  We study the realization of conformal symmetry in the QHE as part of the
$W_\infty$ algebra. Conformal symmetry can be realized already at the classical
level and implies the complexification of coordinate space. Its quantum version
is not unitary. Nevertheless, it can be rendered unitary by a suitable
modification of its definition which amounts to taking proper care of the
quantum measure. The consequences of unitarity for the Chern-Simons theory of
the QHE are also studied, showing the connection of non-unitarity with
anomalies. Finally, we discuss the geometrical paradox of realizing conformal
transformations as area preserving diffeomorphisms.
