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"("H") 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 ("h" takes non-zero values for only finitely many "s").
In other words, a function "F": "G" → "L"("H") is said to be a positive-definite function if the kernel "K": "G" × "G" → "L"("H") defined by "K"("s", "t") = "F"("s"−1"t") is a positive-definite kernel.
Unitary representations.
A unitary representation is a unital homomorphism Φ: "G" → "L"("H") where Φ("s") is a unitary operator for all "s". For such Φ, Φ("s"−1) = Φ("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 Φ: "G" → "L"("H") be a unitary representation of "G". If "P" ∈ "L"("H") is the projection onto a closed subspace "H`" of "H". Then "F"("s") = "P" Φ("s") is a positive-definite function on "G" with values in "L"("H`"). This can be shown readily: 
for every "h": "G" → "H`" with finite support. If "G" has a topology and Φ 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" → "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 "Uah"("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 Φ("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:
where formula_4 denotes the closure of the linear span.
Identify "H" as elements (possibly equivalence classes) 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"("a") for all "a" ∈ "G".
Toeplitz kernels.
Let "G" be the additive group of integers 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") = "Tn" 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 Z, Φ("n") = "U""n" for a unitary operator "U". Moreover, the property "PUaP" = "F"("a") 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.