Metrics on States from Actions of Compact Groups

Let a compact Lie group act ergodically on a unital C∗-algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak-∗ topology. Connes [Co1, Co2, Co3] has shown us that Riemannian metrics on non-commutative spaces (C-algebras) can be specified by generalized Dirac operators. Although in this setting there is no underlying manifold on which one then obtains an ordinary metric, Connes has shown that one does obtain in a simple way an ordinary metric on the state space of the C-algebra, generalizing the Monge-Kantorovich metric on probability measures [Ra] (called the “Hutchinson metric” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees with the underlying weak-∗ topology on the state space. Note that for locally compact spaces their topology agrees with the weak-∗ topology coming from viewing points as linear functionals (by evaluation) on the algebra of continuous functions vanishing at infinity. 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 In this paper we will consider metrics arising from actions of compact groups on Calgebras. For simplicity of exposition we will only deal with “compact” non-commutative spaces, that is, we will always assume that our C-algebras have an identity element. We will explain later what we mean by Dirac operators in this setting (section 4). In terms of this, a brief version of our main theorem is: Theorem 4.2. Let α be an ergodic action of a compact Lie group G on a unital C-algebra A, and let D be a corresponding Dirac operator. Then the metric topology on the state space of A defined by the metric from D agrees with the weak-∗ topology. An important case to which this theorem applies consists of the non-commutative tori [Rf], since they carry ergodic actions of ordinary tori [OPT]. The metric geometry of noncommutative tori has recently become of interest in connection with string theory [CDS, RS, S]. We begin by showing in the first section of this paper that the mechanism for defining a metric on states can be formulated in a very rudimentary Banach space setting (with no algebras, groups, or Dirac operators). In this setting the discussion of agreement between the metric topology and the weak-∗ topology takes a particularly simple form. Then in the second section we will see how length functions on a compact group directly give (without Dirac operators) metrics on the state spaces of C-algebras on which the group acts ergodically. We then prove the analogue in this setting of the main theorem stated above. In the third section we consider compact Lie groups, and show how norms on the Lie algebra directly give metrics on the state space. We again prove the corresponding analogue of our main theorem. Finally, in section 4 we use the results of the previous sections to prove our main theorem, stated above, for the metrics which come from Dirac operators. It is natural to ask about actions of non-compact groups. Examination of [Wv4] suggests that there may be very interesting phenomena there. The considerations of the present paper also make one wonder whether there is an appropriate analogue of length functions for compact quantum groups which might determine a metric on the state spaces of C2 algebras on which a quantum group acts ergodically [Bo, Wn]. This would be especially interesting since for non-commutative compact groups there is only a sparse collection of known examples of ergodic actions [Ws], whereas in [Wn] a rich collection of ergodic actions of compact quantum groups is constructed. Closely related is the setting of ergodic coactions of discrete groups [N, Q]. But I have not explored any of these possibilities. I developed a substantial part of the material discussed in the present paper during a visit of several weeks in the Spring of 1995 at the Fields Institute. I am appreciative of the hospitality of the Fields Institute, and of George Elliott’s leadership there. But it took trying to present this material in a course which I was teaching this Spring, as well as benefit from [P,Wv1, Wv2, Wv3, Wv4], for me to find the simple development given here. 1. Metrics on states Let A be a unital C-algebra. Connes has shown [Co1, Co2, Co3] that an appropriate way to specify a Riemannian metric in this non-commutative situation is by means of a spectral triple. This consists of a representation of A on a Hilbert space H, together with an unbounded self-adjoint operator D on H (the generalized Dirac operator), satisfying certain conditions. The set L(A) of Lipshitz elements of A consists of those a ∈ A such that the commutator [D, a] is a bounded operator. It is required that L(A) be dense in A. The Lipshitz semi-norm, L, is defined on L(A) just by the operator norm L(a) = ‖[D, a]‖. Given states μ and ν of A, Connes defines the distance between them, ρ(μ, ν), by (1.1) ρ(μ, ν) = sup{|μ(a)− ν(a)| : a ∈ L(A), L(a) ≤ 1} . (In the absence of further hypotheses it can easily happen that ρ(μ, ν) = +∞.) The semi-norm L is an example of a general Lipshitz semi-norm, that is [BC, Cu, P, Wv1, Wv2], a semi-norm L on a dense subalgebra L of A satisfying the Leibniz property: (1.2) L(ab) ≤ L(a)‖b‖+ ‖a‖L(b) . Lipshitz norms carry some information about differentiable structure [BC, Cu], but not nearly as much as do spectral triples. But it is clear that just in terms of a given Lipshitz norm one can still define a metric on states by formula (1.1). 3 However, for the purpose of understanding the relationship between the metric topology and the weak-∗ topology, we do not need the Leibniz property (1.2), nor even that A be an algebra. The natural setting for these considerations seems to be the following very rudimentary one. The data is: (1.3a) A normed space A, with norm ‖ ‖, over either C or R. (1.3b) A subspace L of A, not necessarily closed. (1.3c) A semi-norm L on L. (1.3d) A continuous (for ‖ ‖) linear functional, η, on K = {a ∈ L : L(a) = 0} with ‖η‖ = 1. (So K 6 = {0} .) Let A denote the Banach-space dual of A, and set S = {μ ∈ A′ : μ = η on K, and ‖μ‖ = 1} . Thus S is a norm-closed, bounded, convex subset of A, and so is weak-∗ compact. In general S can be quite small; when A is a Hilbert space S will contain only one element. But in the applications we have in mind A will be a unital C-algebra, K will be the onedimensional subspace spanned by the identity element, and η will be the functional on K taking value 1 on the identity element. Thus S will be the full state-space of A. (That K will consist only of the scalar multiples of the identity element in our examples will follow from our ergodicity hypothesis. We treat the case of general K here because this clarifies slightly some issues, and it might possibly be of eventual use, for example in non-ergodic situations.) We do not assume that L is dense in A. But to avoid trivialities we do make one more assumption about our set-up, namely: (1.3e) L separates the points of S. 4 This means that given μ, ν ∈ S there is an a ∈ L such that μ(a) 6= ν(a). (Note that for μ ∈ S there exists a ∈ L with μ(a) 6= 0, since we can just take an a ∈ K such that η(a) 6= 0.) With notation as above, let L̃ = L/K. Then L drops to an actual norm on L̃, which we denote by L̃. But on L̃ we also have the quotient norm from ‖ ‖ on L, which we denote by ‖ ‖ ∼ . The image in L̃ of a ∈ L will be denoted by ã. We remark that when L is a unital algebra (perhaps dense in a C-algebra), and when K is the span of the identity element, then the space of universal 1-forms Ω over L is commonly identified [BC, Br, Co2, Cu] with L⊗L̃, and the differential d : L → Ω is given by da = 1⊗ã. Thus in this setting our L̃ is a norm on the space of universal 1-coboundaries of L. The definition of L which we will use in the examples of section 3 is also closely related to this view. On S we can still define a metric, ρ, by formula (1.1), with L(A) replaced by L. The symmetry of ρ is evident, and the triangle inequality is easily verified. Since we assume that L separates the points of S, so will ρ. But ρ can still take the value +∞. We will refer to the topology on S defined by ρ as the “ρ-topology”, or the “metric topology” when ρ is understood. It will often be convenient to consider elements of A as (weak-∗ continuous) functions on S. At times this will be done tacitly, but when it is useful to do this explicitly we will write â for the corresponding function, so that â(μ) = μ(a) for μ ∈ S. Without further hypotheses we have the following fact. It is closely related to proposition 3.1a of [P], where metrics are defined in terms of linear operators from an algebra into a Banach space. 1.4 Proposition. The ρ-topology on S is finer than the weak-∗ topology. Proof. Let {μk} be a sequence in S which converges to μ ∈ S for the metric ρ. Then it is clear from the definition of ρ that {μk(a)} converges to μ(a) for any a ∈ L with L(a) ≤ 1, and hence for all a ∈ L. This says that â(μk) converges to â(μ) for all a ∈ L. But L̂ is a linear space of weak-∗ continuous functions on S which separates the points of S by assumption (and which 5 contains the constant functions, since they come from any a ∈ K on which η is not 0). A simple compactness argument shows then that L̂ determines the weak-∗ topology of S. Thus {μk} converges to μ in the weak-∗ topology, as desired. There will be some situations in which we want to obtain information about (L, L) from information about S. It is clear that to do this S must “see” all of L. The convenient formulation of this for our purposes is as follows. Let ‖ ‖∞ denote the supremum norm on functions on S. Let it also denote the corresponding semi-norm on L defined by ‖a‖∞ = ‖â‖∞. Clearly ‖â‖∞ ≤ ‖a‖ for a ∈ L. 1.5 Condition. The semi-norm ‖ ‖∞ on L is a norm, and it is equivalent to the norm ‖ ‖, so that there is a constant k with ‖a‖ ≤ k‖â‖∞ for a ∈ L. This condition clearly holds when A is a C-algebra, L is dense in A, and S is the state space of A, so that we are dealing with the usual Kadison functional representation [KR]. But we remark that even in this case the constant k above cannot always be taken to be 1 (bottom of page 263 of [KR]). This suggests that in using formula (1.1) one might want to restrict to using just the self-adjoint elements of L, since there the function representation is isometric. But more experience with examples is needed. We return to the general case. If we are to have the ρ-topology on S agree with the weak-∗ topology, then S must at least have finite ρ-diameter, that is, ρ must be bounded. The following proposition is closely related to theorem 6.2 of [P]. 1.6 Proposition. Suppose there is a constant, r, such that