# 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 *F*: *G* → *L*(*H*) 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 *U _{a}h*(

*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 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 *PU _{a}P* =

*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*) = *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

**Z**, Φ(

*n*) =

*U*

^{n}for a unitary operator

*U*. Moreover, the property

*PU*=

_{a}P*F*(

*a*) now translates to

*PU*=

^{n}P*T*. 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.

^{n}## References

- Christian Berg, Christensen, Paul Ressel
*Harmonic Analysis on Semigroups*, GTM, Springer Verlag. - T. Constantinescu,
*Schur Parameters, Dilation and Factorization Problems*, Birkhauser Verlag, 1996. - B. Sz.-Nagy and C. Foias,
*Harmonic Analysis of Operators on Hilbert Space,*North-Holland, 1970. - Z. Sasvári,
*Positive Definite and Definitizable Functions*, Akademie Verlag, 1994 - Wells, J. H.; Williams, L. R.
*Embeddings and extensions in analysis*. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.