Hyperbolic Group C-algebras and Free-product C-algebras as Compact Quantum Metric Spaces

Let l be a length function on a group G, and let Ml denote the operator of pointwise multiplication by l on l(G). Following Connes, Ml can be used as a “Dirac” operator for C ∗ r (G). It defines a Lipschitz seminorm on C∗ r (G), which defines a metric on the state space of C∗ r (G). We show that if G is a hyperbolic group and if l is a word-length function on G, then the topology from this metric coincides with the weak-∗ topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered C∗-algebras which satisfy a suitable “ Haagerup-type” condition. We also use this framework to prove an analogous fact for certain reduced free products of C∗-algebras. 0. Introduction The group C-algebras of discrete groups provide a much-studied class of “compact non-commutative spaces” (that is, unital C-algebras). In [3] Connes showed that the “Dirac” operator of a spectral triple (i.e. of an unbounded Fredholm module) over a unital C-algebra provides in a natural way a metric on the state space of the algebra. The class of examples most discussed in [3] consists of the group C-algebras of discrete groups, with the Dirac operator coming in a simple way from a word-length function on the group. In [10], [11] the second author pointed out that, motivated by what happens for ordinary compact metric spaces, it is natural to desire that for a spectral triple the topology from the metric on the state space coincides with the weak-∗ topology (for which the state space is compact). This property was verified Date: February 13, 2003. 1991 Mathematics Subject Classification. Primary 46L87; Secondary 20F67, 46L09. The first author was supported by the Japan Society for the Promotion of Science Postdoctral Fellowships for Research Abroad, and the research of the second author was supported in part by National Science Foundation grants DMS99-70509 and DMS-0200591. 1 2 NARUTAKA OZAWA AND MARC A. RIEFFEL in [10] for certain examples. In [12] this property was taken as the defining property for a “compact quantum metric space”. In [13] the second author studied this property for Connes’ original example of discrete groups with Dirac operators coming from a wordlength functions, but was able to verify this property only for the case when the group is Z. This already took a long and interesting argument. We refer the reader to the introduction of [13] for a more extensive discussion of this whole matter. In the present paper we verify the property for the case of hyperbolic discrete groups. In the course of studying this case we discovered that a natural setting was that of filtered C-algebras with faithful trace. Voiculescu had shown earlier [15] how to define an appropriate Dirac operator in that setting. In Section 1 we formulate in that setting a “Haagerup-type condition”, which in Sections 2 and 3 we show is sufficient to imply that the metric from the Dirac operator gives the state space the weak-∗ topology. Then in Section 4 we show that this Haagerup-type condition is satisfied in the case of hyperbolic groups. We mention that quite recently Antonescu and Christensen [1] showed that for non-Abelian free groups the metric on the state space gives the state space finite diameter. Their techniques are close to ours, but make explicit the relationship with Schur multipliers. In Section 5 we show that the Haagerup-type condition fails for the groups Z for n ≥ 2 with their standard length functions, and for groups which contain an amenable group of growth ≥ 4 for the length function in use. Since the approach used in the present paper is entirely different from that used in [13] to successfully treat Z, this raises the interesting question of finding a unified approach which covers both cases. And there remains wide open the question of what happens for other classes of groups, such as the discrete Heisenberg group and other nilpotent discrete groups. Finally, in Section 6 we show that the Haagerup-type condition is satisfied by the reduced free product of any two filtered C-algebras which satisfy the Haagerup-type condition. (Their filtrations give in a natural way a filtration on the free product.) This provides yet more examples of compact quantum metric spaces. We are very much indebted to Gilles Pisier for giving us a proof that for the free group on n generators with its standard word-length function the corresponding metric on the state space gives the state space finite diameter. This showed us how to begin proving things in the direction which we have pursued here. We also warmly thank him for valuable comments on our manuscript. HYPERBOLIC GROUP C ∗ -ALGEBRAS 3 1. Filtered C-algebras We let A be a unital ∗-algebra over C which has a ∗-filtration {An} by finite-dimensional subspaces. Just as in [15] this means that Am ⊂ An if m < n, A = ∪n=0An, An = An and AmAn ⊆ Am+n, and A0 = C1A. We assume further that we are given a faithful state, σ, on A, that is, a linear functional such that σ(aa) > 0 for all a ∈ A unless a = 0, and σ(1A) = 1. Let H = L(A, σ) denote the corresponding GNS Hilbert space. We assume that the left regular representation of A on H is by bounded operators, and we identify A with the corresponding algebra of operators on H. We let ‖ · ‖ denote the operator norm of A. Our notation will not distinguish between a as an operator on H and a as a vector in H, so the context must be examined to see which is intended. We let ‖a‖2 denote the norm of a as a vector in H. We can view each An as a finite-dimensional, thus closed, subspace of H. We let Qn denote the orthogonal projection of H onto An. We then set Pn = Qn−Qn−1, with P0 = Q0. The Pn’s are mutually orthogonal, and ∑ Pn = IH for the strong operator topology. For each a ∈ A and each n we set an = Pn(a), where here a is viewed as a vector. Then an ∈ An, but an / ∈ An−1 unless an = 0. Furthermore a = ∑ an, with at most p non-zero terms in the sum if a ∈ Ap. For the above situation we define, as in [15], an unbounded operator, D, onH byD = ∑∞ n=1 nPn. Notice that A is contained in the domain of D. The following lemma is part of proposition 5.1d of [15]. We include the proof here since we will need a similar argument in Section 3. Lemma 1.1. For any a ∈ A the operator [D, a] has dense domain and is a bounded operator. Proof. Clearly A is contained in the domain of [D, a], and A is dense. Suppose that a ∈ Ap. Then for any given m,n ≥ 0, if PmaPn 6= 0 then there is a ξ ∈ An such that aξ ∈ Am. Since ApAn ⊆ Ap+n, it follows that p + n ≥ m. On taking the adjoint, we see that PmaPn 6= 0, so that p+m ≥ n. Thus |m− n| ≤ p. Consequently, a = ∑ |m−n|≤p PmaPn, converging in the strong operator topology. For each j with |j| ≤ p set Tj = ∑ PmaPm−j . Because the range of the terms PmaPm−j are orthogonal for fixed j, as are the “domains”, we have ‖Tj‖ = sup m ‖PmaPm−j‖ ≤ ‖a‖. 4 NARUTAKA OZAWA AND MARC A. RIEFFEL But for any m,n ≥ 0 we have [D,PmaPn] = (m− n)PmaPn. In particular, [D,PmaPm−j] = jPmaPn. Thus [D, Tj] = jTj . Since a = ∑ Tj, we obtain [D, a] = ∑