Abstract
It is well known that locally defined positive definite functions on Lie groups G generally do not extend to positive definite functions which are defined on the whole group. We introduce two stronger positivity concepts for locally defined functions, and show that they are equivalent to extendability. We then apply this to the case when G is the Heisenberg group. When an additional symmetry is imposed, we obtain a complete spectral analysis of the (locally defined) positive definite functions. The methods of proof are based on unitary dilation techniques (i.e., carefully chosen extensions of some underlying Hilbert space associated to the problem), and on spectral theory for noncommuting operators.
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