We give a practical criterion of invertibility of endomorphisms of \(O_n\) corresponding to unitaries in the normalizer of the diagonal inside the uniformly hyperfinite subalgebra. We also analyze the action of such localized automorphisms on the spectrum of the diagonal thus obtaining criteria of outerness. Unital endomorphisms of the Cuntz algebra \(O_n\) which preserve the canonical uniformly hyperfinite-subalgebra \(F_n ⊆ O_n\) are investigated. We give examples of such endomorphisms \(λ =λ_u\) for which the associated unitary element \(u\) in \(O_n\).
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Abdelgabar Adam Hassan (2024) studied this question.
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