We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected
CW
{\text {CW}}
-complexes, some of which extend
P
P
-localization of nilpotent spaces, at a set of primes
P
P
. We focus our attention on one such functor, whose local objects are
CW
{\text {CW}}
-complexes
X
X
for which the
p
p
th power map on the loop space
Ω
X
\Omega X
is a self-homotopy equivalence if
p
∉
P
p \notin P
. We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield’s homology localization, Bousfield-Kan completion, and Quillen’s plus-construction.