Negative type functions

In this section, we introduce kernels and functions conditionally of negative type. We also make the correspondence between orbits of affine actions and functions conditionally of negative type on groups. We end up this section by defining Property (T) and Haagerup property for groups. We refer to the Appendix C of [BdlHV08] for the proofs and more details about conditionally negative type functions.

Definition 4.1.4. A kernel Ψ : W × W → R on a set W is said to be conditionally of negative type if it satisfies the following properties:

1. Ψ(x, x) = 0, for all x ∈ W ;

2. Ψ(x, y) = Ψ(y, x), for all x, y ∈ W ;

3. For any x₁, . . . , x_n ∈ W and for any α₁, . . . , α_n∈ R satisfying Pn

i=1α_i = 0, we have

n

X

i=1 n

X

j=1

α_iα_jΨ(x_i, x_j) ≤ 0.

It is fairly elementary to produce examples of such kernels. Indeed, take f : W → H to be any map with range H, a Hilbert space. Then, a di-rect computation shows that the kernel defined by Ψ(x, y) = kf (x) − f (y)k² is conditionally of negative type. Interesting examples arise when the set W is endowed with a G-action and when the kernel is G-invariant, i.e.

Ψ(gx, gy) = Ψ(x, y), for all x, y ∈ W and g ∈ G. Considering kernels on G which are invariant for the action on itself leads us to the next definition.

Definition 4.1.5. A continuous function ψ : G → R on a topological group G is said to be conditionally of negative type if the kernel on G defined by (g, h) 7→ ψ(g⁻¹h) is conditionally of negative type.

Going back to the preceding example, if the set W is a G-space and if the map f satisfies kf (gx) − f (gy)k = kf (x) − f (y)k, for all x, y ∈ W and all g ∈ G, then, chosing a basepoint x₀ ∈ W , the associated function ψ on G defined by

ψ(g) = kf (gx₀) − f (x₀)k²

is conditionally of negative type. In particular, if b is a continuous cocycle for some unitary representation π of a group G, then the function on G given by g 7→ kb(g)k² is conditionally of negative type. This example is essen-tially universal, by the so-called GNS construction (see p.63 in [dlHV89] for a proof).

Proposition 4.1.6. (GNS construction) Let ψ be a function conditionally of negative type on G. There exist an orthogonal representation π_ψ of G acting on a real Hilbert space H_ψ and a cocycle b_ψ ∈ Z¹(G, π_ψ) satisfying ψ(g) = kbψ(g)k², for all g ∈ G. Moreover, the triple (Hψ, πψ, bψ) is unique up to isomorphism.

We also note that, if b ∈ Z¹(G, π), then the image of b corresponds to the orbit of 0 under the affine action α = (π, b), that is, b(g) = α(g)0. This explains why cocycles are sometimes referred to as orbital maps.

The following result fills the gap between the geometrical and the algebraic aspects of affine actions on Hilbert spaces. Its proof is a direct consequence of the GNS construction and the Lemma of the centre.

Proposition 4.1.7. Let ψ be a conditionally of negative type function on G, let (H_ψ, π_ψ, b_ψ) be its associated GNS triple and let α_ψ = (π_ψ, b_ψ) be the corresponding affine action. The following properties are equivalent :

1. The function ψ is bounded on G;

2. The map b_ψ is a coboundary;

3. The affine action α_ψ has a fixed point;

4. The affine action α_ψ has only bounded orbits.

In particular, we deduce that an affine isometric action α = (π, b) is un-bounded if and only if b is not a coboundary, and, furthermore, the affine action α is proper if and only if the function on G given by g 7→ kb(g)k is proper.

From all this, it is natural to identify groups for which any affine isometric action on a Hilbert space has a fixed point. These groups have the so-called Property (T). Before stating the precise definition, we recall some important notions of representation theory.

Definition 4.1.8. Let π be a unitary (or orthogonal) representation of G.

1. The representation π strongly contains the trivial representation 1_G (or, π has invariant vectors) if, there exists a non-zero vector ξ ∈ H so that π(g)ξ = ξ, for all g ∈ G. In this case, we write 1_G ⊂ π.

2. The representation π weakly contains the trivial representation 1_G (or, π has almost invariant vectors) if, for any compact subset Q ⊂ G and for any  > 0, there exists a vector ξ ∈ H so that

sup

g∈Q

kπ(g)ξ − ξk < kξk.

In this case, we write 1_G≺ π.

3. The representation π is C₀ if all of its matrix coefficients belong to C₀(G), that is, all the functions of the form ϕ_ξ,η : g 7→ hπ(g)ξ, ηi, for some ξ, η ∈ Hπ, vanish at infinity.

Definition 4.1.9. Let G be a second countable, locally compact group. The group G has Kazhdan’s Property (T) (or simply, Property (T)), if it satisfies one of the following equivalent conditions :

1. Every strongly continuous unitary representation that weakly contains the trivial representation also contains it strongly;

2. Every continuous, conditionally negative type function on G is bounded;

3. Every continuous, affine isometric action of G on a Hilbert space has a fixed point.

The first condition, which was originally introduced by D. Kazhdan in the 1960s, means that the trivial representation 1_G is isolated in bG, the unitary dual of G, for the Fell topology. This explains the notation “(T)”, where the letter T stands for trivial. We remark that, in the above definition, the equivalence between the last two claims is proved by Proposition 4.1.7.

Property (T) has strong structural consequences. Amongst other things, a group G with Property (T) is compactly generated and is unimodular. If G is discrete, then its abelianisation G/[G, G] is finite.

Examples of groups with Property (T) are : compact groups, SL_n+1(Z), SL_n+1(R) and Sp(n, 1), the real rank 1 simple Lie groups of isometries of a quaternionic hermitian form of signature (n, 1), for n ≥ 2. By way of con-trast, the next definition is a strong negation of Property (T).

Definition 4.1.10. Let G be a second countable, locally compact group. The group G has the Haagerup Property (or, is a-T-menable), if it satisfies one of the following equivalent conditions :

1. There exists a strongly continuous unitary C₀-representation which con-tains the trivial representation weakly;

2. There exists a continuous function on G which is conditionally of neg-ative type and proper;

3. There exists a continuous, affine isometric action of G on a Hilbert space which is metrically proper.

This class of groups is quite large and contains : amenable groups, all groups with α^\₂ > 0 encountered in the set of examples appearing in Sub-section 4.1.1 (free groups and, more generally, groups acting properly on trees, Coxeter groups, Baumslag-Solitar groups, Thompson’s group F , ...) and the Lie groups SO(n, 1) and SU (n, 1), which are isometry groups of the n-dimensional real and complex hyperbolic spaces respectively.