We give an algebraic characterization for the conjugate endomorphism ̄ ρ of an en-domorphism ρ of infinite index of a properly infinite von Neumann algebra M such that the set of normal faithful conditional expectations E(M, ρ(M)) is not empty. In the particular case of irreducible endomorphisms we obtain the same result holding in finite index case and in the representation theory of compact groups, that is if ρ is an irreducible endomorphism of an infinite factor, with E(M, ρ(M)) = ∅, then an irreducible endomorphism σ is conjugate to ρ iff σρ id; moreover the identity is contained only once in σρ. Some applications of the above results are also given.