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 C*-module
A complex Hilbert space "H" is a Hilbert C-module under its inner product, the complex numbers being a C*-algebra with an involution given by complex conjugation.