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(H) be the bounded operators on H. A positive-definite function on G is a function that satisfies 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 is said to be a positive-definite function if the kernel defined by is a positive-definite kernel. Such a kernel is G-symmetric, that is, it invariant under left G-action: When 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 that satisfiesfor 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 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, When G = \R^n with the Lebesgue measure, and H = \C^m, a positive-definite function on G is a continuous function such thatfor every continuous function with compact support.

Unitary representations

A unitary representation is a unital homomorphism where \Phi(s) is a unitary operator for all s. For such \Phi,. 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 be a unitary representation of G. If P \in L(H) is the projection onto a closed subspace H' of H. Then is a positive-definite function on G with values in L(H'). This can be shown readily: 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 be the family of functions h: G \to H with finite support. The corresponding positive kernel defines a (possibly degenerate) inner product on. Let the resulting Hilbert space be denoted by V. We notice that the "matrix elements" for all a, s, t in G. So preserves the inner product on V, i.e. it is unitary in L(V). It is clear that the map is a representation of G on V. The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds: 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 for all a \in G.

Toeplitz kernels

Let G be the additive group of integers \mathbb{Z}. The kernel 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}, for a unitary operator U. Moreover, the property 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.