  A natural generalization of the Heisenberg uncertainty principle inequality
holding for non hermitian operators is presented and applied to the fractional
quantum Hall effect (FQHE). This inequality was used in a previous paper to
prove the absence of long range order in the ground state of several 1D systems
with continuous group symmetries. In this letter we use it to rule out the
occurrence of Bose-Einstein condensation in the bosonic representation of the
FQHE wave function proposed by Girvin and MacDonald. We show that the absence
of off-diagonal long range order in this 2D problem is directly connected with
the $q^2$ behavior of the static structure function $S(q)$ at small momenta.
