Outer Automorphisms on Cuntz Algebras

Abstract

We prove that if an automorphism α on the Cuntz algebra On satisfies α(S1) = ιS1 for one isometry S1 of generators S1, …, Sn such that ∑ i = 1 n S i S i * = 1 , S i * S i = 1 , l ⩽ i ⩽ n and a complex number λ with modulus one, then it is outer. In particular, any non trivial automorphism on On leaving one generator fixed is outer.