The space of states of a theory is a representation of the product of the two Virasoro algebras. This space is a Hilbert space if the theory is unitary. This space may contain a vacuum state, or in statistical mechanics, a thermal state. Unless the central charge vanishes, there cannot exist a state that leaves the entire infinite dimensional conformal symmetry unbroken. The best we can have is a state that is invariant under the generators L n ≥ ≥ − − 1 of the Virasoro algebra, whose basis is ( L n ) n ∈ ∈ Z . This contains the generators L − − 1 , L 0 , L 1 of the global conformal transformations. The rest of the conformal group is spontaneously broken.