Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Hilbert transform
The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between "L"(R) and the dual space "L"(R), where "p" and "q" are Hölder conjugates and 1 Banach space of "real"-valued functions in "L"(R), the Hilbert transform defines a linear complex structure on this Banach space. In particular, when "p"=2, the Hilbert transform gives the Hilbert space of real-valued functions in "L"(R) the structure of a "complex" Hilbert space.