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.
be a complex Hilbert space, and
takes non-zero values for only finitely many
is said to be a positive-definite function if the kernel
-symmetric, that is, it invariant under left
is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure
be a finite abelian group and
be the one-dimensional Hilbert space
(This is a special case of unitary representation.)
To show this, recall that a character of a finite group
to the multiplicative group of norm-1 complex numbers.
A unitary representation is a unital homomorphism
are intimately related to unitary representations of
gives rise to a family of positive-definite functions.
Conversely, given a positive-definite function, one can define a unitary representation of
is the projection onto a closed subspace
strongly) continuous, then clearly so is
On the other hand, consider now a positive-definite function
defines a (possibly degenerate) inner product on
Let the resulting Hilbert space be denoted by
We notice that the "matrix elements"
The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds: where
denotes the closure of the linear span.
as elements (possibly equivalence classes) in
, whose support consists of the identity element
be the additive group of integers
is a bounded operator acting on some Hilbert space.
By the discussion from the previous section, we have a unitary representation of
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.