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.
Crossed product
Suppose that "A" is a von Neumann algebra of operators acting on a Hilbert space "H" and "G" is a discrete group acting on "A". We let "K" be the Hilbert space of all square summable "H"-valued functions on "G". There is an action of "A" on "K"
given by 
for "k" in "K", "g", "h" in "G", and "a" in "A",
and there is an action of "G" on "K" given by
The crossed product formula_5 is the von Neumann algebra acting on "K" generated by the actions of "A" and "G" on "K". It does not depend (up to isomorphism) on the choice of the Hilbert space "H".