  We give a local characterization of the existence of Kazhdan projections for
arbitary families of Banach space representations of a compactly generated
locally compact group $G$. We also define and study a natural generalization of
the Fell topology to arbitrary Banach space representations of a locally
compact group. We give several applications in terms of stability of rigidity
under perturbations. Among them, we show a Banach-space version of the
Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent
for $\sigma$-compact locally compact groups. Another kind of applications is
that many forms of Banach strong property (T) are open in the space of marked
groups, and more generally every group with such a property is a quotient of a
compactly presented group with the same property. We also investigate the
notions of central and non central Kazhdan projections, and present examples of
non central Kazhdan projections coming from hyperbolic groups.
