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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 space]]s, and algebraic [[group (mathematics)|group]]s. It can be viewed as a particular type of [[positive-definite kernel]] where the underlying set has the additional group structure.
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| == Definition ==
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| Let ''G'' be a group, ''H'' be a complex Hilbert space, and ''L''(''H'') be the bounded operators on ''H''.
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| A '''positive-definite function''' on ''G'' is a function {{nowrap|''F'': ''G'' → ''L''(''H'')}} that satisfies
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| :<math>\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle \geq 0 ,</math>
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| for every function ''h'': ''G'' → ''H'' with finite support (''h'' takes non-zero values for only finitely many ''s'').
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| 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''<sup>−1</sup>''t'') is a positive-definite kernel.
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| == Unitary representations ==
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| 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''<sup>−1</sup>) = Φ(''s'')*.
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| 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.
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| 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:
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| :<math>\begin{align}
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| \sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle
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| & =\sum_{s,t \in G}\langle P \Phi (s^{-1}t) h(t), h(s) \rangle \\
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| {} & =\sum_{s,t \in G}\langle \Phi (t) h(t), \Phi(s)h(s) \rangle \\
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| {} & = \left\langle \sum_{t \in G} \Phi (t) h(t), \sum_{s \in G} \Phi(s)h(s) \right\rangle \\
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| {} & \geq 0
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| \end{align}
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| </math> | |
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| 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''.
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| 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''<sub>00</sub>(''G'', ''H'') be the family of functions ''h'': ''G'' → ''H'' with finite support. The corresponding positive kernel ''K''(''s'', ''t'') = ''F''(''s''<sup>−1</sup>''t'') defines a (possibly degenerate) inner product on ''C''<sub>00</sub>(''G'', ''H''). Let the resulting Hilbert space be denoted by ''V''.
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| We notice that the "matrix elements" ''K''(''s'', ''t'') = ''K''(''a''<sup>−1</sup>''s'', ''a''<sup>−1</sup>''t'') for all ''a'', ''s'', ''t'' in ''G''. So ''U<sub>a</sub>h''(''s'') = ''h''(''a''<sup>−1</sup>''s'') preserves the inner product on ''V'', i.e. it is unitary in ''L''(''V''). It is clear that the map Φ(''a'') = ''U''<sub>a</sub> is a representation of ''G'' on ''V''.
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| The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:
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| :<math>V = \bigvee_{s \in G} \Phi(s)H \, </math>
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| where <math>\bigvee</math> denotes the closure of the linear span.
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| 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<sub>a</sub>P'' = ''F''(''a'') for all ''a'' ∈ ''G''.
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| == Toeplitz kernels ==
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| 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 matrix|Toeplitz matrices]]. If ''F'' is of the form ''F''(''n'') = ''T<sup>n</sup>'' 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 (operator theory)|contraction]]. By the discussion from the previous section, we have a unitary representation of '''Z''', Φ(''n'') = ''U''<sup>''n''</sup> for a unitary operator ''U''. Moreover, the property ''PU<sub>a</sub>P'' = ''F''(''a'') now translates to ''PU<sup>n</sup>P'' = ''T<sup>n</sup>''. 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.
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| == References ==
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| *Christian Berg, Christensen, Paul Ressel''Harmonic Analysis on Semigroups'', GTM, Springer Verlag.
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| *T. Constantinescu, ''Schur Parameters, Dilation and Factorization Problems'', Birkhauser Verlag, 1996.
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| *B. Sz.-Nagy and C. Foias, ''Harmonic Analysis of Operators on Hilbert Space,'' North-Holland, 1970.
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| *Z. Sasvári, ''Positive Definite and Definitizable Functions'', Akademie Verlag, 1994
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| *Wells, J. H.; Williams, L. R. ''Embeddings and extensions in analysis''. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.
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| [[Category:Operator theory]]
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| [[Category:Representation theory of groups]]
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