L p -Wasserstein distances on state and quasi-state spaces of C ∗ -algebras
aa r X i v : . [ m a t h . OA ] M a y L p -WASSERSTEIN DISTANCES ON STATE AND QUASI-STATE SPACESOF C ∗ -ALGEBRAS DANILA ZAEV
Abstract.
We construct an analogue of the classical L p -Wasserstein distance for the statespace of a C ∗ -algebra. Given an abstract Lipschitz gauge on a C ∗ -algebra A in the senseof Rieffel, one can define the classical L p -Wasserstein distance on the state space of eachcommutative C ∗ -subalgebra of A . We consider a projective limit of these metric spaces,which appears to be the space of all quasi-linear states, equipped with a distance function.We call this distance the projective L p -Wasserstein distance. It is easy to show, that the statespace of a C ∗ -algebra is naturally embedded in the space of its quasi-linear states, hence,the introduced distance is defined on the state space as well. We show that this distance isreasonable and well-behaved. We also formulate a sufficient condition for a Lipschitz gauge,such that the corresponding projective L p -Wasserstein distance metricizes the weak ∗ -topologyon the state space. Contents
1. Introduction 12. Informal discussion 33. Space of quasi-states 84. Wasserstein distances for abelian C ∗ -subalgebras 145. Projective L p -Wasserstein distances 206. Some open problems 26Acknowledgement 26References 271. Introduction
We are going to connect the following three theories.
Quantum compact metric spaces . This theory was basically developed by Marc Rieffel(see, for example, [18], [19], [20]). It studies state spaces of C ∗ -algebras (or, more generally,state spaces of unit-order spaces) equipped with a distance function. This distance functionarise from an abstract Lipschitz gauge, defined on an algebra. Usually it is required for adistance function to metricize the weak ∗ -topology of the state space, and, hence, to provide itwith a structure of a compact metric space. These ideas were pioneered by Alain Connes in the Date : November 15, 2018.
Key words and phrases.
Wasserstein space, Kantorovich problem, Wasserstein distance, state space, quasi-state space, quasi-linear state, C ∗ -algebra, commutative C ∗ -subalgebra, Lip -norm, Lipschitz gauge, compactquantum metric space. ection 1. Introduction noncommutative geometry framework (see [10]), and, as in present-day, it is one of the two mostdeveloped approaches to the noncommutative generalization of the notion of metric space (theother approach is due to Nick Weaver/Greg Kuperberg, [15]). Arguably, the main advantage ofthe Rieffel’s approach is the possibility to define an analogue of a Gromov-Hausdorff distancebetween quantum metric spaces. Kantorovich-Wasserstein spaces . This theory studies spaces of probability measures equippedwith a special type of distances, which are defined as solutions of a variational optimal transportproblem (which is also called Kantorovich or Monge-Kantorovich problem, see [2], [6], [21] for asurvey). These distances depend on the distance on the base space (the space where probabilitymeasures are defined), and are parametrized by a parameter p ∈ [1 , ∞ ). The L p -Wassersteindistance with p = 1 is closely connected with the Connes-Rieffel distance (in the case of acommutative C ∗ -algebra, its state space can be thought of as a space of probability measures).In some sense, Connes generalized Kantorovich construction to the noncommutative case. Butthe L p -Wasserstein distances with p > p = 2, geodesicson the Wasserstein space correspond to a meaningful dynamics (see [2]). Furthermore, the L p -Wasserstein distances are used in quantitative estimations of such functionals as entropy orvariation. There are several attempts to link Kantorovich theory with noncommutative geom-etry ([8], [11], [16]). Some noncommutative analogues of the L - and L -Wasserstein distanceswere given in these papers. But the problem of constructing a meaningful analogue of the L p -Wasserstein distance (for p >
1) for the state space of noncommutative C ∗ -algebra in thegeneral framework of Rieffel is still open. Posets of commutative subalgebras . One of the modern approaches to construct a mathe-matical foundation of quantum theory is to consider, instead of a noncommutative C ∗ -algebraof observables, its poset of commutative C ∗ -subalgebras. This approach is compatible withthe so-called Bohr interpretation of quantum mechanics, and has a beautiful mathematicalformalization in the language of topoi. These ideas were developed and described by differentauthors (see, for example, [9], [13], [14]), but the metric aspect of the theory has not beenexplored yet. When one consider commutative subalgebras instead of original noncommuta-tive algebra, he/she inevitably loses some part of information. However, the state space ofa C ∗ -algebra also contains only a part of the information about the algebra (more precisely,the part that is encoded by self-adjoint elements of the algebra, see [1]). Thus, despite wepotentially lose some data, it seems reasonable to define distances (in particular, analogues ofthe L p -Wasserstein distances) on the state space of a noncommutative algebra, passing throwthe poset of its commutative subalgebras. Further we show that this approach is actually jus-tified. In the topos-theoretical interpretation it is natural to consider a space of all quasi-linearstates (“quasi-states”) instead of a state space itself ([9]). Elements of this space are order-and unit-preserving maps from a C ∗ -algebra A to the field of complex numbers C that arelinear on commutative subalgebras of A . A state space naturally embeds in the correspondingquasi-state space. We do not use the language of topoi in the paper, but we provide a con-struction of (analogues of) the L p -Wasserstein distances for both state and quasi-state spacesof C ∗ -algebras. Structure of the paper . At first, we informally discuss the main ideas and formulatesome problems for the further analysis. Then we provide a rigorous definition of the quasi-state space and establish its connection with the notion of the state space of an algebra. Wedefine the quasi-state space of a C ∗ -algebra A as the projective limit of a projective system2 ection 2. Informal discussion of the state spaces of commutative C ∗ -subalgebras of A . It appears, that the state space of A (equipped with its natural weak ∗ -topology) is topologically embedded in the correspondingquasi-state space.Next step is to add metric information in the form of an abstract Lipschitz gauge intoour framework. We recall some results of Rieffel and provide some new definitions. Thereason for providing new definitions is that we are going to study distances on the state spacesof subalgebras, and to make it possible we need to modify the standard notion of a Lip-seminorm. We establish some easy facts about the L p -Wasserstein metrics on the state spacesof commutative C ∗ -subalgebras (which are actually spaces of probability measures), and thenwe define an analogue of the L p -Wasserstein distance for the (quasi-)state space of the whole C ∗ -algebra A (we call it the projective L p -Wasserstein distance).We formulate conditions on an abstract Lipschitz gauge, such that under them the corre-sponding projective L p -Wasserstein distance metricizes the weak ∗ -topology of the state spaceof A . It appears, that it is sufficient for a gauge to be a Lip-seminorm in the sense of Rieffel, tobe finite on a dense subspace of each maximal commutative C ∗ -subalgebra of A , and to satisfya certain inequality. Unfortunately, there is a lack of non-trivial examples of such Lipschitzgauges.We show that the quasi-state space of a C ∗ -algebra, equipped with the projective L p -Wasserstein distance, can be seen as the projective limit of the state spaces of unital com-mutative C ∗ -subalgebras, equipped with the ordinary L p -Wasserstein distances. Besides themain results, some other facts are proved: for example, the fact, that diameters of the statespace and the quasi-state space coincide under mild assumptions on a Lipschitz gauge, andthe fact that the diagram of all commutative unital C ∗ -subalgebras can be recovered from theself-adjoint part of the corresponding C ∗ -algebra.2. Informal discussion
Let X be a compact metrizable space. Then C ( X ), the algebra of all continuous complex-valued functions, is a commutative C ∗ -algebra. It is unital, because X is compact, and separa-ble, because X is metrizable. Denote by P ( X ) the set of all Borel probability measures on X .Due to the fact, that on a metrizable compact space any Borel measure is a Radon one, P ( X ) isequal to the positive part of the unit sphere of the Banach dual of C ( X ): P ( X ) = S +1 ( C ( X ) ∗ ).Equip P ( X ) with the weak ∗ -topology. By Banach-Alaoglu theorem, it is compact, and, since C ( X ) is separable, P ( X ) is metrizable. It is also known, that P ( X ) is a Choquet simplex (see[17]). Recall the definition of a simplex. Definition 2.1. Choquet boundary ∂ e ( K ) of a compact convex set K is a subset consistedof all extreme points, i.e. such points a ∈ K that ∀ a , a ∈ K, a = a : t · a + (1 − t ) · a = a = ⇒ t = 0 or t = 1 Definition 2.2.
Compact convex set K is a Choquet simplex iff for every element a ∈ K there is a unique measure µ a on K such that • µ a ( ∂ e ( K )) = 1 • a = R K xdµ a Recall, that the existence of a representing measure is guaranteed for any compact convexset (see [17]): 3 ection 2. Informal discussion
Theorem 2.3 (Choquet) . For every element a ∈ K of a compact convex metrizable set K ,there is a probability measure µ a on K such that • µ a ( ∂ e ( K )) = 1 • a = R K xdµ a Bauer simplex is a Choquet simplex with closed (hence compact) Choquet boundary.There are infinitely many non-isomorphic Bauer simplexes. In fact, P ( X ) is an example ofa Bauer simplex. Its Choquet boundary consists of all Dirac measures, and the boundary ishomeomorphic to X via the identification: x → δ x . It is also known, that every Bauer simplexis isomorphic to P ( Y ) for some compact Hausdorff space Y (see [3]).Note, that Dirac measures can be characterized in the different ways:(1) If µ ∈ ∂ e ( P ( X )), then it is a Dirac measure, µ = δ x for some x ∈ X .(2) If µ ∈ P ( X ), such that it has a unique representative measure, ˜ µ ∈ P ( ∂ e ( P ( X ))),which is concentrated on { µ } : ˜ µ ( { µ } ) = 1, then µ is a Dirac measure.(3) If µ ∈ P ( X ) = S +1 ( C ( X ) ∗ ) is a unital C ∗ -homomorphism between algebras C ( X ) and C : µ ( f ( x ) · g ( x )) = µ ( f ( x )) · µ ( g ( x )), where µ ( h ( x )) := R X h ( x ) dµ , then it is a Diracmeasure.The first characterization uses convex structure of P ( X ). It makes sense for every convexcompact set. The second characterization uses measure-theoretic structure of P ( X ), and makessense for even more general cases. In fact, for convex compact sets both the first and the secondcharacterizations coincide (see [12]). Thus, we can define generalized Dirac elements of someconvex compact set as the elements of its Choquet boundary.The third definition uses the fact, that P ( X ) is a subset of a dual space to some algebra.It uses multiplicative structure of predual, and the coincidence of this characterization withthe previous ones is a property of the particular object, P ( X ). One cannot expect suchcoincidence in any other case, but, of course, we are able to provide the following definitionof generalized Dirac measures: for a subset of a Banach dual to some C ∗ -algebra, generalized“Dirac measure” can be defined as an element, which is a unital C ∗ -homomorphism betweenthe algebra and C (its continuity is automatically guaranteed). Recall, that the set of all unital C ∗ -homomorphisms from C ∗ -algebra A to C is called the Gelfand spectrum of the algebra. Itis a subset of the Banach dual of an algebra, and can be equipped with the weak ∗ -topologyinherited by this inclusion: Spec( A ) ⊂ A ∗ . It is also true, that Spec( A ) ⊆ S +1 ( A ∗ ). In the case A = C ( X ), Spec( C ( X )) = ∂ e ( P ( X )).To sum up the above discussion, let us emphasize the following two facts: • Any compact metrizable space X is homeomorphic to ∂ e ( P ( X )) and homeomorphic toSpec( C ( X )), where P ( X ) is the set of all Borel probability measures on X . In the caseof non-metrizable compact Hausdorff space X , we should define P ( X ) as the set of allRadon probability measures. • For any commutative unital C ∗ -algebra A it is true, that P ( X ) = S +1 ( A ∗ ), and ∂ e ( P ( X )) appears to be homeomorphic to Spec( A ) = Spec( C (Spec( A ))). Moreover,Spec( A ) is compact and, if A is separable, it is metrizable.We are going to add some metric information in our framework. It can be done in differentways. One way is to define distance function on X , which metricizes its topology. The otherway is to define a Lipschitz gauge on C ( X ). Following Rieffel, we define Lipschitz gauge as a4 ection 2. Informal discussion partially-defined seminorm on the self-adjoint part of an algebra, which satisfies some naturalrequirements. In the following we provide several rigorous definitions of Lipschitz gauges (or,more precisely, collections of requirements a seminorm should satisfy to be a Lipschitz gauge).As for now, let us consider that L : A sa → [0 , + ∞ ] is an abstract Lipschitz gauge on a separableunital C ∗ -algebra A ( A sa is the subspace of all self-adjoint elements of A ), if it is(1) absolutely homogeneous: L ( af ) = | a | L ( f ), for any a ∈ R , f ∈ A sa ,(2) subadditive: L ( f + g ) ≤ L ( f ) + L ( g ), for any f, g ∈ A sa ,(3) lower semi-continuous: { f ∈ A sa : L ( f ) ≤ t } is norm closed for one, hence every, t ∈ (0 , + ∞ ),(4) L ( a ) = 0 iff a ∈ · R ,(5) d L ( µ, ν ) := sup {| µ ( f ) − ν ( f ) | : f ∈ A sa , L ( f ) ≤ } is a distance function on S +1 ( A ∗ )(positive part of a unit sphere of the Banach dual of A ), which metricizes its weak ∗ -topology.Given an abstract Lipschitz gauge L on A = C ( X ), we are able to define a distance d L on P ( X ) = S +1 ( C ( X ) ∗ ), and, restricting it to ∂ e ( P ( X )) = X , we define a distance function d on X : d ( x, y ) = d L ( δ x , δ y ). Hence, it is possible to recover a distance on X from an abstractLipschitz gauge on A = C ( X ). We can define a new Lipschitz gauge by the formula L ed ( f ) := sup (cid:26) f ( x ) − f ( y ) d ( x, y ) : x = y, x, y ∈ X (cid:27) The distance function induced by L ed coincides with the L -Wasserstein distance , which isdefined “in the dual way”:(1) d e ( µ, ν ) := sup {| µ ( f ) − ν ( f ) | : f ∈ A sa , L ed ( f ) ≤ } == W ( µ, ν ) := inf (cid:26)Z d ( x, y ) dπ : π ∈ P ( X × Y ) , Pr( π ) = ( µ, ν ) (cid:27) where Pr : P ( X × Y ) → P ( X ) × P ( Y ) is defined as Pr( π ) := ((Pr X ) ( π ) , (Pr Y ) ( π )). Thisequality is known as “Kantorovich duality” (or Kantorovich-Rubinstein formula), and it is awell-known result in the mass transportation theory (see [2], [6], [21]). It allows one to definea distance on P ( X ) via minimization over the set of transport plans (probability measureson X × X with fixed marginals). Analogously, we can define the L p -Wasserstein distances on P ( X ) for p ∈ [1 , + ∞ ) using the definition via transport plans: W p ( µ, ν ) := inf ((cid:18)Z d p ( x, y ) dπ (cid:19) p : π ∈ P ( X × Y ) , Pr( π ) = ( µ, ν ) ) It appears, that all these distances metricize the weak ∗ -topology on P ( X ) (see [6]). Some ofthe W p -distances have additional good properties (see [6] for details).It was also proved by Rieffel (see Theorem 8.1 in [19]), that under some additional assump-tions about Lipschitz gauge, L = L ed on C ( X ). This result is important for us, but we shalldiscuss the details later.To sum up, a pair of a commutative separable C ∗ -algebra A and an abstract Lipschitzgauge L on A determines a Wasserstein metric space ( P ( X ) , W p ) for each p ∈ [1 , + ∞ ). In thefollowing sections we shall also consider non-separable C ∗ -algebras, which have non-metrizable5 ection 2. Informal discussion state spaces. In this case the pairs ( P ( X ) , W p ) should be thought as pairs of topological spacesand lower semi-continuous distances on them.Consider a more general case of possibly non-commutative unital separable C ∗ -algebra A .In this case we define the state space as the set S ( A ) := S +1 ( A ∗ ). It is a direct analogue of aset of probability measures. It can be equipped with the corresponding weak ∗ -topology, andappears to be a compact convex set. However, it is known (see [3]), that S ( A ) is a simplex ifand only if A is commutative. Since every space of probability measures P ( X ) is a simplex,in the noncommutative case there is no analogue of a space X : S ( A ) = P ( X ) for any X , if A is not commutative.An element of the Choquet boundary of S ( A ), ∂ e ( S ( A )), is called a pure state on A . Asfollows from the Choquet theorem, every state µ can be represented by ˜ µ ∈ P ( ∂ e ( S ( A ))),bar(˜ µ ) = µ (“bar” = barycenter), but this representation is not unique in general.As we noted earlier, in the commutative case, X can be defined as the boundary of S ( A )or, equivalently, as the Gelfand spectrum Spec( A ). In the noncommutative case, ∂ e ( S ( A ))does not coincide with Spec( A ). Moreover, for the algebra B ( H ) of all bounded operators on aHilbert space, Spec( B ( H )) is empty. It is possible to define “spectrum” of a noncommutative C ∗ -algebra in several different ways (e.g. as a primitive ideal space, as a space of equivalenceclasses of irreducible representations, etc.), but since S ( A ) = P ( X ) for any X , there is no muchsense to do that, at least, if our goal is to construct an analogue of the Wasserstein theory forthe state space.Consider a commutative unital C ∗ -subalgebra A α ⊆ A . Using the results described above,we can define its state space S ( A α ) = P ( X α ) and its Gelfand spectrum X := Spec( A α ), whichis homeomorphic to ∂ e ( P ( X α )). If A α , A β are two commutative unital C ∗ -subalgebras of A and A α ⊆ A β , then there are surjective continuous maps: X β ։ X α (2) P ( X β ) ։ P ( X α )(3)The first map is actually the restriction map from A β to A α of elements of the Gelfandspectrum. The second map is the restriction map of continuous linear functionals. Thesemaps are onto due to the Gelfand duality (each injective C ∗ -homomorphism of commutative C ∗ -algebras corresponds to a surjective continuous map between their Gelfand spectrums). Inthe case of state spaces, the map appears to be affine: t · µ β + (1 − t ) · ν β → t · µ α + (1 − t ) · ν α , t ∈ [0 , . We are able to consider an ordered set of all unital commutative C ∗ -subalgebras A α ofa C ∗ -algebra A , where ordering is defined by the relation of inclusion. It can be thoughtof as a diagram in the category of commutative unital C ∗ -algebras, where morphisms areinjective unital C ∗ -homomorphisms. We can apply Gelfand duality to obtain a diagram in thedual category of Hausdorff compact spaces with surjective continuous maps, and then apply afunctor that maps each compact space to the set of all probability measures on it (equippedwith the weak ∗ -topology) and sends each map f : X β → X α to the map f : P ( X β ) → P ( X α ),which is a pushforward of measures. We obtain a diagram in the category of convex compactspaces and affine continuous maps. It is natural to consider the projective limit of this diagram.It exists and appears to be isomorphic to the space QS ( A ) of all quasi-states (quasi-linear6 ection 2. Informal discussion states) of the algebra A . Here QS ( A ) is supposed to be equipped with a projective topology,inherited from the inclusion QS ( A ) ⊆ Q α S ( A α ). Definition 2.4. A quasi-state on a C ∗ -algebra A is a map µ : A → C , such that it is linear onall unital commutative C ∗ -subalgebras of A , satisfies µ ( a + ib ) = µ ( a ) + iµ ( b ) for all self-adjoint a, b ∈ A sa , µ ( a ∗ a ) ≥ a ∈ A , and µ (1) = 1.It is clear, that the space of all quasi-states is convex. Since all S ( A α ) are compact Hausdorff, Q α S ( A α ) is compact and Hausdorff too, and QS ( A ) ⊆ Q α S ( A α ) is closed, hence compactand Hausdorff. Each quasi-state can be restricted to a subalgebra A α , and this restrictiondefines a continuous affine surjection QS ( A ) ։ S ( A α ).It is clear, that the state space S ( A ) is a subset of the corresponding quasi-state space: S ( A ) ⊆ QS ( A ). It is natural to ask, does the weak ∗ -topology on S ( A ) coincide with thetopology obtained by this inclusion? Question 2.5.
Does the image of the inclusion S ( A ) ⊆ QS ( A ) homeomorphic to S ( A )?The answer is affirmative, as it will be shown in the next section (Corollary 3.7).There are cases, when S ( A ) = QS ( A ). It is stated by the Gleason’s theorem, that, if A isa von Neumann algebra without a direct factor isomorphic to B ( C ), every quasi-state of A isa state. In particular, it is true for every A = B ( H ), if dim H > C ∗ -algebra A . As in the commutative case,we can consider abstract Lipschitz gauge defined on the algebra. In fact, we can use exactlythe same definition of a gauge, as the one given above (as one can note, it does not requirecommutativity of a C ∗ -algebra). As follows from the definition, we are able to define thedistance: d L ( µ, ν ) := sup {| µ ( f ) − ν ( f ) | : f ∈ A sa , L ( f ) ≤ } on S ( A ), which metricizes its weak ∗ -topology. In the noncommutative geometry it is known bythe names “Connes’ distance” or “spectral distance”. The last name is motivated by the fact,that usually L is defined as L ( f ) := || [ D, f ] || for some Dirac operator D (see [10] for definitionsand details). This distance can be thought of as a generalization of the Kantorovich distance.The problem is that there is no clear way to construct an analogue of the L p -Wassersteindistance for S ( A ), when A is a (possibly noncommutative) unital C ∗ -algebra.Let us define a distance d L,α on S ( A α ) for a commutative unital subalgebra A α ⊆ A in thefollowing way: d L,α ( µ α , ν α ) := sup {| µ ( f ) − ν ( f ) | : f ∈ A saα , L ( f ) ≤ } Due to the fact, that S ( A α ) = P (Spec( A α )), we can define the L p -Wasserstein distances on S ( A α ) for p ∈ [1 , + ∞ ) by the formula: W p,α ( µ, ν ) := inf ((cid:18)Z d pL,α ( x, y ) dπ (cid:19) p : π ∈ P (Spec( A α ) × Spec( A α )) , Pr( π ) = ( µ, ν ) ) The natural questions are:(1) Is W p,α a distance on S ( A α ) for any A α ?(2) Is it true, that W p,α ≤ W p,β iff A α ⊆ A β ?(3) Does W ,α = d L,α ?(4) Does W p,α metricize the weak ∗ -topology on S ( A α )?7 ection 3. Space of quasi-states These questions will be discussed later in the paper, and the answers for them can be foundon one of the next sections.Let us define a distance on the quasi-state space QS ( A ) (and, hence, on the state space S ( A ) as well) as follows: W ←− p ( µ, ν ) := sup { W p,α ( µ α , ν α ) : A α ⊆ A} , where A α is a commutative unital C ∗ -subalgebra of A , µ α := µ | A α is the restriction of aquasi-state µ from A to A α .Again, we have a list of natural questions about this just defined object:(1) Is W ←− p a distance function on QS ( A ) (or on S ( A ))?(2) Does W ←− p < ∞ on QS ( A ) ( S ( A ))?(3) Does W ←− = d L on S ( A )?(4) Does W ←− p metricize the weak ∗ -topology on S ( A )?It appears, that under some assumptions about an abstract Lipschitz gauge, the answer tothese questions is affirmative. We call W ←− p the projective L p -Wasserstein distance.We shall also provide a positive answer to the following question: Question 2.6.
Is it possible to define a category, such that ( QS ( A ) , W ←− p ) appears to be aprojective limit of the diagram { ( S ( A α ) , W p,α ) } (where the order is defined in a natural way)?3. Space of quasi-states
In this section we provide a rigorous definition of the quasi-state space of a unital C ∗ -algebra A . It appears to be a projective limit of a projective system of the state spaces ofunital commutative C ∗ -subalgebras of A .Define the category of all unital commutative C ∗ -algebras with injective unital C ∗ -homomorphismsbetween them. Denote it by ucC ∗ in . Note, that due to the C ∗ -structure, these homomorphismsare isometric. It follows from the Gelfand duality, that this category is anti-equivalent to thecategory of all compact Hausdorff spaces with continuous surjections ( CH sur ): ucC ∗ inop CH sur SpecC ( · ) Here Spec is a functor that maps each algebra into its Gelfand spectrum. The (weak)inverse functor to Spec is a functor C ( · ), which sends a compact space to its correspondingcommutative C ∗ -algebra of all C -valued continuous functions: A ≃ C (Spec( A )).Let BS sur be a category of all Hausdorff Bauer simplexes (Choquet simplexes with closedChoquet boundary, see Definition 2.2) with surjective continuous affine maps. Consider afunctor P : CH sur → BS sur , which maps each compact Hausdorff space to the space of allRadon probability measures on it, equipped with the weak ∗ -topology, and each continuoussurjection T : X ։ Y sends to a continuous affine surjection T : P ( X ) ։ P ( Y ), whichis defined as follows: T ( µ )( B ) := µ ( T − ( B )), ∀ B ∈ Bor( Y ), T − ( B ) is the preimage of B , µ ∈ P ( X ). The (weak) inverse of this functor is ∂ e , which associates to every Bauer simplexits Choquet boundary. It is a standard fact, that a space of all Radon probability measureson a compact Hausdorff space is a Hausdorff Bauer simplex (see [1], [7], [17] for details). The8 ection 3. Space of quasi-states fact, that ∂ e ◦ P ≃ Id CH sur follows from the fact, that the Choquet boundary coincides withthe set of all Dirac measures, which are exactly the elements of the Gelfand spectrum. Anarguably non-trivial fact here is the equivalence P ◦ ∂ e ≃ Id BS sur on the level of morphisms.Let us check it. Proposition 3.1.
Let T : P ( X ) ։ P ( Y ) be a morphism in BS sur , T | X be the restriction of T to X , ( T | X ) | : P ( X ) → P ( Y ) be the corresponding pushforward of measures. Then T | X is a morphism between X and Y in CH sur , and ( T | X ) = T . Proof. T | X is, by definition, a continuous map from X to P ( Y ). Since T is affine, it sendsextreme points to extreme points, hence, T | X : X → Y . By surjectivity, contunuity, andaffinity of T , T − ( y ) is a nonempty closed face of P ( X ). It is standard, that extreme pointsof a face are extreme in the whole convex set. Hence, T − ( y ) ∩ X is not empty, and T | X issurjective.By Gelfand duality, ( T | X ) ∗ : C ( Y ) → C ( X ) is a morphism in ucC ∗ in . For every f ∈ C ( Y ),( T | X ) ∗ ( f ) := f ◦ T | X ∈ C ( X ) and R ( f ◦ T | X ) dµ = R f dT ( µ ) for every µ ∈ P ( X ). By the Rietzrepresentation theorem, there is a bijective correspondence between µ ∈ P ( Y ) and positivecontinuous linear functional on C ( Y ). Since(( T | X ) µ )( f ) := Z Y f ( y ) d (( T | X ) µ ) = Z X f ( T | X ( x )) dµ = Z Y f ( y ) dT ( µ ) = T ( µ )( f )is true for every f ∈ C ( Y ), the statement of the proposition follows. (cid:3) Let us call a topology on a convex set K compatible with the convex structure iff theoperation: A t : K × K → K , A t ( a, b ) = t · a + (1 − t ) · b is continuous for every t ∈ [0 , compact convex space we shall mean a convex set equipped with a compatible topology,such that it appears to be compact. Define a category of convex compact Hausdorff spaceswith affine continuous maps between them. We denote it as CCH . It is clear, that BS sur is asubcategory of CCH . Hence, there exists an injective on objects faithful functor from BS sur to CCH , associated with the inclusion. Denote it by i .The following diagram indicates the relationship between categories. Functor S : ucC ∗ inop → BS sur associates with an algebra the unit sphere of its Banach dual space, and sends everymorphism of algebras to the respective dual (adjoint) map. ucC ∗ inop CH sur BS sur CCH
Spec SC ( · ) P ∂ e i Let us state some facts about the category
CCH . Proposition 3.2.
For any set of objects { K α } in CCH , there is a categorical product Q α K α ,which is defined as a topological product with the natural convex structure: t · a + (1 − t ) · b = ( t · a α + (1 − t ) · b α )where a = ( a α ), b = ( b α ), t ∈ [0 , ection 3. Space of quasi-statesProof. A product convex structure is compatible if each factor has a compatible structure. Itfollows from the fact, that the product topology is the topology of pointwise convergence, and( a, b ) → t · a + (1 − t ) · b , t ∈ [0 , a α , b α ) → t · a α + (1 − t ) · b α is, which isexactly our case. It is a standard fact, that the defined product is a compact Hausdorff space.Moreover, projection maps are continuous and affine.For every such K ∈ CCH , that there exists a morphism ϕ α : K → K α in CCH for eachelement K α ∈ { K α } , define ϕ : K → Q α K α as follows: ϕ ( x ) := ( ϕ α ( x )) , ∀ x ∈ K It is straightforward to check, that this map is a continuous affine one, and that ϕ α = P r α ◦ ϕ .Suppose, that there is a morphism ψ : K → Q α K α , such that Pr α ◦ ψ = ϕ α for all α . Thenfor any x ∈ K , Pr α ( ψ ( x )) = ϕ α ( x ), so that ( ϕ ( x )) α = ϕ α ( x ) = ( ψ ( x )) α . Hence, ϕ = ψ , and ϕ is a unique morphism with this property. (cid:3) Proposition 3.3.
For any small diagram ( { K α } , T ), T = { T α,β : K α → K β , α ≤ β } in CCH (a diagram = a functor from a poset to
CCH ) there exists a projective limit. Hence, thecategory
CCH is complete.Projective limit is defined as follows:lim ←− ( { K α } , T ) = ( x ∈ Y α K α : ∀ T α,β ∈ T , x β = T α,β ( x α ) ) where x = ( x α ), and equipped with the induced topological and convex structure. Proof.
A convex structure on a projective limit is well-defined due to the affinity of morphisms T α,β .Let us prove that lim ←−{ K α } is a closed subspace of Q α K α . Let x = ( x α ) be in Q α K α ,but not in lim ←−{ K α } , i.e. we can find ( α, β ) : K α → K β , s.t. x β = T α,β ( x α ). Since K β isHausdorff, we can find disjoint open neighbourhoods V α of x α and U β of T α,β ( x α ). Since T α,β is continuous, U α := T − α,β ( V β ) is an open neighbourhood of x α . Thus U β × U α × Q γ α,β } K γ is an open neighbourhood of x in Q α K α that does not intersect with lim ←− ( { K α } , T ).Suppose we have such an object K and a family of morphisms ϕ α : K → K α , that ϕ β = T α,β ◦ ϕ α iff α ≤ β . Define ϕ : K → lim ←− K α by ϕ ( x ) = ( ϕ α ( x )). Then (Pr α ◦ ϕ )( x ) =Pr α (( ϕ α ( x ))) = ϕ α ( x ), hence Pr α ◦ ϕ = ϕ α . Suppose that there is a morphism ψ : K → lim ←− K α such that Pr α ◦ ψ = ϕ α for all α . Then for any x ∈ K , Pr α ( ψ ( x )) = ϕ α ( x ), so that ( ϕ ( x )) α = ϕ α ( x ) = ( ψ ( x )) α . Hence ϕ = ψ , and ϕ is a unique morphism with this property. (cid:3) Recall, that a unital C ∗ -subalgebra of a C ∗ -algebra A is a subset of A that includes identityelement and appears to be a unital C ∗ -algebra with respect to the inherited multiplication,involution, and linear structure. Since the inclusion of such a subalgebra in A is an injectiveunital C ∗ -homomorphism, which is inevitably isometric, the Banach structure of a subalgebracoincides with the inherited one.For a C ∗ -algebra A we can consider a diagram C ( A ) in ucC ∗ in of all unital commutative C ∗ -subalgebras of A ordered by inclusion. Denote it by C ( A ) := ( {A α } , ⊆ ). It is clear, thatevery inclusion of subalgebras is an injective unital C ∗ -homomorphism of C ∗ -algebras.Using the defined above functors Spec and i ◦ S we obtain a diagram in the category CH sur and a diagram in CCH . We denote them as Spec( C ( A )) := ( { Spec( A α ) } , R ) and S ( C ( A )) :=10 ection 3. Space of quasi-states ( { S ( A α ) } , R ) respectively. For shortness, we shall denote Spec( A α ) as X α , thus, in thisnotation, Spec( C ( A )) := ( { X α } , R ). The ordering is defined by the restriction maps: X α (cid:22) X β iff ∃ R β,α : X β ։ X α , R β,α ( x β ) := x β | A α ; S ( A α ) (cid:22) S ( A β ) iff ∃ ( R β,α ) : S ( A α ) ։ S ( A β ), ( R β,α ) ( µ β ) := µ β | A α . Note, that the restriction maps commute with the functor P :( R β,α ) ◦ P = P ◦ R β,α .Let us define the quasi-state space QS ( A ) for a C ∗ -algebra A as a projective limit of adiagram ( { S ( A α ) } , R ) in CCH . A little bit later we prove, that elements of the quasi-statespace are quasi-states in the sense of Definition 2.4.Recall, that an element a of a C ∗ -algebra is self-adjoint iff a = a ∗ . Denote the orderedBanach space of all self-adjoint elements of A by A sa (Banach and order structures are inheritedfrom A ). It is known (see [1]) that A sa is a Jordan Banach algebra (JB-algebra, for a definitionsee [1]) with respect to the product, defined as an anticommutator a ◦ b := ( a · b + b · a ). Inparticular, it is a commutative non-associative unital Banach algebra over R . This Jordanstructure is defined in terms of the multiplicative structure of A , but it can be also recoveredfrom the order structure of A sa . The order on A sa , defined by the Jordan product ( a (cid:23) b ⇐⇒∃ c ∈ A sa s.t. a − b = c ◦ c ) coincides with the order, inherited from A .Let us summarize some easy facts about self-adjoint part of an algebra in the followingLemma. Lemma 3.4.
Let A be a unital C ∗ -algebra, C ( A ) be a diagram (in ucC ∗ in ) of all unitalcommutative C ∗ -subalgebras of A (with inclusion relation). Then(1) For every A α ∈ C ( A ) its self-adjoint part, A saα , is a Banach unital associative subalgebraof the JB-algebra A sa .(2) A sa is covered by {A saα : A α ∈ C ( A ) } .(3) S ( A ) is determined by A sa .(4) C ( A ) is determined by A sa . Proof.
Let us prove these assertions.(1) By Gelfand duality, A α ≃ C (Spec( A α )), where ≃ is an isomorphism of C ∗ -algebras.The self-adjoint part in this identification corresponds to the set of all continuous real-valued functions on Spec( A α ). It obviously has a structure of a unital commutativeassociative algebra over R , and, being equipped with an inherited norm, it becomesa Banach algebra: the condition || a · b || ≤ || a || · || b || holds since it holds for A , com-pleteness follows from the fact, that the space of all continuous real-valued functionson a Hausdorff compact space is complete w.r.t. uniform topology, which is exactly thetopology determined by the norm || · || . A saα is a Banach subalgebra of A sa , since it isa Banach subalgebra of A , and x ◦ y = x · y for commutative elements of A sa .(2) Every a ∈ A sa generates a unital commutative C ∗ -subalgebra of A (it can be definedas a completion of the set of all polynomials in a over C ). Denote it by A α . It is clear,that a ∈ A saα .(3) Every element µ of S ( A ) is defined by its values on A . Note, that any f ∈ A can berepresented as a linear combination of two self-adjoint elements: a = a + a ∗ − i i ( a − a ∗ )2 .Then µ ( a ) = µ ( a + a ∗ ) − iµ ( i a − a ∗ ). Hence µ is determined by its values on A sa .The weak ∗ -topology on S ( A ) is defined as the weakest one, such that for every f ∈ A , µ → µ ( f ) is a continuous functional on S ( A ). Since µ ( f ) = µ ( a ) + iµ ( b )11 ection 3. Space of quasi-states for some a, b ∈ A sa , the weak ∗ -topology can be equivalently defined as the weakesttopology, such that for every a ∈ A sa , µ → µ ( a ) is a continuous functional.The inverse statement ( A sa can be recovered from S ( A ) as an ordered Banach spaceor, equivalently, as a JB-algebra) is also true, but we do not prove it here (in case ofinterest, see [1]).(4) Consider associative unital Banach subalgebras of A sa . By Theorem 1.12 of [1], eachof them is isometrically isomorphic to the algebra (and ordered Banach space) of allreal-valued continuous functions (equipped with the uniform norm) on some Hausdorffcompact space X . Let F ⊆ A sa be such a subalgebra. We wish to ensure, thatthe minimal C ∗ -subalgebra of A containing F (let us denote it as C ( F )) is a unitalcommutative C ∗ -subalgebra of A with the self-adjoint part F . As a C -vector space, C ( F ) contains the space { f + if : f , f ∈ F } ⊆ A . By axiomatic definition of a C ∗ -algebra, the involution, the multiplication, and the C ∗ -norm are uniquely definedon this set, making it a unital commutative C ∗ -algebra:(a) ( f + if ) ∗ = f − if due to self-adjointness of f , f .(b) ( f + if )( f + if ) = f f + if f + if f − f f due to the distributive law. Thismultiplication is commutative due to commutativity of f k , k = 1 , ... || ( f + if )( f + if ) ∗ || = || f + f || = || f + if || by definition of C ∗ -norm. Dueto positivity of f + f , || f + if || = || ( f + f ) || .(d) { f + if : f , f ∈ F } is complete w.r.t. this norm. Let ( f k + ig k ) be a Cauchysequence. Since || (( f k − f n ) + ( g k − g n ) ) || = || (( f k − f n ) + ( g k − g n ) ) || , || ( f k − f n ) + ( g k − g n ) || < ε implies || ( f k − f n ) || < ε , || ( g k − g n ) || < ε .Using the identity || a || = || a || , which is satisfied in a JB-algebra, || f k − f n || < ε , || g k − g n || < ε , hence ( f k ), ( g k ) are Cauchy sequences in F . Let f, g ∈ F be theirrespective limits. Then it follows from the inequality || ( f − f k ) + ( g − g k ) || ≤|| ( f − f k ) || + || ( g − g k ) || = || ( f − f k ) || + || ( g − g k ) || , that f + ig is a limit for( f k + ig k ).Due to the uniqueness, the defined commutative C ∗ -structure on { f + if : f , f ∈ F } ⊆ A coincides with the inherited one from A . Moreover, it coincides with the C ∗ -structure of the standard Banach complexification of C R ( X ) ≃ F , which is isometricallyisomorphic to C C ( X ). It is clear in this representation, that F is the set of its self-adjoint elements. Combining this result with the first statement of this Lemma, weconclude, that every A α ∈ C ( A ) can be constructed this way. The inclusions of C ∗ -subalgebras correspond to the inclusions of their self-adjoint parts. (cid:3) Proposition 3.5.
The quasi-state space is a set of quasi-states in the sense of Definition 2.4equipped with the natural (element-wise) convex structure and the projective topology: theweakest (coarsest) topology, such that for every unital commutative C ∗ -subalgebra A α ⊆ A ,all linear functionals of the form µ α → µ α ( f ) for all f ∈ A α are continuous. Here µ α is arestriction of µ ∈ QS ( A ) to A α . Proof.
By the definition of an element of QS ( A ), its restriction to A α should be a state on each A α . Since A sa is covered by {A saα : A α ∈ C ( A ) } , for any a ∈ A there exists A α ∈ C ( A ), s.t. µ ( a ∗ a ) = µ | A α ( a ∗ a ) ≥ µ (1) = µ | A α (1) = 1. Moreover, since by Lemma 3.4, A sa determines C ( A ), and, in particular, determines each state space S ( A α ) = S ( A saα ), every element of QS ( A )12 ection 3. Space of quasi-states is uniquely defined by its values on A sa . We can establish an isomorphism between QS ( A )and the set of quasi-states in the sense of Definition 2.4 extending µ ∈ QS ( A ) from A sa to A by the formula: µ ( a + ib ) = µ ( a ) + iµ ( b ), ∀ a, b ∈ A sa . Thus, we conclude that every elementof QS ( A ) can be thought of as a quasi-state in the sense of Definition (2.4).Recall, that a quasi-state on a C ∗ -algebra A is a map µ : A → C , such that it is linearon all commutative subalgebras, satisfies µ ( a + ib ) = µ ( a ) + iµ ( b ) for all self-adjoint a, b ∈ A , µ ( a ∗ a ) ≥ a ∈ A , and µ (1) = 1. If we restrict quasi-state on any unital commutative C ∗ -subalgebra A α , we obtain a positive linear functional s.t. µ (1) = 1. It follows from positivity,that µ α ( f ) ≤
1, if || f || ≤ f ∈ A saα (since 1 − f ≥ || µ α || = 1, µ α isa state on A α , and ( µ α ) ∈ Q S ( A α ). It is straightforward to check that ( µ α ) ∈ lim ←−{ S ( A α ) } .Since QS as a topological space is a projective limit of topological spaces S ( A α ), the topologyon it is defined as the weakest one, s.t. all projections (restriction maps in our case) arecontinuous. Each S ( A α ) is equipped with the weak ∗ -topology, hence, by definition, µ α → µ α ( f )for f ∈ A α should be continuous.Convex structure on QS ( A ) is inherited from Q S ( A α ), and, due to the fact, that A sa is covered by {A saα : A α ∈ C ( A ) } , it coincides with the natural (element-wise) one convexstructure:( tµ + (1 − t ) ν )( f ) = tµ α ( f ) + (1 − t ) ν α ( f ) = tµ ( f ) + (1 − t ) ν ( f ) , ∀ t ∈ [0 , f ∈ A saα ⊆ A sa . (cid:3) Note, that it follows directly from the definition, that QS ( A ) is a compact convex Hausdorffspace. Proposition 3.6. QS ( A ) is equipped with the weakest topology such that ∀ f ∈ A , µ → µ ( f )is a continuous functional. Proof.
By the definition of the quasi-state space, for every unital commutative C ∗ -subalgebra A α ⊆ A , all functionals of the form µ α → µ α ( f ) for f ∈ A α are continuous. Since A sa iscovered by self-adjoint parts of elements of C ( A ), for any a ∈ A sa there exists a commutativeunital C ∗ -subalgebra A α of A , s.t. a ∈ A saα . Hence µ → µ ( a ) = µ | A α ( a ) is continuous for all f ∈ A sa . Since every f can be represented as f = a + ib , where a, b ∈ A sa , a ∈ A saα , b ∈ A saβ for some commutative unital C ∗ -subalgebras A α , A β of A , the functional µ → µ ( f ) = µ ( a ) + iµ ( b ) = µ | A α ( a ) + iµ | A β ( b )is continuous (since it is a linear combination of two continuous functionals) for every f ∈ A .If ∀ f ∈ A , µ → µ ( f ) is continuous, then for any commutative unital C ∗ -subalgebra A α , f ∈ A α ⊆ A implies µ | A α → µ | A α ( f ) = µ ( f ) is continuous. (cid:3) Corollary 3.7.
Topology on S ( A ), induced by the inclusion S ⊆ QS ( A ), coincides with theweak ∗ -topology. Proof. QS ( A ) is equipped with the weakest topology, such that ∀ f ∈ A , µ → µ ( f ) is acontinuous map. It is exactly the definition of the weak ∗ -topology on S ( A ). (cid:3) Let us review the picture. • Both S ( A ) and QS ( A ) are compact convex spaces for any unital C ∗ -algebra A . Thereis a natural inclusion S ( A ) ⊆ QS ( A ), such that S ( A ) is a compact convex subset of QS ( A ). 13 ection 4. Wasserstein distances for abelian C ∗ -subalgebras • QS ( A ) is a projective limit of ( { S ( A α ) } , R ) in the category CCH of compact convexspaces. In some sense this result is dual to the result of [4], where the injective limit of( {A α } , ⊆ ) is described in the category of partial C ∗ -algebras (see its definition there).We do not formalize this duality, since it is not clear how to axiomatically describe QS -spaces. • Obviously, the state space S ( A ) contains not less information than the correspondingquasi-state space QS ( A ). The following picture illustrates the relation between theobjects. The arrow means “the right object is uniquely determined by the left one”, ! means “both the right and the left objects determine each other in the unique way”. S ( A ) A sa ( {A saα } , ⊆ ) QS ( A ) ( {A α } , ⊆ )In the case S ( A ) = QS ( A ), all entities in the picture uniquely determine each other.It is not clear for the author, is there an arrow QS ( A ) ( {A α } , ⊆ ) or not in the generalcase (i.e. is it possible to recover the diagram of all unital commutative C ∗ -subalgebrasknowing only the quasi-state space of an algebra?).4. Wasserstein distances for abelian C ∗ -subalgebras Let A be a unital C ∗ -algebra. Define • C ( A ) := ( {A α } , ⊆ ) as a diagram in ucC ∗ in of all unital commutative C ∗ -subalgebras of A ordered by inclusion, • Spec( C ( A )) := ( { Spec( A α ) } , R ) as a diagram in CH sur of the Gelfand spectra, orderedby restriction maps: R β,α : Spec( A β ) ։ Spec( A α ), R β,α ( ϕ β ) = ϕ α , which are definedfor any ordered pair A α ⊆ A β from C ( A ). For shortness, we shall denote Spec( A α ) as X α , thus, Spec( C ( A )) := ( { X α } , R ), • S ( C ( A )) := ( { S ( A α ) } , R ) as a diagram in CCH of the state spaces, ordered byrestriction maps: ( R β,α ) : S ( A β ) ։ S ( A α ), ( R β,α ) ( µ β ) = µ α , which are defined forany ordered pair A α ⊆ A β from C ( A ). As it was discussed earlier, S ( A α ) ≃ P ( X α ).By abuse of notation, we shall denote by the same letter a state (functional) definedon A α and its corresponding representation as a measure on X α . As it has been shownearlier, ( R β,α ) ( µ β ) = µ β ◦ R − β,α .We are going to use the theory of Lip-seminorms on C ∗ -algebras (or, more generally, on order-unit spaces), which was developed by Rieffel ([18], [19], [20]). The idea is to define an abstractanalogue of a Lipschitz gauge axiomatically. Let A be a unital C ∗ -algebra, L : A sa → [0 , + ∞ ]be an abstract analogue of a Lipschitz gauge. Define the following notation: • B ( A ) := { a ∈ A sa : L ( a ) < + ∞}• B ( A ) := { a ∈ A sa : L ( a ) = 1 }• N ( A ) := { a ∈ A sa : L ( a ) = 0 } If it does not lead to a confusion, we shall use B , B , N instead of B ( A ), B ( A ) and N ( A )respectively.Let us provide the following definition. Definition 4.1. L : A sa → [0 , + ∞ ] is an L -seminorm on a unital C ∗ -algebra A iff it is14 ection 4. Wasserstein distances for abelian C ∗ -subalgebras (1) absolutely homogeneous: L ( af ) = | a | L ( f ), for any a ∈ R , f ∈ A sa ,(2) subadditive: L ( f + g ) ≤ L ( f ) + L ( g ), for any f, g ∈ A sa ,(3) L (1) = 0,(4) B ( A ) separates points in S ( A ): for any two distinct states µ, ν ∈ S ( A ) there exists f ∈ B ( A ) s.t. µ ( f ) = ν ( f ). Remark 4.2.
We are going to use the term Lip-seminorm in the same sense as Rieffel do.Since the axioms above are weaker then the axioms of Lip-seminorm, we gave another namefor this object.
Lemma 4.3.
The property “ B ( A ) separates points in S ( A )” is equivalent to “ B ( A ) is a weakdense subspace of A ”, where weak density means S ( A ) = S +1 ( B ( A ) ∗ ) ( B ( A ) is assumed to beequipped with the topology induced by the inclusion in A ). Proof. If B := B ( A ) does not separate points, there are two µ, ν ∈ S ( A ), µ = ν s.t. µ ( f ) = ν ( f ), ∀ f ∈ B . Hence the restriction map S ( A ) → S +1 ( B ∗ ), µ → µ | B is not injective, becausedifferent µ and ν has the same image.If S ( A ) = S +1 ( B ∗ ) then S ( A ) → S +1 ( B ∗ ), µ → µ | B is not injective (since it is surjective dueto Hahn-Banach theorem, but not one-to-one). Hence there are two distinct points µ, ν ∈ S ( A )s.t. ∀ f ∈ B : µ ( f ) = ν ( f ), which contradicts with the separation of points. (cid:3) Sometimes it is useful to have a weaker definition of an abstract Lipschitz gauge:
Definition 4.4. L : A sa → [0 , + ∞ ] is a partially-defined L -seminorm on a unital C ∗ -algebra A if it is(1) absolutely homogeneous: L ( af ) = | a | L ( f ), for any a ∈ R , f ∈ A sa ,(2) subadditive: L ( f + g ) ≤ L ( f ) + L ( g ), for any f, g ∈ A sa ,(3) L (1) = 0.Here we do not require any type of separation of points. Thus, it is possible for a partially-defined L -seminorm to be infinite everywhere except 1 · R . The important (but obvious) factis that a restriction of any L -seminorm from A to any unital C ∗ -subalgebra A α of A is apartially-defined L -seminorm on A α .The following map d L : S ( A ) × S ( A ) → [0 , + ∞ ] d L ( µ, ν ) := sup {| µ ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A sa } is called a spectral distance on S ( A ). Let us review some known facts about this map.Recall that a pseudo-distance function is the same thing as a distance function, except itmay vanish on pairs of non-equal points. Proposition 4.5. If L is a partially-defined seminorm on a unital C ∗ -algebra A , then d L is a [0 , + ∞ ]-valued lower semi-continuous pseudo-distance function on S ( A ). If L is an L -seminorm, d L is [0 , + ∞ ]-valued distance function on S ( A ). Proof.
Similar statements were proved by many authors (see, for example, [10], [18]). Forconvenience, we provide a proof here. • Since L (1) = 0, there is at least one f s.t. L ( f ) ≤
1. For every such f , F ( µ, ν ) := | µ ( f ) − ν ( f ) | is a continuous map from S ( A ) × S ( A ) to [0 , + ∞ ) (since every µ → µ ( f )15 ection 4. Wasserstein distances for abelian C ∗ -subalgebras is continuous by definition of the weak ∗ -topology on S ( A )). Hence d L is a lower semi-continuous map from S ( A ) × S ( A ) to [0 , + ∞ ] as a pointwise supremum of continuousmaps. • Obviously, d L ( µ, µ ) = 0. If L is L -seminorm, for µ = ν , due to the weak density of B , µ | B = ν | B . Hence, there is an f ∈ B s.t. | µ ( f ) − ν ( f ) | >
0. We can scale this f bysome α >
0, s.t. L ( αf ) ≤
1, and obtain that d L ( µ, ν ) ≥ | α || µ ( f ) − ν ( f ) | >
0. Thus, inthe case of L -seminorm, d L ( µ, ν ) = 0 iff µ = ν . • d L ( µ, ν ) = d L ( ν, µ ) obviously. • Check the triangle inequality: d L ( µ, ν ) + d L ( ν, η ) == sup {| µ ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A sa } + sup {| η ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A sa } ≥≥ sup {| µ ( f ) − ν ( f ) | + | η ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A sa } ≥≥ sup {| µ ( f ) − ν ( f ) + ν ( f ) − η ( f ) | : L ( f ) ≤ , f ∈ A sa } ≥≥ sup {| µ ( f ) − η ( f ) | : L ( f ) ≤ , f ∈ A sa } = d L ( µ, η ) (cid:3) The important result of Rieffel ([19], Th. 4.1, Th 4.2) is that in the case L is an L -seminorm, L d ( f ) := sup (cid:26) f ( µ ) − f ( ν ) d L ( µ, ν ) : µ = ν, µ, ν ∈ S ( A ) (cid:27) is also an L -seminorm, and it induces the same distance on S ( A ) as L does: d L = d L d .Moreover, L d Ld = L d . If L is a lower semi-continuous L -seminorm: { f ∈ A sa : L ( f ) ≤ t } isnorm closed for one, hence every, t ∈ (0 , + ∞ ), then L = L d . We do not require lower semi-continuity in the definition of L -seminorm, but it is clear, that we are always able to consider L d instead of the original L -seminorm L .In the case L is an L -seminorm, it is also true ([18], Pr. 1.4), that topology generated by d L is not weaker than the weak ∗ -topology on S ( A ). In the case of separable A , and, hence,metrizable S ( A ), it is natural to ask these two topologies to coincide. The following definitionis due to Rieffel (see [19]). Definition 4.6. An L -seminorm L on a unital separable C ∗ -algebra A is called Lip -seminorm iff (1) L ( f ) = 0 ⇐⇒ f = a · , a ∈ R ,(2) a distance function d L on S ( A ) metricizes the weak ∗ -topology on S ( A ).It is possible to formulate a criterion for an L -seminorm L to be a Lip-seminorm. Let B := B ( A ), B := B ( A ), N := N ( A ). Note that B/N is a norm space equipped with thequotient norm || · ||
B/N defined as || f + N || B/N := inf n ∈ N || f + n || B , where || · || B is a norm on B inherited from the inclusion B ⊆ A . Proposition 4.7 (Criterion for L to be a Lip-seminorm, [18], Th. 1.8) . An L -seminorm L ona unital separable C ∗ -algebra A is a Lip-seminorm iff N = R · B in B/N is totally bounded w.r.t. the quotient norm || · ||
B/N .16 ection 4. Wasserstein distances for abelian C ∗ -subalgebras Now we consider a unital commutative C ∗ -subalgebra A α ⊆ A . Let us define d L,α : S ( A α ) × S ( A α ) → [0 , ∞ ] as follows: d L,α ( µ α , ν α ) := sup {| µ ( f ) − ν ( f ) | : f ∈ A saα , L ( f ) ≤ } Proposition 4.8.
For any C ∗ -algebra A and any partially-defined L -seminorm L on A , d L,α is a lower semi-continuous [0 , + ∞ ]-valued pseudo-distance function. If B α := B ( A ) ∩ A α separates points in S ( A α ) (or, equivalently, restriction of L to A α is an L -seminorm), then itis a measurable [0 , + ∞ ]-valued distance function. Proof. • Since L (1) = 0, and A α is a unital subalgebra, there exists at least one f ∈ A saα s.t. L ( f ) ≤
1. For every such f , F ( µ α , ν α ) := | µ α ( f ) − ν α ( f ) | is a continuous mapfrom S ( A α ) × S ( A α ) to [0 , + ∞ ). Hence, d L,α is a lower semi-continuous map from S ( A ) × S ( A α ) to [0 , + ∞ ] as a pointwise supremum of continuous maps. • Obviously, d L,α ( µ α , µ α ) = 0. • If B α is weak dense in A α , for every µ α = ν α , µ α | B = ν α | B . Hence, there is an f ∈ B s.t. | µ α ( f ) − ν α ( f ) | >
0. We can scale this f by some t >
0, s.t. L ( tf ) ≤
1, and obtainthat d L,α ( µ α , ν α ) ≥ | t || µ α ( f ) − ν α ( f ) | >
0. Thus, d L,α ( µ α , ν α ) = 0 iff µ α = ν α . • d L,α ( µ α , ν α ) = d L,α ( ν α , µ α ) obviously. • Check the triangle inequality: d L,α ( µ α , ν α ) + d L,α ( ν α , η α ) == sup {| µ α ( f ) − ν α ( f ) | : L ( f ) ≤ , f ∈ A saα } + sup {| η α ( f ) − ν α ( f ) | : L ( f ) ≤ , f ∈ A saα } ≥≥ sup {| µ α ( f ) − ν α ( f ) | + | η ( f ) − ν α ( f ) | : L ( f ) ≤ , f ∈ A saα } ≥≥ sup {| µ α ( f ) − ν α ( f ) + ν α ( f ) − η α ( f ) | : L ( f ) ≤ , f ∈ A saα } ≥≥ sup {| µ α ( f ) − η α ( f ) | : L ( f ) ≤ , f ∈ A saα } = d L,α ( µ α , η α ) (cid:3) Note, that the restriction of d L,α on X α ⊆ S ( A α ) remains to be a lower semi-continuous[0 , + ∞ ]-valued pseudo-distance function. Lemma 4.9. If B ∩ A α separates points in S ( A α ), where B := { f ∈ A sa : L ( f ) < ∞} , A α is a unital commutative C ∗ -subalgebra of A (equivalently, the restriction of L to A α isan L -seminorm), and L is a Lip-seminorm on A , then the restriction of L to A α is also aLip-seminorm. Proof.
Since the restriction of L to A α is an L -seminorm, it remains to check, that the distance d L,α metricizes the weak ∗ -topology on A α . Let us use the criterion from Proposition 4.7.Since N = R · N ⊆ A α , C ∗ -norm on A α coincides with the norm inherited from A , and( B ∩A α ) / ( N ∩A α ) = ( B ∩A α ) /N ⊆ B/N , it follows that the norms ||·||
B/N = ||·|| ( B ∩A α ) / ( N ∩A α ) on their common domain of definition. The image of B ∩ A α in ( B ∩ A α ) / ( N ∩ A α ) coincideswith the image of B ∩ A α in B/N , and, in fact, it is a subset of a totally bounded image of B in B/N . Hence, it is totally bounded. (cid:3)
Let us define W p,α : S ( A α ) × S ( A α ) → [0 , ∞ ] for p ∈ [1 , + ∞ ) by the formula: W p,α ( µ α , ν α ) := inf ((cid:18)Z d pL,α ( x, y ) dπ (cid:19) p : π ∈ P ( X α × X α ) , Pr( π ) = ( µ α , ν α ) ) ection 4. Wasserstein distances for abelian C ∗ -subalgebras Here P ( X α × X α ) is the set of all Radon probability measures on X α × X α (recall, thateach X α is equipped with the weak ∗ -topology, and all σ -algebras are supposed to be Borel),Pr( π ) := ((Pr ) ( π ) , (Pr ) ( π )).In the formulation of the next statement we shall use the following notion, introduced in[19]. Definition 4.10. ([19], Definition 4.5) Let L be a Lip-seminorm on a C ∗ -algebra A , B := { a ∈ A sa : L ( a ) < ∞} , B := { a ∈ A sa : L ( a ) ≤ } , ¯ B , ¯ B be their closures w.r.t. C ∗ -norm on A . The closure of L , ¯ L : ¯ B → [0 , + ∞ ] is defined as¯ L ( a ) := inf { t ∈ [0 , + ∞ ] : ∃ b ∈ ¯ B s.t. a = t · b } Proposition 4.11.
In the case(1) A is a separable unital C ∗ -algebra,(2) L is a Lip-seminorm on A ,(3) B ∩ A α is dense in A α ,(4) for L α := L | A α the inequality¯ L α (sup( f, g )) ≤ max( ¯ L α ( f ) , ¯ L α ( g ))is satisfied for all f, g ∈ ¯ B ∩ A α ,we obtain a valid equality: d L,α ( µ α , ν α ) = inf (cid:26)Z d L,α ( x, y ) dπ : π ∈ P ( X α × X α ) , Pr( π ) = ( µ α , ν α ) (cid:27) =: W ,α ( µ α , ν α )for all µ α , ν α ∈ S ( A α ). Proof.
Since S ( A α ) is a Bauer simplex, each element µ α ∈ S ( A α ) has a unique representationas a Radon probability measure on ∂ e ( S ( A α )) ≃ X α , i.e. ˜ µ α ∈ P ( X α ), bar(˜ µ α ) = µ α . ByLemma 4.9, the restriction of L on A α is a Lip-seminorm, thus d L metricizes the weak ∗ -topologyon S ( A ).Due to the inequality from the hypothesis of the Lemma and density of B ∩ A α , by Theorem8.1 of [19], L α := L | A α coincides with the Lipschitz seminorm induced by the distance d L on X α : L α ( f ) = L ed L ( f ) := sup (cid:26) f ( x ) − f ( y ) d L ( x, y ) : x = y, x, y ∈ X α (cid:27) , ∀ f ∈ A saα Hence we can use the Kantorovich-Rubinstein duality (see [6]) to obtain: W ,α ( µ α , ν α ) := inf (cid:26)Z d L,α ( x, y ) dπ : π ∈ P ( X α × X α ) , Pr( π ) = (˜ µ α , ˜ ν α ) (cid:27) == sup { µ α ( f ) − ν α ( f ) : f ∈ A saα , L ed L ( f ) ≤ } == sup { µ α ( f ) − ν α ( f ) : f ∈ A saα , L ( f ) ≤ } =: d L,α ( µ α , ν α ) (cid:3) Proposition 4.12.
For any C ∗ -algebra A and any partially-defined L -seminorm L on A , W p,α is a [0 , + ∞ ]-valued pseudo-distance function on S ( A α ). If B α := B ∩A α separates points in A α (or, equivalently, the restriction of L to A α is an L -seminorm), then W p,α is a [0 , + ∞ ]-valueddistance function. 18 ection 4. Wasserstein distances for abelian C ∗ -subalgebrasProof. By a slight abuse of notation, we denote as µ α both the linear functional (state) µ restricted to A α and the measure on X α that corresponds to this state. By the Rietz repre-sentation theorem, it is a Radon probability measure. By Π( µ α , ν α ) we denote the set of allRadon probability measures on X α × X α with the marginals µ α and ν α . • W p,α ( µ α , µ α ) = 0, because d pL,α ( x, x ) = 0, and (Id , Id) µ α is an element of Π( µ α , µ α ),which is concentrated on the diagonal set Diag := { ( x, x ) : x ∈ X α } . • If B α is weak dense in B , d pL,α is a [0 , + ∞ ]-valued distance function. Hence d pL,α ( x, x ) =0 iff x = y . Since Π( µ α , ν α ) contains no measures π s.t. π (Diag) = 1, when µ α = ν α ,it follows that W p,α ( µ α , ν α ) > • W p,α ( µ α , ν α ) = W p,α ( ν α , µ α ), because ∀ π ∈ Π( µ α , ν α ) there exists π T ∈ Π( ν α , µ α )defined as R f ( x, y ) dπ = R f ( y, x ) dπ T for all measurable functions f on X α × X α , andbecause d pL,α is Borel measurable and symmetric. • By Lemma 1.1.6 from [6] for any two Radon measures π , π on X α × X α s.t. the secondmarginal of the first measure coincides with the first marginal of the second measure,there exists a Radon measure on X α × X α × X α , such that its projection on the firsttwo factors is equal to π , and the projection on the last two factors is equal to π . Letfor any n ∈ N , π n ∈ Π( µ α , ν α ) s.t. W p,α ( µ α , ν α ) ≥ (cid:16)R d pL,α π n (cid:17) p − n , π n ∈ Π( ν α , η α ) s.t. W p,α ( ν α , η α ) ≥ (cid:16)R d pL,α π n (cid:17) p − n , γ n be a measure in Π( µ α , ν α , η α ) as in the describedLemma with projections π n , π n . Then W p,α ( µ α , η α ) ≤ (cid:18)Z d pL,α ( x, z ) d Pr , ( γ n ) (cid:19) p = (cid:18)Z d pL,α ( x, z ) dγ n (cid:19) p ≤≤ (cid:18)Z ( d L,α ( x, y ) + d L,α ( y, z )) p dγ n (cid:19) p ≤ (cid:18)Z ( d L,α ( x, y ) dγ n (cid:19) p + (cid:18)Z ( d L,α ( y, z ) dγ n (cid:19) p == (cid:18)Z d pL,α π n (cid:19) p + (cid:18)Z d pL,α π n (cid:19) p ≤ W p,α ( µ α , ν α ) + W p,α ( ν α , η α ) + 2 n Taking a limit as n → ∞ in both parts of the inequality, we obtain the requiredstatement. (cid:3) Proposition 4.13.
Let L be a partially-defined L -seminorm on a unital C ∗ -algebra A . Forany µ, ν ∈ QS ( A ) and any pair A α ⊆ A β of unital commutative C ∗ -subalgebras of A , it is truethat W p,β ( µ β , ν β ) ≥ W p,α ( µ α , ν α ) , Proof.
Note, that d L,β ( µ β , ν β ) ≥ d L,α ( µ α , ν α ) as follows directly from the definition of d L,α .Since the restriction map R β,α : X β ։ X α is surjective (as follows from Gelfand duality), thedirect product map ( R × R ) β,α : X β × X β ։ X α × X α , ( R × R ) β,α ( x β , y β ) = ( R β,α ( x β ) , R β,α ( y β )),and the corresponding pushforward of measures (( R × R ) β,α ) : P ( X β × X β ) ։ P ( X α × X α ) arealso surjective. Since projection operator commutes with the restriction maps, (( R × R ) β,α ) :Π( µ β , ν β ) ։ Π( µ α , ν α ) is well-defined and surjective too. Thus, every π α ∈ P ( µ α , ν α ) is equal19 ection 5. Projective L p -Wasserstein distances to (( R × R ) β,α ) π β for some π β ∈ P ( µ β , ν β ). It follows that R d L,α dπ α = R d L,α ◦ ( R × R ) β,α dπ β .But d L,β ( µ β , ν β ) ≥ d L,α ( µ α , ν α ) = ( d L,α ◦ ( R × R ) β,α )( µ β , ν β )and, hence, R d L,α dπ α ≤ R d L,β dπ β . Thus, W p,β ( µ β , ν β ) ≥ W p,α ( µ α , ν α ). (cid:3) Proposition 4.14.
Let L be a partially-defined L -seminorm on a unital C ∗ -algebra A , A α ⊆ A be a unital commutative C ∗ -subalgebra. Then W p,α ≤ W q,α if p ≤ q . Proof.
By definition, W p,α ( µ α , ν α ) = inf {|| d p,α || L p ( π ) : π ∈ Π( µ α , ν α ) } . The statement followsdirectly from the fact, that || · || L p ( π ) ≤ || · || L q ( π ) for p ≤ q and any probability measure π . (cid:3) Proposition 4.15.
Let A be a separable unital C ∗ -algebra, L be a Lip-seminorm, and B ∩ A α separates points in S ( A α ), where B := { f ∈ A sa : L ( f ) < ∞} , A α is a unital commutative C ∗ -subalgebra of A (equivalently, the restriction of L to A α is an L -seminorm). Then W p,α metricizes the weak ∗ -topology on S ( A α ) for every p ∈ [1 , ∞ ). Proof.
By Proposition 4.9, L , being restricted to A α , remains to be a Lip-seminorm. Hence(by definition of a Lip-seminorm) d L,α metricizes the weak ∗ -topology on S ( A α ).By definition, W pp,α ( µ α , ν α ) := inf nR d pL,α dπ : π ∈ Π( µ α , ν α ) o is the L p -Wasserstein distancebetween measures µ α and ν α on a compact metric space ( X α , d L,α ). Hence, by the standardtheory of Wasserstein distances (see any of [2], [6], [21]), it metricizes the weak ∗ -topology on P ( X α ) = S ( A α ). (cid:3) Projective L p -Wasserstein distances Let A be a unital C ∗ -algebra. Define C ( A ) := ( {A α } , ⊆ ), Spec ( C ( A )) := ( { Spec( A α ) } , R ), S ( C ( A )) := ( { S ( A α ) } , R ) as in the previous section.We are going to construct a distance function on the quasi-state space of A using distancefunctions on the state spaces of its unital commutative subalgebras. As we have already seen,distance functions W p,α on S ( A α ) are well-behaved (they are actually distances and metricizethe weak ∗ -topologies) in the case L is finite on a large enough subspace of A α , or, moreprecisely, if B ⊆ A α separates points in S ( A α ) (here B := { a ∈ A sa : L ( a ) < ∞} as earlier).It motivates the following definition. Definition 5.1. An L -seminorm L on a unital C ∗ -algebra A is called solid iff B ∩A α separatespoints in S ( A α ) for any maximal unital commutative subalgebra A α of A .In other words, we ask an L -seminorm to remain to be an L -seminorm, when it is restricted toa maximal subalgebra. It appears (as Proposition 4.9 states), that in this case a Lip-seminormremains to be a Lip-seminorm when restricted to a maximal subalgebra. Example 5.2. If L -seminorm L on A has only finite values, it is solid. Remark 5.3.
As it was shown in Lemma 3.4, for every unital C ∗ -algebra A , its self-adjoint part A sa is covered by self-adjoint parts of unital commutative C ∗ -subalgebras of A . Let us fix some f ∈ A sa . Commutative unital C ∗ -subalgebras containing f constitute a poset w.r.t. inclusion.Each chain of this poset has an upper bound (union of the corresponding subalgebras). UsingZorn’s lemma, we conclude, that f is a self-adjoint element of some maximal commutativeunital C ∗ -subalgebra. Hence, the self-adjoint part of a unital C ∗ -algebra is covered by theself-adjoint parts of its maximal unital commutative C ∗ -subalgebras.20 ection 5. Projective L p -Wasserstein distances As follows from the results of the previous section, if L is a solid L -seminorm on A , thenfor every maximal unital commutative C ∗ -subalgebra A α , W p,α is a [0 , + ∞ ]-valued distancefunction. If, moreover, L is a Lip-seminorm (which is possible only if A is separable), W p,α metricizes the weak ∗ -topology on S ( A α ). Proposition 5.4. If L is a solid L -seminorm on A , B ( A ) separates points in QS ( A ). Proof.
Let µ, ν ∈ QS ( A ), µ = ν , but µ | B = ν | B . It follows that µ | B ∩A α = ν | B ∩A α for everymaximal commutative unital C ∗ -subalgebra A α . Since L is solid, µ | A α = ν | A α . Since maximalunital commutative subalgebras of A covers A sa , it follows that µ = ν , which contradicts theassumption. (cid:3) This fact allows us to extend distance function d L from the state space S ( A ) to the quasi-state space QS ( A ). Definition 5.5.
Let A be a unital C ∗ -algebra, L be a partially-defined L -seminorm on A .Then d L : QS ( A ) × QS ( A ) → [0 , ∞ ] is defined by the formula: d L ( µ, ν ) := sup {| µ ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A sa } Proposition 5.6. d L is a [0 , + ∞ ]-valued lower semi-continuous pseudo-distance function on QS ( A ). If L is a solid L -seminorm, d L is a [0 , + ∞ ]-valued lower semi-continuous distancefunction on QS ( A ). Proof.
The proof is analogous to the proof of Proposition 4.5. We only provide an argumentfor the lower semi-continuity of d L on the new domain of definition. The rest part of the proofcan be copy-pasted directly from the proof of Proposition 4.5.Note, that F ( µ, ν ) := | µ ( f ) − ν ( f ) | is a continuous map from QS ( A ) × QS ( A ) to [0 , + ∞ ),because every µ → µ ( f ) is continuous by the definition of the projective topology on QS ( A ).Hence d L is a lower semi-continuous map from QS ( A ) × QS ( A ) to [0 , + ∞ ] as a pointwisesupremum of continuous maps. In case L is solid, by Proposition 5.4, B ( A ) separates pointsin QS ( A ), and, hence, d L ( µ, ν ) = 0 implies µ = ν (as in 4.5). (cid:3) Definition 5.7.
For any p ∈ [1 , ∞ ] define the projective L p -Wasserstein distance W ←− p : QS ( A ) × QS ( A ) → [0 , ∞ ] by the formula: W ←− p ( µ, ν ) := sup α { W p,α ( µ α , ν α ) } where µ α := µ | A α . Proposition 5.8.
For any unital C ∗ -algebra A and any partially-defined L -seminorm L , W ←− p is a [0 , + ∞ ]-valued pseudo-distance function. If L is a solid L -seminorm, then W ←− p is a [0 , + ∞ ]-valued distance function. Proof.
According to Proposition 4.13, for any two µ, ν ∈ QS ( A ), W p,β ( µ β , ν β ) ≥ W p,α ( µ α , ν α )if A α ⊆ A β . It follows that W ←− p ( µ, ν ) = sup { W p,β ( µ β , ν β ) : A β is a maximal element of ( {A α } , ⊆ ) } In other words, we may take supremum over the set consisted of only maximal subalgebras.Since for each of these subalgebras, W p,α is a [0 , + ∞ ]-valued pseudo-distance function on S ( A α ) (by Proposition 4.12), we may think of it as a [0 , + ∞ ]-valued pseudo-distance function21 ection 5. Projective L p -Wasserstein distances on S ( A ): W p,α ( µ, ν ) := W p,α ( µ α , ν α ). It is straightforward to check that W ←− p is a [0 , + ∞ ]-valuedpseudo-distance function.If L is solid, by Proposition 5.4, B ( A ) separates points in QS ( A ). Hence, µ = ν = ⇒ µ ( f ) = ν ( f ) for some f ∈ A . Since a self-adjoint part of any C ∗ -algebra is covered by its maximalunital commutative subalgebras (and states of C ∗ -algebra are determined by their values onself-adjoint elements), it follows that µ = ν = ⇒ µ α = ν α for some maximal A α ∈ C ( A ).Hence W ←− p ( µ, ν ) > µ = ν , and W ←− p is a [0 , + ∞ ]-valued distance function. (cid:3) Definition 5.9.
A Lip-seminorm L on a unital C ∗ -algebra A is called Wasserstein-compatible iff for any maximal unital commutative C ∗ -subalgebra A α of A (1) B ∩ A α is dense in A α ,(2) for the closure of L α := L | A α , ¯ L α , the inequality¯ L α (sup( f, g )) ≤ max( ¯ L α ( f ) , ¯ L α ( g ))is satisfied for all f, g ∈ ¯ B ∩ A α . Here B := { a ∈ A sa : L ( a ) ≤ } .Since a dense subspace of a Banach space separates points in its dual, every Wasserstein-compatible Lip-seminorm is a solid Lip-seminorm. Remark 5.10.
If Lip-seminorm L on A has only finite values and L (sup( f, g )) ≤ max( L ( f ) , L ( g )) , ∀ f, g ∈ A sa , then it is Wasserstein-compatible. More generally, if this inequality is satisfied, B := { a ∈A sa : L ( a ) < ∞} is closed under finite lattice operations, and each B ∩ A α is dense in A α forany maximal unital commutative C ∗ -subalgebra A α of A , then L is Wasserstein-compatible.It follows from Corollary 8.3 of [19] and closedness of C ∗ -subalgebras.As we shall see later, if Lip-seminorm is Wasserstein-compatible, the projective Wassersteindistances metricize the weak ∗ -topology of the state space. Unfortunately, this assumptionseems to be too strong, and there are no known non-trivial examples of Wasserstein-compatibleLip-seminorms. Proposition 5.11.
For any unital separable C ∗ -algebra A and any Wasserstein-compatibleLip-seminorm L on A , d L = W ←− on QS ( A ). Proof.
Since self-adjoint part of every C ∗ -algebra is covered by its maximal unital commutativesubalgebras, d L ( µ, ν ) := sup {| µ ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A sa } == sup α { sup {| µ ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A saα }} , ∀ µ, ν ∈ QS ( A ) , where α parametrizes the set of all maximal unital commutative C ∗ -subalgebras of A . It followsby the definition of Wasserstein-compatible Lip-seminorm, that all assumptions of Proposition4.11 are satisfied, hence W ,α = d L,α for every such α . Thussup α { sup {| µ ( f ) − ν ( f ) | : L ( f ) ≤ , f ∈ A saα }} == sup α { sup {| µ α ( f ) − ν α ( f ) | : L ( f ) ≤ , f ∈ A saα }} == sup α { d L,α ( µ α , ν α ) } = sup α { W ,α ( µ α , ν α ) } =: W ←− ( µ, ν )22 ection 5. Projective L p -Wasserstein distances where µ α = µ | A α ∈ S ( A α ) is a restriction of a quasi-state to a subalgebra. (cid:3) Let us define diameters of the quasi-state space and the state space of A as follows.diam( QS ( A )) := sup { d L ( µ, ν ) : µ, ν ∈ QS ( A ) } (4) diam( S ( A )) := sup { d L ( µ, ν ) : µ, ν ∈ S ( A ) } (5)Here A is a unital C ∗ -algebra, L is an L -seminorm. It is clear, that in the case of Lip-seminorm L , diam( S ( A )) < ∞ . Let(6) diam( S ( A α )) := sup { d L,α ( µ α , ν α ) : µ α , ν α ∈ S ( A α ) } Proposition 5.12.
Let A be a unital C ∗ -algebra, L be a partially-defined L -seminorm on A .Then sup α { diam( S ( A α )) } = diam( S ( A )) = diam( QS ( A ))where α parametrizes all maximal unital commutative C ∗ -subalgebras of A . Proof.
Since the restriction map ( R α ) : S ( A ) ։ S ( A α ), ( R α ) ( µ ) = µ α := µ | A α is surjective,and the analogous map, defined on QS ( A ),( R α ) : QS ( A ) ։ S ( A α ) , is surjective too, the equality sup α { d L,α ( µ | A α , ν | A α ) } = d L ( µ, ν ), where α parametrizes allmaximal unital commutative C ∗ -subalgebras, implies thatsup α { diam( S ( A α )) } = sup α { sup { d L,α ( µ α , ν α ) : µ α , ν α ∈ S ( A α ) }} == sup α { sup { d L,α ( µ | A α , ν | A α ) : µ, ν ∈ S ( A ) }} == sup { sup α { d L,α ( µ | A α , ν | A α ) } : µ, ν ∈ S ( A ) } == sup { d L ( µ, ν ) : µ, ν ∈ S ( A ) } =: diam( S ( A ))By exactly the same argument,sup α { diam( S ( A α )) } = sup α { sup { d L,α ( µ α , ν α ) : µ α , ν α ∈ S ( A α ) }} == sup α { sup { d L,α ( µ | A α , ν | A α ) : µ, ν ∈ QS ( A ) }} == sup { sup α { d L,α ( µ | A α , ν | A α ) } : µ, ν ∈ QS ( A ) } == sup { d L ( µ, ν ) : µ, ν ∈ QS ( A ) } =: diam( QS ( A )) (cid:3) Proposition 5.13.
Let A be a unital C ∗ -algebra, L be a partially-defined L -seminorm on A .Then for every p ∈ [1 , ∞ ) and every µ, ν ∈ QS ( A ) W ←− pp ( µ, ν ) ≤ diam( S ( A α )) p − · W ←− ( µ, ν ) Proof.
Recall, that by definition, W pp,α ( µ α , ν α ) := inf (cid:26)Z d pL,α dπ : π ∈ Π( µ α , ν α ) (cid:27) ection 5. Projective L p -Wasserstein distances For any π ∈ P ( X α × X α ) it is true, that Z X α × X α d pL,α ( x, y ) dπ ≤ sup { d p − L,α ( x, y ) : x, y ∈ X α } · Z X α × X α d L,α ( x, y ) dπ ≤≤ sup { d p − L,α (˜ µ α , ˜ ν α ) : ˜ µ α , ˜ ν α ∈ S ( A α ) } · Z X α × X α d L,α ( x, y ) dπ == diam( S ( A α )) p − · Z X α × X α d L,α ( x, y ) dπ Passing to the infimum over all π ∈ Π( µ α , ν α ), we obtain: W pp,α ( µ α , ν α ) ≤ diam( S ( A α )) p − · W ,α ( µ α , ν α )As follows from Proposition 5.12, W pp,α ( µ α , ν α ) ≤ diam( S ( A )) p − · W ,α ( µ α , ν α ). Let us fixsome µ, ν ∈ QS ( A ) and take the supremum over all maximal unital commutative subalgebras A α . We obtain W ←− pp ( µ, ν ) ≤ diam( S ( A )) p − · W ←− ( µ, ν ) (cid:3) Theorem 5.14.
Let A be a unital separable C ∗ -algebra, L be a Wasserstein-compatible Lip-seminorm on A . Then W ←− p metricizes the weak ∗ -topology on S ( A ) for every p ∈ [1 , ∞ ). Proof.
Since d L metricizes the weak ∗ -topology on S ( A ), and S ( A ) is compact, diam( S ( A )) isfinite. By Proposition 5.11, W ←− ( µ, ν ) = d L ( µ, ν ) on QS ( A ), hence W ←− ( µ, ν ) metricizes theWeak ∗ -topology on S ( A ). By Proposition 5.13, W ←− p ≤ diam( S ( A α )) p − p · W ←− p By Proposition 4.14, W ,α ≤ W p,α for p ∈ [1 , + ∞ ], A α ∈ C ( A ). It follows, that W ←− ≤ W ←− p ≤ diam( S ( A α )) p − p · W ←− p Thus, W ←− p metricizes the same topology as W ←− does. (cid:3) Let us look on the projective L p -Wasserstein distance from the categorical point of view.It is possible to define a category of all pairs ( K, d ), where K is a Hausdorff compact convexset and d is a [0 , ∞ ]-valued pseudo-distance function on K . Morphisms between ( K , d ) and( K , d ) in this category are defined as continuous affine maps f between K and K such that d ◦ f ≤ d (in other words, f is a contraction). Denote this category as CCHpd con . As
CCH does, it contains all small projective limits.
Proposition 5.15.
For any small diagram ( { ( K α , d α ) } , T ), T = { T α,β : K α → K β , α ≤ β } in CCHpd con there is a projective limit. The category
CCHpd con is complete. Projectivelimit in
CCHpd con is defined as a projective limit in
CCH , equipped with the followingpseudo-distance function: d := sup α { d α ◦ P r α } . Projections Pr α are the projection maps fromthe definition of the projective limit in CCH .24 ection 5. Projective L p -Wasserstein distancesProof. It is obvious, that d ≥ d α ◦ Pr α , hence all Pr α are morphisms in CCHpd con .Suppose we have an object (
K, d K ) and a family of morphisms ϕ α : K → K α , such that ϕ α = T β,α ◦ ϕ β iff α ≤ β and d α ◦ ϕ α ≤ d K . Define ϕ : K → lim ←− K α as ϕ ( x ) = ( ϕ α ( x )). Then(Pr α ◦ ϕ )( x ) = Pr α ( ϕ α ( x )) = ϕ α ( x ), hence Pr α ◦ ϕ = ϕ α and d α ◦ Pr α ◦ ϕ = d α ◦ ϕ α ≤ d K Take the supremum over all α in both sides of the inequality: d K ≥ sup α { d α ◦ Pr α ◦ ϕ } = d ◦ ϕ Hence ϕ is a morphism in CCHpd con . The uniqueness of this morphism follows from itsuniqueness in
CCH (see the proof of Proposition 3.3). (cid:3)
Let us define the category of all pairs ( A , L ), where A is a unital commutative C ∗ -algebraand L is a partially-defined L -seminorm on A . Morphisms between ( A , L ) and ( A , L ) areinjective unital C ∗ -homomorphisms between A and A , such that L ( f ( a )) = L ( a ) ∀ a ∈ A .Denote it by ucC ∗ L in .Define a functor ( S, d ) from ucC ∗ L opin to CCHpd con that acts on objects as follows:(
S, d )( A , L ) = ( S ( A ) , d L )and sends every morphism f to its Banach adjoint f ∗ restricted to S ( A ). Functoriality ofthis map follows from the fact that d L is a [0 , + ∞ ]-valued pseudo-distance function, and that d L,α ◦ f ∗ ≤ d L,β , if f : A α ֒ → A β is a morphism in ucC ∗ L in .Analogously, we can define functors ( S, Wass p ) from ucC ∗ L opin to CCHpd con for every p ∈ [1 , ∞ ) that act on objects as follows:( S, Wass p )( A , L ) := ( S ( A ) , W p )and maps every morphism f to its Banach adjoint f ∗ restricted to S ( A ). Here W p is the L p -Wasserstein distance on ( S ( A ) , W p ) w.r.t. pseudo-distance function d L on Spec( A ). Thefact, that it is a functor, follows from Proposition 4.12, which asserts that W p is a [0 , + ∞ ]-valued pseudo-distance function, and Proposition 4.13, which implies that W p ◦ f ∗ ≤ W p if f : A α ֒ → A β is a morphism in ucC ∗ L in .Recall, that for a C ∗ -algebra A we can consider a diagram C ( A ) (in the category ucC ∗ in ) ofall unital commutative C ∗ -subalgebras of A ordered by inclusion. Let us consider a partially-defined Lip-seminorm L on A and equip each A α ∈ C ( A ) with the restriction of L to A α . Weobtain a diagram ( { ( A α , L α ) } , ⊆ ) in ucC ∗ L in .Using the defined above functors, ( S, d ) and ( S, Wass p ), we can obtain diagrams in thecategory CCHpd con : ( { ( S ( A α ) , d L,α ) } , R ) and ( { ( S ( A α ) , W p,α ) } , R ). Projective limits ofthese diagrams are exactly the quasi-state spaces ( QS ( A ) , d L ) and ( QS, W ←− p ).Resume the already known facts about these spaces: • If L is a partially-defined L -seminorm on A , then ( QS ( A ) , d L ) and ( QS, W ←− p ) for every p ∈ [1 , ∞ ) are compact convex Hausdorff spaces equipped with [0 , ∞ ]-valued lowersemi-continuous pseudo-distance functions. Proof: Prop. 5.6 and 5.8. • If L is an L -seminorm on A , then ( QS ( A ) , d L ) is a compact convex space equippedwith [0 , ∞ ]-valued lower semi-continuous distance function. Proof: Prop. 5.6.25 ection 6. Some open problems • If L is a solid L -seminorm on A , then ( QS ( A ) , W ←− p ) for every p ∈ [1 , ∞ ) is a com-pact convex space equipped with [0 , ∞ ]-valued lower semi-continuous distance function.Proof: Prop. 5.8. • If L is a Wasserstein-compatible Lip-seminorm on A , then ( S ( A ) , W ←− p ) for every p ∈ [1 , ∞ ) is a compact convex metric space, and ( QS ( A ) , d L ) = ( QS ( A ) , W ←− ). Proof:Prop. 5.11 and 5.14. 6. Some open problems
We provided a rigorous definition for a space of quasi-states, QS ( A ), where A is a unital C ∗ -algebra. It appears to be a compact convex set. It is natural to ask, how to characterizecompact convex sets that arise this way. Recall, that for the state spaces of C ∗ -algebras thereis a complete characterization in the terms of their convex structure (see [1]). It seems thatthere should be a connection (duality of some sort) between partial C ∗ -algebras, introduced in[4], and the desired notion of QS -spaces.It is known, that the self-adjoint part of a C ∗ -algebra can be recovered from its state space(which is considered as an ordered Banach space). How much information is contained in aquasi-state space? Is it possible to recover a diagram of all unital commutative C ∗ -subalgebrasknowing only the corresponding quasi-state space?There is a lack of non-trivial examples of solid Lip-seminorms. Obviously, all pairs ( A , L ),where A is a unital C ∗ -algebra and L is a finite Lip-seminorm, are acceptable examples of solidLip-seminorms. The problem is to find a pair of a separable infinite-dimensional noncommuta-tive C ∗ -algebra and a solid Lip-seminorm defined on it, which is not everywhere finite. Recallthat, by definition, Lip-seminorm is solid iff it is finite on a “weak dense” (in the sense ofseparation of points) subset of every maximal abelian C ∗ -subalgebra. This assumption seemsnatural, but it is difficult to verify. The explicit description of maximal abelian C ∗ -subalgebrasis possible for some von Neumann algebras, but only finite-dimensional von Neumann algebrasare separable as topological spaces. It is interesting to verify, whether the classical examplesof quantum compact metric spaces, introduced by Connes and Rieffel, satisfy this property.Speaking about Wasserstein-compatible Lip-seminorms, there is a lack of examples of suchseminorms even in finite-dimensional case. It is interesting to know, do the matrix algebrasallow a Lip-seminorm of this type. Acknowledgement
The author is extremely grateful to Dr. Frederic Latremoliere for pointing out a seriousmistake in the first version of this paper.The paper was written during an internship of the author in Scuola Normale Superiore diPisa. The author is grateful to SNS for its hospitality, Dr. Luigi Ambrosio for interesting andfruitful mathematical conversations, and Higher School of Economics for a sponsorship of thisvisit.The author wish to thank Alexander Kolesnikov for a constant support during the author’sresearch activity. 26 ection 6. Some open problems
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Faculty of Mathematics, Higher School of Economics, Moscow
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