  We prove uniqueness for continuity equations in Hilbert spaces $H$. The
corresponding drift $F$ is assumed to be in a first order Sobolev space with
respect to some Gaussian measure. As in previous work on the subject, the proof
is based on commutator estimates which are infinite dimensional analogues to
the classical ones due to DiPerna-Lions. Our general approach is, however,
quite different since, instead of considering renormalized solutions, we prove
a dense range condition implying uniqueness. In addition, compared to known
results by Ambrosio-Figalli and Fang-Luo, we use a different approximation
procedure, based on a more regularizing Ornstein-Uhlenbeck semigroup and
consider Sobolev spaces of vector fields taking values in $H$ rather than the
Cameron-Martin space of the Gaussian measure. This leads to different
conditions on the derivative of $F$, which are incompatible with previous work
on the subject. Furthermore, we can drop the usual exponential integrability
conditions on the Gaussian divergence of $F$, thus improving known uniqueness
results in this respect.
