Cyclic vectors for Hilbert algebras

In this paper, we give a simple proof of the Stone-von Neumann theorem about the irreducible unitary representations of Heisenberg groups. First of all, we prove a result about the representations of Hilbert algebras of Hilbert-Schmidt operators on a Hilbert space. Then we use it to give a simple proof of the Stone-von Neumann theorem. Applications to theoretical physics are mentioned.