  A function of positive type can be defined as a positive functional on a
convolution algebra of a locally compact group. In the case where the group is
abelian, by Bochner's theorem a function of positive type is, up to
normalization, the Fourier transform of a probability measure. Therefore,
considering the group of translations on phase space, a suitably normalized
phase-space function of positive type can be regarded as a realization of a
classical state. Thus, it may be called a function of classical positive type.
Replacing the ordinary convolution on phase space with the twisted convolution,
one obtains a noncommutative algebra of functions whose positive functionals we
may call functions of quantum positive type. In fact, by a quantum version of
Bochner's theorem, a continuous function of quantum positive type is, up to
normalization, the (symplectic) Fourier transform of a Wigner quasi-probability
distribution; hence, it can be regarded as a phase-space realization of a
quantum state. Playing with functions of positive type, classical and quantum,
one is led in a natural way to consider a class of semigroups of operators, the
classical-quantum semigroups. The physical meaning of these mathematical
objects is unveiled via quantization, so obtaining a class of quantum dynamical
semigroups that, borrowing terminology from quantum information science, may be
called classical-noise semigroups.
