  We generalize the well known $C^*$-algebraic result of Kawamura-Tomiyama and
Archbold-Spielberg for crossed products of discrete transformation groups to
the realm of Banach algebras and twisted actions. Namely we show that
topological freeneess is equivalent to the intersection property for all
reduced twisted Banach algebra crossed products coming from subgroups, and in
the untwisted case to a generalized intersection property for a full
$L^p$-operator algebra crossed product for some (and hence any) $p\in
[1,\infty]$. This gives efficient simplicity criteria for various Banach
algebra crossed products. We also use it to identify the prime ideal space of
some crossed products as the quasi-orbit space of the action. For amenable
actions we prove that the full and reduced twisted $L^p$-operator algebras
coincide.
