clifford algebra. They satisfy γ µ γ ν + γ ν γ µ = 2 g µν I M . (2.8)With π the multiplicative representation of C ( M ) on the Hilbert space H of square integrablespinors on M , ( π ( f ) ψ ) ( x ) = f ( x ) ψ ( x ) ∀ ψ ∈ H , x ∈ M , (2.9)one easily checks that [ ∂/, π ( f )] acts as multiplication by (cid:80) µ γ µ ∂ µ f , since by the Leibniz rule[ ∂/, π ( f )] ψ = (cid:88) µ γ µ ∂ µ f ψ − f γ µ ∂ µ ψ = (cid:32)(cid:88) µ γ µ ∂ µ f (cid:33) ψ. (2.10)Hence for real functions f , using the property of the C ∗ -norm and Einstein summation on repeatedindices, one gets (cid:107) [ ∂/, π ( f )] (cid:107) = (cid:107) ( γ µ ∂ µ f ) ∗ γ ν ∂ ν f (cid:107) = (cid:107) γ µ γ ν ∂ µ f ∂ ν f (cid:107) = 12 (cid:107) ( γ µ γ ν + γ ν γ µ ) ∂ µ f ∂ ν f (cid:107) (2.11)= (cid:107) g µν ∂ µ f ∂ ν f (cid:107) = (cid:107) grad f (cid:107) = (cid:107) f (cid:107) Lip . (2.12)The Lipschitz norm of f can also be retrieved from the Laplacian ∆ (see e.g. [11, § (cid:107) f (cid:107) Lip = (cid:107) [[∆ , π ( f )] , π ( f )] (cid:107) (2.13)where π denotes the representation of C ( M ) on the Hilbert space of square integrable functionson M . In the noncommutative setting, one could be tempted to define the Lipschitz ball with abi-commutator similar to (2.13) instead of (1.6). However, it is easier to generalize to the noncom-mutative case a first order differential operator than a second order one, which justifies the choiceof (1.6).As proposed by Rieffel, one could even work with the unit ball B L . = { a ∈ A , L ( a ) ≤ } (2.14)with L a seminorm not necessarily coming from the commutator with an operator. In this con-tribution however, having Connes’ reconstruction theorem in minds (see the next section) we willstick to the definition (1.6), and view D as a noncommutative generalization of the Dirac operator. II A Monge-Kantorovich metric in noncommutative geometry
III.1 Spectral triple
To extend formula (1.4) to the noncommutative setting (1.5), the starting point is to choose asuitable algebra A , a suitable representation π on some Hilbert space H , and a suitable operator D . For (1.5) to make sense as a distance, one needs that [ D, π ( a )] be in B ( H ) for any a ∈ A , orat least for a dense subset of A ; otherwise one may have d D ( ϕ, ˜ ϕ ) = 0 although ϕ (cid:54) = ˜ ϕ . FollowingConnes [8] one further asks that0. π ( a )[ D − λ I ] is compact for any a ∈ A and λ in the resolvent set of D .When the algebra is unital, this simply means that D has compact resolvent. A triplet ( A , H , D )satisfying the conditions above is called a spectral triple .Although for our purposes we do not need the full machinery of noncommutative geometry, itis interesting to recall the general context. By imposing five extra-conditions c Connes is able toextend Gelfand duality beyond topology [9]: if ( A , H , D ) is a spectral triple satisfying i-v with A unital & commutative, then there exists a compact (connected, without boundary) Riemannianmanifold M such that A (cid:39) C ∞ ( M ). Conversely, to any such M is associated the spectral triple( C ∞ ( M ) , Ω • ( M ) , d + d † ) which satisfies i-v. With two more conditions (vi real structure, viiPoincar´e duality), the reconstruction theorem extends to spin manifolds.A noncommutative geometry is intended as a geometrical object whose set of functions definedon it is a noncommutative algebra. As such it is not a usual space (otherwise its algebra offunctions would be commutative, by Gelfand theorem), so it requires new mathematical tools tobe investigated. Spectral triples provide such tools: first by formulating in purely algebraic termsall the aspects of Riemannian geometry (Connes reconstruction theorem), second by giving thema sense in the noncommutative context (properties i-vii still makes sense for noncommutative A ).commutative spectral triple → noncommutative spectral triple (cid:108) ↓ Riemannian geometry non-commutative geometrySpecifically, the formula (1.5) of the spectral distance is a way to export to the noncommutativesetting the usual notion of Riemannian geodesic distance. Notice the change of point of view: thedistance is no longer the infimum of a geometrical object (i.e. the length of the paths betweenpoints), but the supremum of an algebraic quantity (the difference of the valuations of two states).This is interesting for physics, for it provides a notion of distance no longer based on objects illdefined in a quantum context: Heisenberg uncertainty principle makes the notions of points andpath between points highly problematic.A natural question is whether one looses any trace of the distance-as-an-infimum by passing tothe noncommutative side. More specifically, is there some “noncommutative Kantorovich duality”allowing to view the spectral distance as the minimization of some “noncommutative cost” ?distance as a supremum: d d + d † commutative case → d D noncommutative case ↑ | Kantorovich duality: d d + d † = W d D = W D ? ↓ ↓ distance as an infimum: W with cost d geo noncommutative cost ?The (very preliminary) elements of answer we give in the next section comes from the followingobservation: in the commutative case, the cost function is retrieved as the Monge-Kantorovichdistance between pure states of C ( M ). So in the noncommutative case, if the spectral distancewere to coincide with some “Monge-Kantorovich”-like distance W D on S ( A ), then the associatedcost should be the spectral distance on the pure state space P ( A ). c They are quite technical and we do not need them here. Let us simply mention that can be viewed as an algebraictranslation of the following properties of a manifold: i. the dimension, ii. the signature operator being a first orderdifferential operator iii. the smoothness of the coordinates, iv. orientability, v. existence of the tangent bundle. II.2 Optimal transport on the pure state space
Let ( A , H , D ) be a spectral triple. We aim at defining a “Monge-Kantorovich”-like distance W D on the state space S ( A ), taking as a cost function the spectral distance d D on the pure state space P ( A ). A first idea is to mimic formula (2.1) with X = P ( A ), that is W ( µ, ˜ µ ) = inf π (cid:90) P ( A ) ×P ( A ) d π d D ( ω, ˜ ω ) . (3.1)For this to make sense as a distance on S ( A ), we should restrict to states ϕ ∈ S ( A ) that are givenby a probability measures on P ( A ). This is possible (at least) when A is separable and unital: S ( A ) is then metrizable [2, p. 344] so that by Choquet theorem any state ϕ ∈ S ( A ) is given by aprobability measure µ ∈ Prob( P ( A )). One should be careful however that the correspondence isnot 1 to 1: S ( A ) → Prob( P ( A )) is injective, but two distinct probability measures µ , µ may yieldthe same state ϕ . This is because A is not an algebra of continuous functions on P ( A ) (otherwise A would be commutative). We give an explicit example of such a non-unique decomposition insection IV.2.Thus W D is not a distance on Prob( P ( A )), but on a quotient of it, precisely given by S ( A ).This forbids us to define W D by formula (3.1). A possibility is to consider the infimuminf µ, ˜ µ W ( µ, ˜ µ ) (3.2)on all the probability measures µ, ˜ µ ∈ Prob( P ( A )) such that ϕ ( a ) = (cid:90) P ( A ) ω ( a ) d µ, ˜ ϕ ( a ) = (cid:90) P ( A ) ω ( a ) d˜ µ. (3.3)However it is not yet clear that (3.2) is a distance on S ( A ).Here we explore another way, consisting in viewing A as an “noncommutative algebra of func-tions” on P ( A ), a ( ω ) . = ω ( a ) ∀ ω ∈ P ( A ) , a ∈ A ; (3.4)and define the set of “ d D -Lipschitz noncommutative functions” in analogy with (2.3)Lip D ( A ) . = { a ∈ A such that | a ( ω ) − a ( ω ) | ≤ d D ( ω , ω ) ∀ ω , ω ∈ P ( A ) } . (3.5)By mimicking (2.2) we then defines for any ϕ, ˜ ϕ ∈ S ( A ) W D ( ϕ, ˜ ϕ ) . = sup a ∈ Lip D ( A ) | ϕ ( a ) − ˜ ϕ ( a ) | . (3.6) Proposition III.1 W D is a distance, possibly infinite, on S ( A ) . Moreover for any ϕ, ˜ ϕ ∈ S ( A ) , d D ( ϕ, ˜ ϕ ) ≤ W D ( ϕ, ˜ ϕ ) . (3.7) The equation above is an equality on the set of convex linear combinations ϕ λ . = λ ω + (1 − λ ) ω of any two given pure states ω , ω : namely for any λ, ˜ λ ∈ [0 , one has d D ( ϕ λ , ϕ ˜ λ ) = | λ − ˜ λ | d D ( ω , ω ) = W D ( ϕ λ , ϕ ˜ λ ) . (3.8) Proof.
We first check that W D is a distance. Symmetry in the exchange ϕ ↔ ˜ ϕ is obvious, as wellas ϕ = ˜ ϕ ⇒ W D ( ϕ, ˜ ϕ ) = 0. The triangle inequality is immediate: for any ϕ (cid:48) ∈ S ( A ) one has W D ( ϕ, ˜ ϕ ) = sup a ∈ Lip D ( A ) | ϕ ( a ) − ˜ ϕ ( a ) | ≤ sup a ∈ Lip D ( A ) ( | ϕ ( a ) − ϕ (cid:48) ( a ) | + | ϕ (cid:48) ( a ) − ˜ ϕ ( a ) | ) (3.9) ≤ sup a ∈ Lip D ( A ) | ϕ ( a ) − ˜ ϕ (cid:48) ( a ) | + sup a ∈ Lip D ( A ) | ϕ (cid:48) ( a ) − ˜ ϕ ( a ) | = W D ( ϕ, ϕ (cid:48) ) + W D ( ϕ (cid:48) , ϕ ) . A bit less immediate is W D ( ϕ, ˜ ϕ ) = 0 ⇒ ϕ = ˜ ϕ . To show this, let us first observe that B D ( A ) ⊂ Lip D ( A ) (3.10) therwise there would exist a ∈ B D ( A ) and ω , ω ∈ P ( A ) such that ω ( a ) − ω ( a ) > d D ( ω , ω ) , which would contradict the definition of the spectral distance. Let us now assume W D ( ϕ, ˜ ϕ ) = 0.This means ϕ ( a ) = ˜ ϕ ( a ) for all a ∈ Lip D ( A ). For a / ∈ Lip D ( A ), denote λ a . = inf ω ,ω ∈P ( A ) d D ( ω , ω ) ω ( a ) − ω ( a ) . (3.11)One has 1 (cid:107) [ D, π ( a )] (cid:107) ≤ λ a < . (3.12)The r.h.s. inequality comes from a / ∈ Lip D ( A ): there exists at least one pair ω , ω such that ω ( a ) − ω ( a ) > d D ( ω , ω ). The l.h.s. inequality follows from the definition of the spectraldistance: any pair ω , ω ∈ P ( A ) satisfies d D ( ω , ω ) ≥ ω ( a ) − ω ( a ) (cid:107) [ D, π ( a )] (cid:107) , (3.13)which is well defined because a / ∈ B D ( A ) by (3.10) so that (cid:107) [ D, π ( a )] (cid:107) (cid:54) = 0. In other terms λ a isfinite and non-zero, so that λ a a is in Lip D ( A ), meaning that ϕ ( λ a a ) − ˜ ϕ ( λ a a ) - hence ϕ ( a ) − ˜ ϕ ( a )- vanish. So ϕ = ˜ ϕ and W D is a distance.Eq. (3.7) follows from (3.10): the supremum for d D is searched on a smaller set than for W D .The first part of (3.8) comes from ϕ λ ( a ) − ϕ ˜ λ ( a ) = ( λ − ˜ λ ) ( ω ( a ) − ω ( a )) , (3.14)for this means d D ( ϕ λ , ϕ ˜ λ ) = sup a ∈B D ( A ) | λ − ˜ λ | ( ω ( a ) − ω ( a )) = | λ − ˜ λ | sup a ∈B D ( A ) ( ω ( a ) − ω ( a )) (3.15)= | λ − ˜ λ | d D ( ω , ω ) . (3.16)The second part of (3.8) is obtained noticing that (3.14) together with the definition of Lip D ( A )imply W D ( ϕ λ , ϕ ˜ λ ) ≤ | λ − ˜ λ | d D ( ω , ω ) , (3.17)that is W D ( ϕ λ , ϕ ˜ λ ) ≤ d D ( ϕ λ , ϕ ˜ λ ) by (3.15), and the result by (3.7). (cid:4) The difference between W D and d D - if any - is entirely contained in the difference betweenthe D -Lipschitz ball (1.6) and Lip D ( A ) defined in (3.5). In the commutative case A = C ( M ),these two notions of Lipschitz function coincide with the usual one, so that d D = W D = W . Forthe moment, we let as an open question whether in the noncommutative case d D = W D in fullgenerality. In the next section we illustrate the equality (3.8) with various examples, including anoncommutative one A = M ( C ). IV Examples
IV.1 A two-point space
The spectral triple A = C , H = C , D = (cid:18) m ¯ m (cid:19) (4.1)where m ∈ C and a = ( z , z ) ∈ A is represented by π ( z , z ) = (cid:18) z z (cid:19) (4.2)describes a two-point space, for the algebra C has only two pure states, δ ( z , z ) . = z , δ ( z , z ) . = z . (4.3) ence any non-pure state is of the form ϕ λ = λδ + (1 − λ ) δ and by proposition III.1 one knowsthat that d D = W D on the whole of S ( A ).It is easy to check explicitly that the two notions of Lipschitz element coincide: one has (cid:107) [ D, a ] (cid:107) = | m ( z − z ) | , (4.4)hence by (1.5) d D ( δ , δ ) = 1 | m | . (4.5)Therefore a = ( z , z ) ∈ Lip D ( C ) means | z − z | ≤ | m | , which is equivalent to (cid:107) [ D, a ] (cid:107) ≤
1. HenceLip D ( C ) = B D ( C ).Although very elementary (and commutative !), this example illustrates interesting propertiesof the spectral distance: P ( A ) is a discrete space (hence there is no notion of geodesic) but still thedistance is finite; the spectral distance on non-pure states is “Monge-Kantorovich”-like with costthe spectral distance on pure states. IV.2 The sphere
Let us now come to the slightly more involved (and noncommutative) example A = M ( C ).Identifying a 2 × C , any unit vector ξ ∈ C defines apure state ω ξ ( a ) = ( ξ, a ξ ) = Tr( s ξ a ) ∀ a ∈ A (4.6)where ( · , · ) denotes the usual inner product in C and s ξ ∈ M ( C ) is the projection on ξ (in Diracnotation s ξ = | ξ (cid:105)(cid:104) ξ | ). Two vectors equal up to a phase define the same state, and any pure stateis obtained in this way. In other terms, the set of pure states of M ( C ) is the complex projectiveplane C P , which is in 1-to-1 correspondence with the two-sphere via ξ = (cid:18) ξ ξ (cid:19) ∈ C P ↔ x ξ = x ξ = Re( ξ ξ ) y ξ = Im( ξ ξ ) z ξ = | ξ | − | ξ | ∈ S . (4.7)A non-pure state ϕ is determined by a probability distribution φ on S : ϕ ( a ) = (cid:90) S φ ( x ξ ) ω ξ ( a ) d x ξ (4.8)for any a ∈ M ( C ), with d x ξ the SU (2) invariant measure on S . However the correspondencebetween Prob( S ) and S ( M ( C )) is not 1-to-1. One computes [3, § s ϕ such that ϕ ( a ) = Tr ( s ϕ a ) (4.9)actually depends on the barycenter of the probability measure φ only: s ϕ = (cid:32) z φ +12 x φ − iy φ x φ + iy φ − z φ (cid:33) (4.10)where x φ = ( x φ , y φ , z φ ) with x φ . = (cid:90) S φ ( x ξ ) x ξ d x ξ (4.11)and similar notation for y φ , z φ .With the equivalence relation φ ∼ φ (cid:48) ⇐⇒ x φ = x φ (cid:48) , the state space S ( M ( C )) = S ( C ( S )) / ∼ = Prob( S ) / ∼ (4.12)is homeomorphic to the Euclidean 2-ball: S ( M ( C )) (cid:51) ϕ −→ x φ ∈ B . (4.13) his means that any two states ϕ, ϕ (cid:48) are convex linear combinations of the same two pure states.So by proposition III.1 one has d D = W D on the whole of S ( M ( C )).Depending on the choice of the representation and of the Dirac operator, one deals with com-pletely different cost functions: viewing A = M ( C ) acting on H = M ( C ) as a truncation of theMoyal plane [3], one inherits a Dirac operator such that d D is finite on P ( A ) (hence on S ( A )): d D ( x φ , x φ (cid:48) ) = (cid:114) θ × (cid:40) cos α d Ec ( x φ , x φ (cid:48) ) when α ≤ π ,
12 sin α d Ec ( x φ , x φ (cid:48) ) when α ≥ π , (4.14)where d Ec ( x φ , x φ (cid:48) ) = | x φ − x φ (cid:48) | is the Euclidean distance in the ball and α is the angle betweenthe segment [ x φ , x φ (cid:48) ] and the horizontal plane z ξ = constant (see figure 1). Α x Φ x Φ (cid:162) eq x Φ (cid:162) Figure 1: The vertical plane inside the unit ball that contains x φ , x φ (cid:48) . We denote x eqφ (cid:48) the projectionof x φ (cid:48) in the “equatorial” plane z φ = cst On the contrary, making A = M ( C ) act on H = C , with D a 2-by-2 matrix with distinctnon-zero eigenvalues D , D , one obtains [12] d D ( x φ , x φ (cid:48) ) = (cid:26) | D − D | d Ec ( x φ , x φ (cid:48) ) if z φ = z φ (cid:48) , + ∞ if z φ (cid:54) = z φ (cid:48) . (4.15)Here the eigenvectors of D - chosen as a basis of H - are mapped to the north and south poles of S . IV.3 Product of the continuum by the discrete
We summarize here the discussion developed in [11, § M by the spectral triple (4.1) is the spectral triple ( A (cid:48) , H (cid:48) , D (cid:48) ) where [8] A (cid:48) . = C ∞ ( M ) ⊗ C , H (cid:48) . = Ω • ( M ) ⊗ C , D (cid:48) . = ( d + d † ) ⊗ I + Γ ⊗ D (4.16)with Γ a graduation of Ω • ( M ). An element of A (cid:48) is a pair a (cid:48) = ( f, g ) of functions in C ∞ ( M ), andpure states of (the C ∗ -closure of) A (cid:48) are pairs x i . = ( δ x , δ i ) (4.17) here δ x ∈ P ( C ( M )) is the evaluation at x ∈ M while δ i =1 , is one of the two pure states of C defined in (4.3). Thus P ( A (cid:48) ) appears as the disjoint union of two copies of M . Explicitly, theevaluation on an element of A (cid:48) reads x ( a (cid:48) ) = f ( x ) , y ( a (cid:48) ) = g ( y ) . (4.18)The spectral distance in this two-sheet model coincides with the geodesic distance d (cid:48) geo in themanifold M (cid:48) = M × [0 ,
1] with Riemannian metric (cid:18) g µν | m | (cid:19) (4.19)where g µν is the Riemannian metric on M . Namely one has [17] d D (cid:48) ( x , y ) = d (cid:48) geo (( x, , ( y, . (4.20)Non-pure states of A (cid:48) are given by pairs of measures ( µ, ν ) on M , normalized to (cid:90) M d µ + (cid:90) M d ν = 1 , whose evaluation on a (cid:48) = ( f, g ) is ϕ ( a ) = (cid:90) M f d µ + (cid:90) M g d ν. (4.21)By proposition III.1 one has d D (cid:48) ≤ W D (cid:48) where W D (cid:48) is the Monge-Kantorovich distance on M ∪ M associated to the cost d D (cid:48) . The equality holds for states localized on the same copy: ϕ = (0 , ν ) , ˜ ϕ = (0 , ˜ ν ) or ϕ = ( µ, , ˜ ϕ = (˜ µ, , since one then has d D (cid:48) ( ϕ, ˜ ϕ ) = d d + d † ( ϕ, ˜ ϕ ) = W ( ϕ, ˜ ϕ ) = W D (cid:48) ( ϕ, ˜ ϕ ) . (4.22)For two states localized on distinct copies, the question is open. It is interesting to notice thatone may project back the problem on a single copy of M , using the cost function c ( x, y ) . = d D (cid:48) ( x , y ) . = (cid:115) d geo ( x, y ) + 1 | m | defined on M , rather than d D (cid:48) defined on M ∪ M . The particularity of this single-sheet cost c isthat it does not vanish on the diagonal, c ( x, x ) = | m | (cid:54) = 0.This might yield interesting perspective in physics: in the description of the standard modelof elementary particles in noncommutative geometry [4], the recently discovered Higgs field [5]appears as an extra-component of the metric similar to | m | , except that it is no longer a constantbut a function on M . From this perspective the Higgs field represents the cost to stay at the samepoint of space-time, but jumping from one copy to the other. V Conclusion
In this contribution we presented preliminary steps towards a definition of a Monge-Kantorovichdistance in noncommutative geometry. Given a spectral triple ( A , H , D ), we introduced a newdistance W D on the space of state S ( A ), which is the exact translation in the noncommutativesetting of Kantorovich dual formula, taking as a cost function Connes spectral distance d D onthe the pure state space P ( A ). By construction, W D = d D on P ( A ), and we showed that thesame is true for non-pure states given by convex linear combinations of the same two pure states.Although very restrictive, this condition applies to the interesting example A = M ( C ), showingthat W D = d D on a unit ball. At this point two questions remain open and will be the object offuture works: Is W D equal to d D on the whole of S ( A ) or only on part of it ? • Is there a dual formula to d D and/or W D as an infimum (a kind of reverse Kantorovichduality), for instance formula (3.2), or the Wasserstein distance in free probabilities introducedby Biane and Voiculescu [1] ?These questions are not necessarily linked. If both answers turn out to be positive, this wouldindicate that computing the spectral distance is exactly a problem of optimal transport, and spectraltriples could be used as a factory of cost functions. References [1] P. Biane and D. Voivulescu,
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