Connes metric for states in group algebras

We are interested in the case when Γ is a discrete group with identity element e and the algebra A is the reduced C * -algebra C * r (Γ). The Hilbert space is ℓ 2 (Γ), with C * r (Γ) acting as left convoluters (i.e. the left regular representation). The Dirac operator is defined in terms of a length function on Γ. A length function is a map L : Γ → IR + satisfying 1. L(gh) ≤ L(g) + L(h) for all g, h ∈ Γ.

2. L(g -1 ) = L(g) for all g ∈ Γ.

3. L(e) = 0.

If Γ is given by generators and relations, the prototypical length function is the map which assigns to each word its (minimal) length. We shall fix this data L, and we will make the further assumption that the sets {g ∈ Γ : L(g) ≤ c} are finite for any c > 0. The Dirac operator [Connes2] is then defined as follows:

where {δ g : g ∈ Γ} is the canonical orthonormal basis of ℓ 2 (Γ). As is custom, we shall denote by λ g the element δ g regarded as an operator in ℓ 2 (Γ). The metric (of the non commutative metric space) is defined in the state space S(C * r (Γ)) of C * r (Γ) by means of the formula

Here [ , ] denotes the usual conmutator of operators. This d is not necessarily finite. In this note we study situations in which it is finite, and consider a problem posed by M. Rieffel, asking under which assumptions the metric thus defined induces on the state space a topology which is equivalent to the w * topology.

The basic example of this situation, which even justifies the name “non commutative metric space”, occurs when A is C(M ), the algebra of continuous functions on a spin manifold M [Connes2], [GL]. M Rieffel found [Rieffel] a natural triple associated to the noncommutative tori. Also he pointed out that one can find a positive answer for matrix algebras.

In this note we consider this problem for group algebras arising from discrete groups and triples arising from length functions. Instead of dealing with the d metric directly, we refer it to two metrics, d ∞ and d 2 , related with the asymptotic behaviour of the family { 1 L(g) : e = g ∈ Γ}:

and

First note that d ∞ is a well defined metric and that d ∞ (ϕ, ψ) ≤ d(ϕ, ψ). The first fact is apparent.

To prove the second, note that [D, λ g ] = L(g), and therefore

Also note that d 2 may fail to be finite. Indeed, consider Γ = Z Z × Z Z. Then the family { 1 L(g) : g = e} does not belong to ℓ 2 (Z Z × Z Z). Consider the positive definite functions f (g) = 1 for all g and h = δ e . These functions induce states ϕ f and ϕ h on

Denote by K(Γ) the group algebra of Γ, i.e. the set of elements of the form g∈F α g λ g , where

which induces a topology equivalent to the w * -topology.

have their norms bounded (by 1), and since

On the other hand, there exists n 0 such that for all n ≥ n 0 ,

Here we establish the basic inequality for these metrics, namely

because Dλ g δ e = Dδ g = L(g)δ g , and in particular Dδ e = 0. Therefore

Proof. Pick a = g∈F α g λ g ∈ K(Γ), with [D, a] ≤ 1 (note that for any a ∈ K(Γ), [D, a] is a bounded operator). Then

which by the Cauchy-Schwartz inequality is less than or equal to

The proof finishes by observing that the set of elements We emphasize that d 2 might be infinite. It would be finite if for example the family { 1 L(g) : e = g ∈ Γ} would lie in ℓ 2 (Γ). This imposes a strong condition on Γ, namely, that the group Γ has linear growth (polinomial growth with degree 1), see [Gromov] and [Connes2]. This means, that there exists constants k, l such that #{g ∈ Γ : L(g) ≤ c} ∼ kc + l.

Example 2.3 Let us consider the following examples, of groups Γ wich satisfy that the family { 1 L(g) : e = g ∈ Γ} lies in ℓ 2 . 1. Let Γ = Z Z. Here the length function is L(m) = |m|, m ∈ Z Z. The group C * -algebra equals in this case C(S 1 ). 2. Let Γ be a finite extension of Z Z, i.e. a group Γ which has a copy of Z Z inside, as a normal subgroup, and the quotient F = Γ/Z Z is finite. Then, as a set, Γ is Z Z×F . Let F = {f 1 , …, f n }.

Then the classes of (1, f 1 ), …, (1, f n ) (i.e. these elements regarded as elements of Γ) are generators for Γ. Let us consider the length function L given by word length with respect to this set of generators. Note that for this L, there are at most 2n elements of Γ with any given length. It follows that { 1 L(g) : e = g ∈ Γ} lies in ℓ 2 . The (reduced) C * -algebra of such Γ can be computed. They consist of algebras of n × n matrices with entries in C(S 1 ), see chapter VIII of [Davidson] for a complete descrition of this computation. Let us point out two special cases of this type (a) Γ = Z Z × F with the usual product for pairs. In this case the C * -algebra is

The algebra C * r (F ) is finite dimensional, therefore in this case C * r (Γ) consists of a direct sum of full matrix alge

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