# Positive definite function on a group

In

operator theory , a**positive definite function on a group**relates the notions of positivity, in the context ofHilbert space s, and algebraic groups. It can be viewed as a particular type ofpositive 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:$sum\_\{s,t\; in\; G\}langle\; F(s^\{-1\}t)\; h(t),\; h(s)\; angle\; geq\; 0\; ,$

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 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**An

is an unital homomorphism Φ: "G" → "L"("H") where Φ("s") is an unitary operator for all "s". For such Φ, Φ("s"unitary representation ^{-1}) = Φ("s")*.Positive functions on "G" is 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:

:$egin\{array\}\{rl\}sum\_\{s,t\; in\; G\}langle\; F(s^\{-1\}t)\; h(t),\; h(s)\; angle\; =sum\_\{s,t\; in\; G\}langle\; P\; Phi\; (s^\{-1\}t)\; h(t),\; h(s)\; angle\; \backslash \; \{\}\; =sum\_\{s,t\; in\; G\}langle\; Phi\; (t)\; h(t),\; Phi(s)h(s)\; angle\; \backslash \; \{\}\; =\; langle\; sum\_\{t\; in\; G\}\; Phi\; (t)\; h(t),\; sum\_\{s\; in\; G\}\; Phi(s)h(s)\; angle\; \backslash \; \{\}\; geq\; 0end\{array\}$

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". An 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:

:$V\; =\; igvee\_\{s\; in\; G\}\; Phi(s)H$

where $igvee$ 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 acontraction . By the discussion from the previous section, we have a unitary representation of**Z**, Φ("n") = "U"^{"n"}for an unitary operator "U". Moreover, the property "PU_{a}P" = "F"("a") now translates to "PU^{n}P" = "T^{n}". This is preciselySz.-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***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

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