We present some exact equiconsistency results on the preservation
of the property of L(R) being a Solovay model under various classes of
projective forcing extensions. As an application we build models in which MA
holds for $\Sigma^1_n$ partial orderings, but it fails for the
$\Sigma^1_{n+1}$.

It is proved that the relation of isomorphism between separable Banach
spaces is a complete analytic equivalence relation, i.e., that any
analytic equivalence relation Borel reduces to it. Thus, separable Banach
spaces up to isomorphism provide complete invariants for a great number of
mathematical structures up to their corresponding notion of isomorphism.