  Generalizing results of our earlier paper, we investigate the following
question. Let $\pi(\lambda) : A \to B$ be an analytic family of surjective
homomorphisms between two Banach algebras, and $q(\lambda)$ an analytic family
of idempotents in $B$. We want to find an analytic family $p(\lambda)$ of
idempotents in $A$, lifting $q(\lambda)$, i.e., such that
$\pi(\lambda)p(\lambda) = q(\lambda)$, under hypotheses of the type that the
elements of $\text{Ker}\, \pi(\lambda)$ have small spectra. For spectra which
do not disconnect $\Bbb C$ we obtain a local lifting theorem. For real analytic
families of surjective $^*$-homomorphisms (for continuous involutions) and
self-adjoint idempotents we obtain a local lifting theorem, for totally
disconnected spectra. We obtain a global lifting theorem if the spectra of the
elements in $\text{Ker}\, \pi(\lambda)$ are $\{0\}$, both in the analytic case,
and, for $^*$-algebras (with continuous involutions) and self-adjoint
idempotents, in the real analytic case. Here even an at most countably infinite
set of mutually orthogonal analytic families of idempotents can be lifted to
mutually orthogonal analytic families of idempotents. In the proofs, spectral
theory is combined with complex analysis and general topology, and even a
connection with potential theory is mentioned.
