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Positive-definite function on a group

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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

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Let be a group, be a complex Hilbert space, and be the bounded operators on . A positive-definite function on is a function that satisfies

for every function with finite support ( takes non-zero values for only finitely many ).

In other words, a function is said to be a positive-definite function if the kernel defined by is a positive-definite kernel. Such a kernel is -symmetric, that is, it invariant under left -action: When is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure . A positive-definite function on is a continuous function that satisfiesfor every continuous function with compact support.

Examples

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The constant function , where is the identity operator on , is positive-definite.

Let be a finite abelian group and be the one-dimensional Hilbert space . Any character is positive-definite. (This is a special case of unitary representation.)

To show this, recall that a character of a finite group is a homomorphism from to the multiplicative group of norm-1 complex numbers. Then, for any function , When with the Lebesgue measure, and , a positive-definite function on is a continuous function such thatfor every continuous function with compact support.

Unitary representations

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A unitary representation is a unital homomorphism where is a unitary operator for all . For such , .

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

Let be a unitary representation of . If is the projection onto a closed subspace of . Then is a positive-definite function on with values in . This can be shown readily:

for every with finite support. If has a topology and is weakly(resp. strongly) continuous, then clearly so is .

On the other hand, consider now a positive-definite function on . A unitary representation of can be obtained as follows. Let be the family of functions with finite support. The corresponding positive kernel defines a (possibly degenerate) inner product on . Let the resulting Hilbert space be denoted by .

We notice that the "matrix elements" for all in . So preserves the inner product on , i.e. it is unitary in . It is clear that the map is a representation of on .

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

where denotes the closure of the linear span.

Identify as elements (possibly equivalence classes) in , whose support consists of the identity element , and let be the projection onto this subspace. Then we have for all .

Toeplitz kernels

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Let be the additive group of integers . The kernel is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If is of the form where is a bounded operator acting on some Hilbert space. One can show that the kernel is positive if and only if is a contraction. By the discussion from the previous section, we have a unitary representation of , for a unitary operator . Moreover, the property now translates to . 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

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  • 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.