Positive-definite function on a group

Definition

Let G be a group, H be a complex Hilbert space, and L(H) be the bounded operators on H. A positive-definite function on G is a function F: G \to L(H) that satisfies

:\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle \geq 0 ,

for every function h: G \to H with finite support (h takes non-zero values for only finitely many s).

In other words, a function F: G \to L(H) is said to be a positive-definite function if the kernel K: G \times G \to L(H) defined by K(s, t) = F(s^{-1}t) is a positive-definite kernel. Such a kernel is G-symmetric, that is, it invariant under left G-action: K(s, t) = K(rs, rt) , \quad \forall r \in GWhen G is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure \mu. A positive-definite function on G is a continuous function F: G \to L(H) that satisfies\int_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle ; \mu(ds) \mu(dt) \geq 0 ,for every continuous function h: G \to H with compact support.

Examples

The constant function F(g) = I , where I is the identity operator on H, is positive-definite.

Let G be a finite abelian group and H be the one-dimensional Hilbert space \mathbb{C}. Any character \chi: G \to \mathbb{C} is positive-definite. (This is a special case of unitary representation.)

To show this, recall that a character of a finite group G is a homomorphism from G to the multiplicative group of norm-1 complex numbers. Then, for any function h: G \to \mathbb{C}, \sum_{s,t \in G}\chi(s^{-1}t)h(t)\overline{h(s)} = \sum_{s,t \in G}\chi(s^{-1})h(t)\chi(t)\overline{h(s)} = \sum_{s}\chi(s^{-1})\overline{h(s)}\sum_{t}h(t)\chi(t) = \left|\sum_{t}h(t)\chi(t)\right|^2 \geq 0.When G = \R^n with the Lebesgue measure, and H = \C^m, a positive-definite function on G is a continuous function F : \R^n \to \C^{m\times m} such that\int_{x, y \in \R^n} h(x)^\dagger F(x-y) h(y); dxdy \geq 0for every continuous function h: \R^n \to \C^m with compact support.

Unitary representations

A unitary representation is a unital homomorphism \Phi: G \to L(H) where \Phi(s) is a unitary operator for all s. For such \Phi, \Phi(s^{-1}) = \Phi(s)^*.

Positive-definite functions on G are intimately related to unitary representations of G. Every unitary representation of G gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of G in a natural way.

Let \Phi: G \to L(H) be a unitary representation of G. If P \in L(H) is the projection onto a closed subspace H' of H. Then F(s) = P \Phi(s) is a positive-definite function on G with values in L(H'). This can be shown readily:

:\begin{align} \sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle & =\sum_{s,t \in G}\langle P \Phi (s^{-1}t) h(t), h(s) \rangle \ {} & =\sum_{s,t \in G}\langle \Phi (t) h(t), \Phi(s)h(s) \rangle \ {} & = \left\langle \sum_{t \in G} \Phi (t) h(t), \sum_{s \in G} \Phi(s)h(s) \right\rangle \ {} & \geq 0 \end{align}

for every h: G \to H' with finite support. If G has a topology and \Phi is weakly(resp. strongly) continuous, then clearly so is F.

On the other hand, consider now a positive-definite function F on G. A unitary representation of G can be obtained as follows. Let C_{00}(G, H) be the family of functions h: G \to H with finite support. The corresponding positive kernel K(s, t) = F(s^{-1}t) defines a (possibly degenerate) inner product on C_{00}(G, H). Let the resulting Hilbert space be denoted by V.

We notice that the "matrix elements" K(s, t) = K(a^{-1}s, a^{-1}t) for all a, s, t in G. So U_ah(s) = h(a^{-1}s) preserves the inner product on V, i.e. it is unitary in L(V). It is clear that the map \Phi(a) = U_a is a representation of G on V.

The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:

:V = \bigvee_{s \in G} \Phi(s)H

where \bigvee denotes the closure of the linear span.

Identify H as elements (possibly equivalence classes) in V, whose support consists of the identity element e \in G, and let P be the projection onto this subspace. Then we have PU_aP = F(a) for all a \in G.

Toeplitz kernels

Let G be the additive group of integers \mathbb{Z}. The kernel K(n, m) = F(m - n) is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If F is of the form F(n) = T^n where T is a bounded operator acting on some Hilbert space, one can show that the kernel K(n, m) is positive if and only if T is a contraction. By the discussion from the previous section, we have a unitary representation of \mathbb{Z}, \Phi(n) = U^n for a unitary operator U. Moreover, the property PU_aP = F(a) now translates to PU^nP = T^n. This is precisely Sz.-Nagy's dilation theorem and hints at an important dilation-theoretic characterization of positivity that leads to a parametrization of arbitrary positive-definite kernels.

References