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.

Let ${\displaystyle G}$ be a group, ${\displaystyle H}$ be a complex Hilbert space, and ${\displaystyle L(H)}$ be the bounded operators on ${\displaystyle H}$. A positive-definite function on ${\displaystyle G}$ is a function ${\displaystyle F:G\to L(H)}$ that satisfies

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

for every function ${\displaystyle h:G\to H}$ with finite support (${\displaystyle h}$ takes non-zero values for only finitely many ${\displaystyle s}$).

In other words, a function ${\displaystyle F:G\to L(H)}$ is said to be a positive-definite function if the kernel ${\displaystyle K:G\times G\to L(H)}$ defined by ${\displaystyle K(s,t)=F(s^{-1}t)}$ is a positive-definite kernel. Such a kernel is ${\displaystyle G}$-symmetric, that is, it invariant under left ${\displaystyle G}$-action: $${\displaystyle K(s,t)=K(rs,rt),\quad \forall r\in G}$$When ${\displaystyle G}$ is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure ${\displaystyle \mu }$. A positive-definite function on ${\displaystyle G}$ is a continuous function ${\displaystyle F:G\to L(H)}$ that satisfies$${\displaystyle \int _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \;\mu (ds)\mu (dt)\geq 0,}$$for every continuous function ${\displaystyle h:G\to H}$ with compact support.

The constant function ${\displaystyle F(g)=I}$, where ${\displaystyle I}$ is the identity operator on ${\displaystyle H}$, is positive-definite.

Let ${\displaystyle G}$ be a finite abelian group and ${\displaystyle H}$ be the one-dimensional Hilbert space ${\displaystyle \mathbb {C} }$. Any character ${\displaystyle \chi :G\to \mathbb {C} }$ is positive-definite. (This is a special case of unitary representation.)

To show this, recall that a character of a finite group ${\displaystyle G}$ is a homomorphism from ${\displaystyle G}$ to the multiplicative group of norm-1 complex numbers. Then, for any function ${\displaystyle h:G\to \mathbb {C} }$, $${\displaystyle \sum _{s,t\in G}\chi (s^{-1}t)h(t){\overline {h(s)}}=\sum _{s,t\in G}\chi (s^{-1})h(t)\chi (t){\overline {h(s)}}=\sum _{s}\chi (s^{-1}){\overline {h(s)}}\sum _{t}h(t)\chi (t)=\left|\sum _{t}h(t)\chi (t)\right|^{2}\geq 0.}$$When ${\displaystyle G=\mathbb {R} ^{n}}$ with the Lebesgue measure, and ${\displaystyle H=\mathbb {C} ^{m}}$, a positive-definite function on ${\displaystyle G}$ is a continuous function ${\displaystyle F:\mathbb {R} ^{n}\to \mathbb {C} ^{m\times m}}$ such that$${\displaystyle \int _{x,y\in \mathbb {R} ^{n}}h(x)^{\dagger }F(x-y)h(y)\;dxdy\geq 0}$$for every continuous function ${\displaystyle h:\mathbb {R} ^{n}\to \mathbb {C} ^{m}}$ with compact support.

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

Positive-definite functions on ${\displaystyle G}$ are intimately related to unitary representations of ${\displaystyle G}$. Every unitary representation of ${\displaystyle G}$ gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of ${\displaystyle G}$ in a natural way.

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

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

for every ${\displaystyle h:G\to H'}$ with finite support. If ${\displaystyle G}$ has a topology and ${\displaystyle \Phi }$ is weakly(resp. strongly) continuous, then clearly so is ${\displaystyle F}$.

On the other hand, consider now a positive-definite function ${\displaystyle F}$ on ${\displaystyle G}$. A unitary representation of ${\displaystyle G}$ can be obtained as follows. Let ${\displaystyle C_{00}(G,H)}$ be the family of functions ${\displaystyle h:G\to H}$ with finite support. The corresponding positive kernel ${\displaystyle K(s,t)=F(s^{-1}t)}$ defines a (possibly degenerate) inner product on ${\displaystyle C_{00}(G,H)}$. Let the resulting Hilbert space be denoted by ${\displaystyle V}$.

We notice that the "matrix elements" ${\displaystyle K(s,t)=K(a^{-1}s,a^{-1}t)}$ for all ${\displaystyle a,s,t}$ in ${\displaystyle G}$. So ${\displaystyle U_{a}h(s)=h(a^{-1}s)}$ preserves the inner product on ${\displaystyle V}$, i.e. it is unitary in ${\displaystyle L(V)}$. It is clear that the map ${\displaystyle \Phi (a)=U_{a}}$ is a representation of ${\displaystyle G}$ on ${\displaystyle V}$.

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

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

where ${\displaystyle \bigvee }$ denotes the closure of the linear span.

Identify ${\displaystyle H}$ as elements (possibly equivalence classes) in ${\displaystyle V}$, whose support consists of the identity element ${\displaystyle e\in G}$, and let ${\displaystyle P}$ be the projection onto this subspace. Then we have ${\displaystyle PU_{a}P=F(a)}$ for all ${\displaystyle a\in G}$.

Let ${\displaystyle G}$ be the additive group of integers ${\displaystyle \mathbb {Z} }$. The kernel ${\displaystyle K(n,m)=F(m-n)}$ is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If ${\displaystyle F}$ is of the form ${\displaystyle F(n)=T^{n}}$ where ${\displaystyle T}$ is a bounded operator acting on some Hilbert space, one can show that the kernel ${\displaystyle K(n,m)}$ is positive if and only if ${\displaystyle T}$ is a contraction. By the discussion from the previous section, we have a unitary representation of ${\displaystyle \mathbb {Z} }$, ${\displaystyle \Phi (n)=U^{n}}$ for a unitary operator ${\displaystyle U}$. Moreover, the property ${\displaystyle PU_{a}P=F(a)}$ now translates to ${\displaystyle PU^{n}P=T^{n}}$. 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.

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