Positive definite function on a group

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" → "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 unitary representation is an unital homomorphism Φ: "G" → "L"("H") where Φ("s") is an unitary operator for all "s". For such Φ, Φ("s"-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 \ {} & =sum_{s,t in G}langle Phi (t) h(t), Phi(s)h(s) angle \ {} & = langle sum_{t in G} Phi (t) h(t), sum_{s in G} Phi(s)h(s) angle \ {} & 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 "Uah"("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 "PUaP" = "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") = "Tn" 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 Z, Φ("n") = "U""n" for an unitary operator "U". Moreover, the property "PUaP" = "F"("a") now translates to "PUnP" = "Tn". 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

*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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