The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism where U[subset of or equal to]X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators L[rho] which may occur for given by [alpha](f)=f[circle, open][sigma]. We obtain examples of partial dynamical systems (XA,[sigma]A) such that the construction of the covariance algebra C*(XA,[sigma]A), proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism , 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 [rho][set membership, variant]C(U) such that .