Normal Generation of Unitary Groups of Cuntz Algebras by Involutions

In purely infinite factors, P. de la Harpe proved that a normal subgroup of the unitary group which contains a non-trivial self-adjoint unitary contains all self-adjoint unitaries of the factor. Also he proved the same result in finite continuous factors. In a previous work the author proved a similar result in some types of unital AF-algebras. In this paper we extend the result of de la Harpe, concerning the purely infinite factors to a main example of purely infinite C∗-algebras called the Cuntz algebras On(2 ≤ n ≤ ∞) and we prove that U(On) is normally generated by some non-trivial involution. In particular, in the Cuntz algebra O∞ we prove that U(O∞) is normally generated by self-adjoint unitary of odd type.