  Unital endomorphisms of the Cuntz algebra O_n which preserve the canonical
UHF-subalgebra F_n of O_n are investigated. We give examples of such
endomorphisms for which the associated unitary element in O_n does not belong
to F_n. One such example, in the case where n=2, arises from a construction of
a unital endomorphism on O_2 which preserves the canonical UHF-subalgebra and
where the relative commutant of the image in O_2 contains a copy of O_2.
