The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism σ :U →X where U ⊆ X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators Lρ which may occur for α :C(X)→C(U) given by α(f ) = f ◦ σ. We obtain examples of partial dynamical systems (XA,σA) such that the construction of the covariance algebra C∗(XA,σA), proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism O(XA,α,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 ρ ∈ C(U) such that O(XA,α,Lρ)∼= C∗(XA,σA). © 2005 Elsevier Inc. All rights reserved.