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 that satisfies

:$\sum_\langle F(s^t) h(t), h(s) \rangle \geq 0 ,$

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-definite 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

A unitary representation is a unital homomorphism Φ: ''G'' → ''L''(''H'') where Φ(''s'') is a unitary operator for all ''s''. For such Φ, Φ(''s''⁻¹) = Φ(''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 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:

:$\begin \sum_\langle F(s^t) h(t), h(s) \rangle & =\sum_\langle P \Phi (s^t) h(t), h(s) \rangle \\ & =\sum_\langle \Phi (t) h(t), \Phi(s)h(s) \rangle \\ & = \left\langle \sum_ \Phi (t) h(t), \sum_ \Phi(s)h(s) \right\rangle \\ & \geq 0 \end$

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''. A 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_ah''(''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:

:$V = \bigvee_{s \in G} \Phi(s)H \,$

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'' ∈ ''G'', and let ''P'' be the projection onto this subspace. Then we have ''PU_aP'' = ''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 a unitary operator ''U''. Moreover, the property ''PU_aP'' = ''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

*Christian Berg, Christensen, Paul Ressel'', Harmonic Analysis on Semigroups'', GTM, Springer Verlag.
*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
*J. H. Wells, L. R. Williams, ''Embeddings and extensions in analysis'', Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.

Operator theory
Representation theory of groups