Positive-definite function on a group


In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.

Definition

Let G be a group, H be a complex Hilbert space, and L be the bounded operators on H.
A positive-definite function on G is a function that satisfies
for every function h: G → H with finite support.
In other words, a function F: G → L is said to be a positive-definite function if the kernel K: G × G → L defined by K = F is a positive-definite kernel.

Unitary representations

A unitary representation is a unital homomorphism Φ: G → L where Φ is a unitary operator for all s. For such Φ, Φ = Φ*.
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 Φ: G → L be a unitary representation of G. If P ∈ L is the projection onto a closed subspace H` of H. Then F = P Φ is a positive-definite function on G with values in L. This can be shown readily:
for every h: G → H` with finite support. If G has a topology and Φ is weakly 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 C00 be the family of functions h: G → H with finite support. The corresponding positive kernel K = F defines a inner product on C00. Let the resulting Hilbert space be denoted by V.
We notice that the "matrix elements" K = K for all a, s, t in G. So Uah = h preserves the inner product on V, i.e. it is unitary in L. It is clear that the map Φ = Ua is a representation of G on V.
The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:
where denotes the closure of the linear span.
Identify H as elements in V, whose support consists of the identity element e ∈ G, and let P be the projection onto this subspace. Then we have PUaP = F for all a ∈ G.

Toeplitz kernels

Let G be the additive group of integers Z. The kernel K = F is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If F is of the form F = Tn where T is a bounded operator acting on some Hilbert space. One can show that the kernel K is positive if and only if T is a contraction. By the discussion from the previous section, we have a unitary representation of Z, Φ = Un for a unitary operator U. Moreover, the property PUaP = F now translates to PUnP = Tn. 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.