Metrics on State Spaces

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the whole state space as being those which are lower semicontinuous. We show that their domain of Lipschitz elements can be enlarged so as to form a dual Banach space, which generalizes the situation for ordinary Lipschitz seminorms. We give a characterization of the metrics on state spaces which come from Lipschitz seminorms. The natural (broader) setting for these results is provided by the “function spaces” of Kadison. A variety of methods for constructing Lipschitz seminorms is indicated. In non-commutative geometry (based on C-algebras), the natural way to specify a metric is by means of a suitable “Lipschitz seminorm”. This idea was first suggested by Connes [C1] and developed further in [C2, C3]. Connes pointed out [C1, C2] that from a Lipschitz seminorm one obtains in a simple way an ordinary metric on the state space of the C-algebra. This metric generalizes the Monge–Kantorovich metric on probability measures [KA, Ra, RR]. In this article we make more precise the relationship between metrics on the state space and Lipschitz seminorms. Let ρ be an ordinary metric on a compact space X . The Lipschitz seminorm, Lρ, determined by ρ is defined on functions f on X by (0.1) Lρ(f) = sup{|f(x)− f(y)|/ρ(x, y) : x 6= y}. 1991 Mathematics Subject Classification. Primary 46L87; Secondary 58B30, 60B10. The research reported here was supported in part by National Science Foundation Grant DMS–96–13833. Typeset by AMS-TEX 1 (This can take value +∞.) It is known that one can recover ρ from Lρ by the relationship ρ(x, y) = sup{|f(x)− f(y)| : Lρ(f) ≤ 1}. But a slight extension of this relationship defines a metric, ρ̄, on the space S(X) of probability measures on X , by (0.2) ρ̄(μ, ν) = sup{|μ(f)− ν(f)| : Lρ(f) ≤ 1}. This is the Monge–Kantorovich metric. The topology which it defines on S(X) coincides with the weak-∗ topology on S(X) coming from viewing it as the state space of the Calgebra C(X). The extreme points of S(X) are identified with the points of X . On the extreme points ρ̄ coincides with ρ. Thus relationship (0.1) can be viewed as saying that Lρ can be recovered just from the restriction of its metric ρ̄ on S(X) to the set of extreme points of S(X). Suppose now that A is a unital C-algebra with state space S(A), and let L be a Lipschitz seminorm on A. (Precise definitions are given in Section 2.) Following Connes [C1, C2], we define a metric, ρ, on S(A) by the evident analogue of (0.2). We show by simple finite dimensional examples determined by Dirac operators that L may well not be determined by its restriction to the extreme points of S(A). It is then natural to ask whether L is determined by ρ̄ on all of S(A), by a formula analogous to (0.1). One of our main theorems (Theorem 5.1) states that the Lipschitz seminorms for which this is true are exactly those which are lower semicontinuous in a suitable sense. For ordinary compact metric spaces (X, ρ) it is known that the space of Lipschitz functions with a norm coming from the Lipschitz seminorm is the dual of a certain other Banach space. Another of our main theorems (Theorem 4.1) states that the same is true in our non-commutative setting, and we give a natural description of this predual. We also characterize the metrics on S(A) which come from Lipschitz seminorms (Theorem 9.11). We should make precise that we always require that our Lipschitz seminorms be such that the metric on S(A) which they determine gives the weak-∗ topology on S(A). An elementary characterization of exactly when this happens was given in [Rf]. (See also [P].) This property obviously holds for finite-dimensional C-algebras. It is known to hold in many situations for commutative C-algebras, as well as for C-algebras obtained by combining commutative ones with finite dimensional ones. But this property has not been verified for many examples beyond those. However in [Rf] this property was verified for some interesting infinite-dimensional non-commutative examples, such as the non-commutative tori, and I expect that eventually it will be found to hold in a wide variety of situations. Actually, we will see below that the natural setting for our study is the broader one of order-unit spaces. The theory of these spaces was launched by Kadison in his memoire [K1]. For this reason it is especially appropriate to dedicate this article to him. (In [K2] 2 Kadison uses the terminology “function systems”, but we will follow [Al] in using the terminology “order-unit space” as being a bit more descriptive of these objects.) On the other hand, most of the interesting constructions currently in view of Lipschitz seminorms on non-commutative C-algebras, such as those from Dirac operators, or those in [Rf], also provide in a natural way seminorms on all the matrix algebras over the algebras. Thus it is likely that “matrix Lipschitz seminorms” in analogy with the matrix norms of [Ef] will eventually be of importance. But I have not yet seen how to use them in a significant way, and so we do not deal with them here. Let us mention here that a variety of metrics on the state spaces of full matrix algebras have been employed by the practitioners of quantum mechanics. A recent representative paper where many references can be found is [ZS]. We will later make a few comments relating some of these metrics to the considerations of the present paper. The last three sections of this paper will be devoted to a discussion of the great variety of ways in which Lipschitz seminorms can arise, even for commutative algebras. We do discuss here some non-commutative examples, but most of our examples are commutative. I hope in a later paper to discuss and apply some other important classes of non-commutative examples. Some of the applications which I have in mind will require extending the theory developed here to quotients and sub-objects. Finally, we should remark that while we give here considerable attention to how Dirac operators give metrics on state spaces, Connes has shown [C2] that Dirac operators encode far more than just the metric information. In particular they give extensive homological information. But we do not discuss this aspect here. 1. Recollections on order-unit spaces We recall [Al] that an order-unit space is a real partially-ordered vector space, A, with a distinguished element e, the order unit, which satisfies: 1) (Order unit property) For each a ∈ A there is an r ∈ R such that a ≤ re. 2) (Archimedean property) If a ∈ A and if a ≤ re for all r ∈ R, then a ≤ 0. For any a ∈ A we set ‖a‖ = inf{r ∈ R : −re ≤ a ≤ re}. We obtain in this way a norm on A. In turn, the order can be recovered from the norm, because 0 ≤ a ≤ e iff ‖a‖ ≤ 1 and ‖e − a‖ ≤ 1. The primary source of examples consists of the linear spaces of all self-adjoint elements in unital C-algebras, with the identity element serving as order unit. But any linear space of bounded self-adjoint operators on a Hilbert space will be an order-unit space if it contains the identity operator. We expect that this broader class of examples will be important for the applications of metrics on state spaces. We will not assume that A is complete for its norm. This is important for us because the domains of Lipschitz norms will be dense, but usually not closed, in the completion. 3 (The completion is always again an order-unit space.) This also accords with the definition in [Al]. By a state of an order-unit space (A, e) we mean a continuous linear functional, μ, on A such that μ(e) = 1 = ‖μ‖. States are automatically positive. We denote the collection of all the states of A, i.e. the state space of A, by S(A). It is a w-compact convex subset of the Banach space dual, A, of A. To each a ∈ A we get a function, â, on S(A) defined by â(μ) = μ(a). Then â is an affine function on S(A) which is continuous by the definition of the w-topology. The basic representation theorem of Kadison [K1, K2, K3] (see Theorem II.1.8 of [Al]) says that for any order-unit space the representation a → â is an isometric order isomorphism of A onto a dense subspace of the space Af(S(A)) of all continuous affine functions on S(A), equipped with the supremem norm and the usual order on functions (and with e clearly carried to the constant function 1). In particular, if A is complete, then it is isomorphic to all of Af(S(A)). Thus we can view the order-unit spaces as exactly the dense subspaces containing 1 inside Af(K), where K is any compact convex subset of a topological vector space. This provides an effective view from which to see many of the properties of order-unit spaces. Most of our theoretical discussion will be carried out in the setting of order-unit spaces and Af(K), though our examples will usually involve specific C-algebras. We let C(K) denote the real C-algebra of all continuous functions on K, in which Af(K) sits as a closed subspace. It will be important for us to work on the quotient vector space à = A/(Re). We let ‖ ‖ denote the quotient norm on à from ‖ ‖. This quotient norm is easily described. For a ∈ A set max(a) = inf{r : a ≤ re} min(a) = sup{r : re ≤ a}, so that ‖a‖ = (max(a)) ∨ (−min(a)). Then it is easily seen that ‖ã‖∼ = (max(a)−min(a))/2. 2. The radius of the state space Let A be an order-unit space. Since the term “Lipschitz seminorm” has somewhat wide but imprecise usage, we will not use this term for our main objects of precise study (which we will define in the next section). Almost the minimal requirement for a Lipschitz seminorm is that its null-space be exactly the scalar multiples of the order unit. We will use the term “Lipschitz seminorm” in this general sense. We emphasize that a Lipschitz seminorm will usually not be continuous for ‖ ‖. 4 Let L be a Lipschitz seminorm on A. For μ, ν ∈ S(A) we can define a metric, ρL, on S(A) by ρL(μ, ν) = sup{|μ(a)− ν(a)| : L(a) ≤ 1} (which may be +∞). Then ρL determines a topology on S(A). We want this topology to agree with the weak-∗ topology. Since S(A) is compact, ρL must then give S(A) finite diameter. We examine this aspect here. It is actually more convenient for us to work with “radius” (half the diameter), since this will avoid factors of 2 in various places. We would like to use the properties of orderunit spaces to express the radius in terms of L in a somewhat more precise way than was implicit in [Rf] in its more general context. The following considerations will also be used extensively later. As in [Rf] and in the previous section, we denote the quotient vector space A/(Re) by Ã, with its quotient norm ‖ ‖. But in addition to this norm, the quotient seminorm L̃ from L is also a norm on Ã, since L takes value 0 only on Re. The dual Banach space to à for ‖ ‖ is just A′, the subspace of A consisting of those λ ∈ A such that λ(e) = 0. We denote the norm on A dual to ‖ ‖ still by ‖ ‖. The dual norm on A′ is just the restriction of ‖ ‖ to A′. If we view A as a dense subspace of Af(K) ⊆ C(K), then by the Hahn–Banach theorem λ extends (not uniquely) to C(K) with same norm. There we can take the Jordan decomposition into disjoint non-negative measures. Note that for positive measures their norm on C(K) equals their norm on A, since e ∈ A. Thus we find μ, ν ≥ 0 such that λ = μ − ν and ‖λ‖ = ‖μ‖ + ‖ν‖. But 0 = λ(e) = μ(e)− ν(e) = ‖μ‖ − ‖ν‖. Consequently ‖μ‖ = ‖ν‖ = ‖λ‖/2. Thus if ‖λ‖ ≤ 2 we have ‖μ‖ = ‖ν‖ ≤ 1. If ‖λ‖ < 2 set t = ‖μ‖ < 1, and rescale μ and ν so that they are in S(A). Then λ = tμ− tν = μ− (tν + (1− t)μ). Now (tν+(1−t)μ) is no longer disjoint from μ, but we have obtained the following lemma, which will be used in a number of places. 2.1 Lemma. The ball D2 of radius 2 about 0 in A 0 coincides with {μ−ν : μ, ν ∈ S(A)}. 2.2 Proposition. With notation as earlier, the following conditions are equivalent for an r ∈ R: 1) For all μ, ν ∈ S(A) we have ρL(μ, ν) ≤ 2r. 2) For all a ∈ A we have ‖ã‖ ≤ rL(ã). Proof. Suppose that condition 1 holds. Let a ∈ A and λ ∈ D2. Then by the lemma λ = μ− ν for some μ, ν ∈ S(A). Thus |λ(a)| = |(μ− ν)(a)| ≤ L(a)ρL(μ, ν) ≤ L(a)2r. 5 Since λ(e) = 0, thus inequality holds whenever a is replaced by a + se for s ∈ R. Thus condition 2 holds. Conversely, suppose that condition 2 holds. Then for any μ, ν ∈ S(A) and a ∈ A with L(a) ≤ 1 we have |μ(a)− ν(a)| = |(μ− ν)(a)| ≤ 2‖ã‖∼ ≤ 2r. Thus ρL(μ, ν) ≤ 2r as desired. We caution that just because a metric space has radius r, it does not follow that there is a ball of radius r which contains it, as can be seen by considering equilateral triangles in the plane. We remark that just because ρL gives S(A) finite radius, it does not follow that ρL gives the weak-∗ topology. Perhaps the simplest example arises when A is infinite dimensional and L(a) = ‖ã‖. 3. Lower semicontinuity for Lipschitz norms We assume from now on that L is a Lipschitz seminorm which gives S(A) finite radius. Let L1 = {a : L(a) ≤ 1}. From theorem 1.8 of [Rf] we know that ρL will give the weak-∗ topology on S(A) exactly if the image of L1 in à is totally bounded for ‖ ‖. Equivalently, by theorem 1.9 of [Rf], for one, hence all, t ∈ R, t > 0, the set Bt = {a : L(a) ≤ 1 and ‖a‖ ≤ t} must be totally bounded in A for ‖ ‖. (This uses the assumption of finite radius.) We remark that this implies that if {an} is a sequence (or net) in A converging pointwise on S(A) to a ∈ A, and if there is a constant k such that ‖an‖ ≤ k and L(an) ≤ k for all n, then an converges to a in norm. This is because {an} is contained in kB1 whose closure in the completion Ā of A is compact. Let b be any norm limit point of {an} in Ā. Then a subsequence of {an} converges in norm to b. But it still converges pointwise on S(A) to a. Consequently b = a, and a is the only norm limit point of {an}. We would like to show that L and ρL contain the same information, and more specifically that we can recover L from ρL as being the usual Lipschitz seminorm for ρL. By this we mean the following. Let ρ be any metric on S(A). Define Lρ on C(S(A)) by (3.1) Lρ(f) = sup{|f(μ)− f(ν)|/ρ(μ, ν) : μ 6= ν}, where this may take value +∞. Let Lipρ = {f : Lρ(f) < ∞}. We can restrict Lρ to Af(S(A)). In general, few elements of Af(S(A)) will be in Lipρ. However, on viewing the elements of A as elements of Af(S(A)), we have: 6 3.2 Lemma. Let L be a Lipschitz norm on A with corresponding metric ρL on S(A). Then A ⊆ LipρL . Proof. For μ, ν ∈ S(A) and a ⊆ A we have |â(μ)− â(ν)| = |μ(a)− ν(a)| ≤ L(a)ρL(μ, ν). We would like to recover L on A from ρL by means of formula (3.1). However, the seminorms defined by (3.1) have an important continuity property: 3.3 Definition. Let A be a normed vector space, and let L be a seminorm on A, except that we permit it to take value +∞. Then L is lower semicontinuous if for any sequence {an} in A which converges in norm to a ∈ A we have L(a) ≤ lim inf{L(an)}. Equivalently for one, hence every, t ∈ R, t > 0, the set Lt = {a ∈ A : L(a) ≤ t} is norm-closed in A. 3.4 Proposition. Let A be an order-unit space, and let ρ be a metric on S(A) which gives the weak-∗ topology. Define Lρ on C(S(A)) by formula (3.1). Then Lρ is lower semicontinuous. Consequently, the restriction of Lρ to any subspace of C(S(A)), such as A or Af(C(S(A))), will be lower semicontinuous. Proof. When we view Lρ as a function of f , the formula (3.1) says that Lρ is the pointwise supremum of a collection of functions (labeled by pairs μ, ν with μ 6= ν) which are clearly continuous on C(S(A)) for the supremum norm. But the pointwise supremum of continuous functions is lower semicontinuous. Lower semicontinuity is the last property which we need for our present purposes. So we now define our main object of study: 3.5 Definition. Let A be an order-unit space. By a Lip-norm on A we mean a seminorm, L, on A (taking finite values) with the following properties: 1) For a ∈ A we have L(a) = 0 if and only if a ∈ Re. 2) L is lower semicontinuous. 3) {a ∈ A : L(a) ≤ 1} has image in à which is totally bounded for ‖ ‖. 3.6 Example. Here is an example of a seminorm L which satisfies conditions 1) and 3) above but is not lower semicontinuous. Let I = [−1, 1], and let A = C(I), the algebra of functions which have continuous first derivatives on I. Define L on A by L(f) = ‖f ′‖∞ + |f ′(0)|. 7 It is easily seen that properties 1) and 3) above are satisfied. For each n let gn be the function defined by gn(t) = n|t| for |t| ≤ 1/n, and gn(t) = 1 elsewhere. Let fn(t) = ∫ t −1 gn(s)ds. Then the sequence {fn} converges uniformly to the function f given by f(t) = t+ 1. But L(fn) = 1 for each n, whereas L(f) = 2. We remark that the example considered at the end of Section 2, in which L = ‖ ‖, satisfies conditions 1 and 2 but not 3. A substantial supply of examples of lower semicontinuous seminorms can be obtained from the W -derivations of Weaver [W2, W3]. These derivations will in general have large null spaces, and the seminorms from them need not give the weak-∗ topology on the state space. But many of the specific examples of W -derivations which Weaver considers do in fact give Lip-norms. In terms of Weaver’s terminology, which we do not review here, we have: 3.7 Proposition. Let M be a von Neumann algebra and let E be a normal dual operator M -bimodule. Let δ : M → E be a W -derivation, and denote the domain of δ by L, so that L is an ultra-weakly dense unital ∗-subalgebra of M . Define a seminorm, L, on L by L(a) = ‖δ(a)‖E. Then L is lower semicontinuous, and L1 = {a ∈ L : L(a) = 1} is norm-closed in M itself. Proof. Let {an} be a sequence in L which converges in norm to b ∈ M . To show that L is lower semicontinuous, it suffices to consider the case in which {an} is contained in L1. Then the set {(an, δ(an))} is a bounded subset of the graph of δ for the norm max{‖ ‖M , ‖ ‖E}. Since the graph of a W -derivation is required to be ultra-weakly closed, and since bounded ultraweakly closed subsets are compact for the ultra-weak topology, there is a subnet which converges ultra-weakly to an element (c, δ(c)) of the graph of δ. Then necessarily c = b, so that b ∈ L, and δ(b) is in the ultra-weak closure of {δ(an)}. Consequently L(b) = ‖δ(b)‖ ≤ 1. Because of the importance of Dirac operators, it is appropriate to verify lower semicontinuity for the Lipschitz norms which they determine. This is close to a special case of Proposition 3.7, but does not require any kind of completeness, nor an algebra structure on A. 3.8 Proposition. Let A be a linear subspace of bounded self-adjoint operators on a Hilbert space H, containing the identity operator. Let D be an essentially self-adjoint operator on H whose domain, D(D), is carried into itself by each element of A. Assume that [D, a] is a bounded operator on D(D) for each a ∈ A (so that [D, a] extends uniquely to a bounded operator on H). Define L on A by L(a) = ‖[D, a]‖. Then L is lower semicontinuous. Proof. Let {an} be a sequence in A which converges in norm to a ∈ A. Suppose that there is a constant, k, such that L(an) ≤ k for all n. For any ξ, η ∈ D(D) with ‖ξ‖ = 1 = ‖η‖ we have 〈[D, a]ξ, η〉 = 〈aξ,Dη〉 − 〈Dξ, aη〉 = lim〈[D, an]ξ, η〉. 8 But |〈[D, an]ξ, η〉| ≤ k for each n, and so ‖[D, a]‖ ≤ k. We remark that the Lipschitz norms constructed in [Rf] by means of actions of compact groups are easily seen to be lower semicontinuous, and so to be Lip-norms. We return to considering general seminorms. Suppose that L is a seminorm satisfying conditions 1 and 3 of Definition 3.5. Let L1 be as at the beginning of this section, with L̄1 its norm closure in A. Then L̄1 is a closed convex balanced (unbounded) subset of A. We can thus define the corresponding Minkowski functional, L0, for L̄1, that is, L0(a) = inf{r ∈ R + : a ∈ rL̄1}. Then it is easy to see that L0 is lower semicontinuous. In fact: 3.9 Proposition. With notation as above, L0 is the largest Lip-norm on A smaller than L. Proof. Since the image of L1 in à is totally bounded, the same is true of L̄1, so condition 3 of Definition 3.5 is satisfied. If a ∈ A and a / ∈ Re, then there is an s ∈ R such that sã is not in the image of L̄1. Thus a / ∈ (1/s)L̄1 and L0(a) 6= 0, so that condition 1 is satisfied. The statement about “largest” is clear. Suppose now that L is a Lip-norm on an order-unit space A. Let Ā denote the completion of A for ‖ ‖, and let L̄1 denote now the closure of L1 in Ā rather than just A. Let L̄ denote the corresponding “Minkowski functional” on Ā obtained by setting, for b ∈ Ā, L̄(b) = inf{r ∈ R : b ∈ rL̄1}. Since there may be no such r, we must allow the value +∞. With this understanding, L̄ will be a seminorm on Ā. It is easily seen that L̄(b) ≤ 1 exactly if b ∈ L̄1, and that L̄ is lower semicontinuous because L̄1 is closed. Up to this point we did not use the lower semicontinuity of L. It’s import is given by: 3.10 Proposition. Let L be a Lip-norm on the order-unit space A, and let L̄ on Ā be defined as above. Then L̄ is an extension of L, that is, for a ∈ A we have L̄(a) = L(a). Furthermore, the restriction of L̄ to {b ∈ Ā : L̄(b) < ∞} is a Lip-norm. Proof. Suppose that a ∈ A and L(a) = 1. Then a ∈ L1 ⊆ L̄1 and so clearly L̄(a) ≤ 1. Conversely, if L̄(a) ≤ 1, then a ∈ L̄1. Thus there is a sequence {an} in L1 which converges to a, with L(an) ≤ 1 for every n. From the lower semicontinuity of L it follows that L(a) ≤ 1. The proof that L̄ gives a Lip-norm is similar to the proof of Proposition 3.9.