Endomorphisms of O_n which preserve the canonical UHF-subalgebra

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.