  Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed)
in I, a *-homomorphism \al:A --> M(I) and a map L:J --> A with some properties,
based on [3] and [9] we define a C*-algebra O(A,\al,L) which we call the
"Crossed Product by a Partial Endomorphism." In the second section we introduce
the Crossed Product by a Partial Endomorphism O(X,\al,L) induced by a local
homeomorphism \sigma:U --> X where X is a compact Hausdorff space and U is an
open subset of X.The main result of this section is that every nonzero gauge
invariant ideal of O(X,\al,L) has nonzero intersection with C(X). We present
the example which motivated this work, the Cuntz-Krieger algebra for infinite
matrices. We show in the third section a bijection between the gauge invariant
ideals of O(X,\al,L) and the \sigma,\sigma^{-1} -invariant open subsets of X.
The last section is dedicated to the study of O(X,\al,L) in the case where the
pair (X,\sigma) has an extra property, wich we call topological freeness. We
prove that in this case every nonzero ideal of O(X,\al,L) has nonzero
intersection with C(X). If moreover (X,\sigma) has the property that
(X',\sigma_X') is topologically free for each closed
sigma,\sigma^{-1}-invariant subset X' of X then we obtain a bijection between
the ideals of O(X,\al,L) and the open \sigma,\sigma^{-1}-invariant subsets of
X. We conclude this section by showing a simplicity criteria for the
Cuntz-Krieger algebras for infinite matrices.
