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

$\sum _{s,t\in G}⟨F(s^{-1}t)h(t),h(s)⟩\ge 0,$

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: $$K(s,t)=K(rs,rt),\mathrm{\forall }r\in G$$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$$\underset{s,t\in G}{\int }⟨F(s^{-1}t)h(t),h(s)⟩\mu (ds)\mu (dt)\ge 0,$$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 $h:G\to \mathbb{ℂ}$, $$\sum _{s,t\in G}\chi (s^{-1}t)h(t)\overline{h(s)}=\sum _{s,t\in G}\chi (s^{-1})h(t)\chi (t)\overline{h(s)}=\sum _s\chi (s^{-1})\overline{h(s)}\sum _{t}h(t)\chi (t)={|\sum _{t}h(t)\chi (t)|}^2\ge 0.$$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$$\underset{x,y\in {\mathbb{ℝ}}^n}{\int }h(x{)}^{†}F(x-y)h(y)dxdy\ge 0$$for every continuous function $h:{\mathbb{ℝ}}^n\to {\mathbb{ℂ}}^m$ with compact support.

Unitary representations

A unitary representation is a unital homomorphism $\mathrm{\Phi }:G\to L(H)$ where $\mathrm{\Phi }(s)$ is a unitary operator for all $s$. For such $\mathrm{\Phi }$, $\mathrm{\Phi }(s^{-1})=\mathrm{\Phi }(s{)}^{*}$. 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 $\mathrm{\Phi }:G\to L(H)$ be a unitary representation of $G$. If $P\in L(H)$ is the projection onto a closed subspace $H^{\prime }$ of $H$. Then $F(s)=P\mathrm{\Phi }(s)$ is a positive-definite function on $G$ with values in $L(H^{\prime })$. This can be shown readily:

$\begin{array}{rl}\sum _{s,t\in G}⟨F(s^{-1}t)h(t),h(s)⟩& =\sum _{s,t\in G}⟨P\mathrm{\Phi }(s^{-1}t)h(t),h(s)⟩\\ & =\sum _{s,t\in G}⟨\mathrm{\Phi }(t)h(t),\mathrm{\Phi }(s)h(s)⟩\\ & =⟨\sum _{t\in G}\mathrm{\Phi }(t)h(t),\sum _{s\in G}\mathrm{\Phi }(s)h(s)⟩\\ & \ge 0\end{array}$

for every $h:G\to H^{\prime }$ with finite support. If $G$ has a topology and $\mathrm{\Phi }$ 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 $\mathrm{\Phi }(a)=U_a$ is a representation of $G$ on $V$. The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:

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

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

Toeplitz kernels

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