View source for 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 space]]s, and algebraic [[group (mathematics)|group]]s. It can be viewed as a particular type of [[positive-definite kernel]] where the underlying set has the additional group structure.
== Definition ==
Let <math>G</math> be a group, <math>H</math> be a complex Hilbert space, and <math>L(H)</math> be the bounded operators on <math>H</math>.
A '''positive-definite function''' on <math>G</math> is a function <math>F: G \to L(H)</math> that satisfies
:<math>\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle \geq 0 ,</math>
for every function <math>h: G \to H</math> with finite support (<math>h</math> takes non-zero values for only finitely many <math>s</math>).
In other words, a function <math>F: G \to L(H)</math> is said to be a positive-definite function if the kernel <math>K: G \times G \to L(H) </math> defined by <math>K(s, t) = F(s^{-1}t)</math> is a positive-definite kernel. Such a kernel is <math>G</math>-symmetric, that is, it invariant under left <math>G</math>-action: <math display="block">K(s, t) = K(rs, rt) , \quad \forall r \in G</math>When <math>G</math> is a [[locally compact group]], the definition generalizes by integration over its left-invariant [[Haar measure]] <math>\mu</math>. A positive-definite function on <math>G</math> is a continuous function <math>F: G \to L(H)</math> that satisfies<math display="block">\int_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle \; \mu(ds) \mu(dt) \geq 0 ,</math>for every continuous function <math>h: G \to H</math> with [[compact support]].
=== Examples ===
The constant function <math>F(g) = I </math>, where <math>I</math> is the identity operator on <math>H</math>, is positive-definite.
Let <math>G</math> be a finite abelian group and <math>H</math> be the one-dimensional Hilbert space <math>\mathbb{C}</math>.  Any [[Character theory|character]] <math>\chi: G \to \mathbb{C}</math> is positive-definite. (This is a special case of [[Positive-definite function on a group#Unitary representations|unitary representation]].)
To show this, recall that a character of a finite group <math>G</math> is a homomorphism from <math>G</math> to the multiplicative group of norm-1 complex numbers. Then, for any function <math>h: G \to \mathbb{C}</math>, <math display="block">\sum_{s,t \in G}\chi(s^{-1}t)h(t)\overline{h(s)} = \sum_{s,t \in G}\chi(s^{-1})h(t)\chi(t)\overline{h(s)}
= \sum_{s}\chi(s^{-1})\overline{h(s)}\sum_{t}h(t)\chi(t) = \left|\sum_{t}h(t)\chi(t)\right|^2 \geq 0.</math>When <math>G = \R^n</math> with the [[Lebesgue measure]], and <math>H = \C^m</math>, a positive-definite function on <math>G</math> is a continuous function <math>F : \R^n \to \C^{m\times m}</math> such that<math display="block">\int_{x, y \in \R^n} h(x)^\dagger F(x-y) h(y)\; dxdy \geq 0</math>for every continuous function <math>h: \R^n \to \C^m</math> with compact support.
== Unitary representations ==
A '''[[unitary representation]]''' is a unital homomorphism <math>\Phi: G \to L(H)</math> where <math>\Phi(s)</math> is a unitary operator for all <math>s</math>. For such <math>\Phi</math>, <math>\Phi(s^{-1}) = \Phi(s)^*</math>.
Positive-definite functions on <math>G</math> are intimately related to unitary representations of <math>G</math>. Every unitary representation of <math>G</math> gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of <math>G</math> in a natural way.
Let <math>\Phi: G \to L(H)</math> be a unitary representation of <math>G</math>. If <math>P \in L(H)</math> is the projection onto a closed subspace <math>H'</math> of <math>H</math>. Then <math>F(s) = P \Phi(s)</math> is a positive-definite function on <math>G</math> with values in <math>L(H')</math>. This can be shown readily:
:<math>\begin{align}
\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle
& =\sum_{s,t \in G}\langle P \Phi (s^{-1}t) h(t), h(s) \rangle \\
{} & =\sum_{s,t \in G}\langle \Phi (t) h(t), \Phi(s)h(s) \rangle \\
{} & = \left\langle \sum_{t \in G} \Phi (t) h(t), \sum_{s \in G} \Phi(s)h(s) \right\rangle \\
{} & \geq 0
\end{align}
</math>
for every <math>h: G \to H'</math> with finite support. If <math>G</math> has a topology and <math>\Phi</math> is weakly(resp. strongly) continuous, then clearly so is <math>F</math>.
On the other hand, consider now a positive-definite function <math>F</math> on <math>G</math>. A unitary representation of <math>G</math> can be obtained as follows. Let <math>C_{00}(G, H)</math> be the family of functions <math>h: G \to H</math> with finite support. The corresponding positive kernel <math>K(s, t) = F(s^{-1}t)</math> defines a (possibly degenerate) inner product on <math>C_{00}(G, H)</math>. Let the resulting Hilbert space be denoted by <math>V</math>.
We notice that the "matrix elements" <math>K(s, t) = K(a^{-1}s, a^{-1}t)</math> for all <math>a, s, t</math> in <math>G</math>. So <math>U_ah(s) = h(a^{-1}s)</math> preserves the inner product on <math>V</math>, i.e. it is unitary in <math>L(V)</math>. It is clear that the map <math>\Phi(a) = U_a</math> is a representation of <math>G</math> on <math>V</math>.
The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:
:<math>V = \bigvee_{s \in G} \Phi(s)H</math>
where <math>\bigvee</math> denotes the closure of the linear span.
Identify <math>H</math> as elements (possibly equivalence classes) in <math>V</math>, whose support consists of the identity element <math>e \in G</math>, and let <math>P</math> be the projection onto this subspace. Then we have <math>PU_aP = F(a)</math> for all <math>a \in G</math>.
== Toeplitz kernels ==
Let <math>G</math> be the additive group of integers <math>\mathbb{Z}</math>. The kernel <math>K(n, m) = F(m - n)</math> is called a kernel of ''Toeplitz'' type, by analogy with [[Toeplitz matrix|Toeplitz matrices]]. If <math>F</math> is of the form <math>F(n) = T^n</math> where <math>T</math> is a bounded operator acting on some Hilbert space. One can show that the kernel <math>K(n, m)</math> is positive if and only if <math>T</math> is a [[Contraction (operator theory)|contraction]]. By the discussion from the previous section, we have a unitary representation of <math>\mathbb{Z}</math>, <math>\Phi(n) = U^n</math> for a unitary operator <math>U</math>. Moreover, the property <math>PU_aP = F(a)</math> now translates to  <math>PU^nP = T^n</math>. 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.
== References ==
*{{cite book |last1=Berg |first1=Christian |last2=Christensen |first2=Paul |last3=Ressel |title=Harmonic Analysis on Semigroups |series=Graduate Texts in Mathematics |volume=100 |publisher=Springer Verlag |year=1984}}
*{{cite book |last=Constantinescu |first=T. |title=Schur Parameters, Dilation and Factorization Problems |publisher=Birkhauser Verlag |year=1996}}
*{{cite book |last1=Sz.-Nagy |first1=B. |last2=Foias |first2=C. |title=Harmonic Analysis of Operators on Hilbert Space |publisher=North-Holland |year=1970}}
*{{cite book |last=Sasvári |first=Z. |title=Positive Definite and Definitizable Functions |publisher=Akademie Verlag |year=1994}}
*{{cite book |last1=Wells |first1=J. H. |last2=Williams |first2=L. R. |title=Embeddings and extensions in analysis |series=Ergebnisse der Mathematik und ihrer Grenzgebiete |volume=84 |publisher=Springer-Verlag |location=New York-Heidelberg |year=1975 |pages=vii+108}}
[[Category:Operator theory]]
[[Category:Representation theory of groups]]