The Crossed Product by a Partial Endomorphism
  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. 
R. Exel, D. Royer