The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism sigma: U -> X where U subset of X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators L-p which may occur for alpha : C (X) -> C (U) given by alpha(f) = f circle sigma. We obtain examples of partial dynamical systems (X-A, sigma(A)) such that the construction of the covariance algebra C*(X-A, U-A), proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism O(X-A, alpha, L), recently introduced by the author and R. Exel, associated to this system are not equivalent, in the sense that there does not exist an invertible function p epsilon C(U) such that O(X-A, alpha, L-p) congruent to C* (X-A, sigma(A)). (c) 2005 Elsevier Inc. All rights reserved.