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 $F:G\to L(H)$ that satisfies

$\sum _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \geq 0,$

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 $F:G\to L(H)$ is said to be a positive-definite function if the kernel $K:G\times G\to L(H)$ defined by $K(s,t)=F(s^{-1}t)$ is a positive-definite kernel. Such a kernel is $G$-symmetric, that is, it invariant under left $G$-action: $$K(s,t)=K(rs,rt),\quad \forall r\in G$$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 $F:G\to L(H)$ that satisfies$$\int _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \;\mu (ds)\mu (dt)\geq 0,$$for 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 $\chi :G\to \mathbb {C}$ 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 $h:G\to \mathbb {C}$, $$\sum _{s,t\in G}\chi (s^{-1}t)h(t){\overline {h(s)}}=\sum _{s,t\in G}\chi (s^{-1})h(t)\chi (t){\overline {h(s)}}=\sum _{s}\chi (s^{-1}){\overline {h(s)}}\sum _{t}h(t)\chi (t)=\left|\sum _{t}h(t)\chi (t)\right|^{2}\geq 0.$$When $G=\mathbb {R} ^{n}$ with the Lebesgue measure, and $H=\mathbb {C} ^{m}$, a positive-definite function on $G$ is a continuous function $F:\mathbb {R} ^{n}\to \mathbb {C} ^{m\times m}$ such that$$\int _{x,y\in \mathbb {R} ^{n}}h(x)^{\dagger }F(x-y)h(y)\;dxdy\geq 0$$for every continuous function $h:\mathbb {R} ^{n}\to \mathbb {C} ^{m}$ with compact support.

Unitary representations

A unitary representation is a unital homomorphism $\Phi :G\to L(H)$ where $\Phi (s)$ is a unitary operator for all $s$. For such $\Phi$, $\Phi (s^{-1})=\Phi (s)^{*}$.

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 $\Phi :G\to L(H)$ be a unitary representation of $G$. If $P\in L(H)$ is the projection onto a closed subspace $H'$ of $H$. Then $F(s)=P\Phi (s)$ is a positive-definite function on $G$ with values in $L(H')$. This can be shown readily:

${\begin{aligned}\sum _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle &=\sum _{s,t\in G}\langle P\Phi (s^{-1}t)h(t),h(s)\rangle \\{}&=\sum _{s,t\in G}\langle \Phi (t)h(t),\Phi (s)h(s)\rangle \\{}&=\left\langle \sum _{t\in G}\Phi (t)h(t),\sum _{s\in G}\Phi (s)h(s)\right\rangle \\{}&\geq 0\end{aligned}}$

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 $C_{00}(G,H)$ be the family of functions $h:G\to H$ 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 $\Phi (a)=U_{a}$ is a representation of $G$ on $V$.

The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:

$V=\bigvee _{s\in G}\Phi (s)H$

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 $PU_{a}P=F(a)$ for all $a\in G$.

Toeplitz kernels

Let $G$ be the additive group of integers $\mathbb {Z}$. 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 $\mathbb {Z}$, $\Phi (n)=U^{n}$ for a unitary operator $U$. Moreover, the property $PU_{a}P=F(a)$ now translates to $PU^{n}P=T^{n}$. 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.

References

• Berg, Christian; Christensen, Paul; Ressel (1984). Harmonic Analysis on Semigroups. Graduate Texts in Mathematics. Vol. 100. Springer Verlag.
• Constantinescu, T. (1996). Schur Parameters, Dilation and Factorization Problems. Birkhauser Verlag.
• Sz.-Nagy, B.; Foias, C. (1970). Harmonic Analysis of Operators on Hilbert Space. North-Holland.
• Sasvári, Z. (1994). Positive Definite and Definitizable Functions. Akademie Verlag.
• Wells, J. H.; Williams, L. R. (1975). Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 84. New York-Heidelberg: Springer-Verlag. pp. vii+108.