  Assume sigma is a face of a Gorenstein* simplicial complex D. We investigate
the question of whether the Weak Lefschetz Property of the Stanley-Reisner ring
k[D] (over an infinite field k) is equivalent to the same property of the
Stanley-Reisner ring k[D_sigma] of the stellar subdivision D_sigma. We prove
that this is the case if the dimension of sigma is big compared to the
codimension.
