On Endomorphisms of the Cuntz Algebra which Preserve the Canonical UHF-Subalgebra, II

Abstract

It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms ${\lambda }_u$ of the Cuntz algebra ${\mathcal{O}}_n$ such that ${\lambda }_u({\mathcal{F}}_n)\subseteq {\mathcal{F}}_n$ but $u\notin {\mathcal{F}}_n$, and a question was raised if for such a $u$ there must always exist a unitary $v\in {\mathcal{F}}_n$ with ${\lambda }_u{|}_{{\mathcal{F}}_n}={\lambda }_v{|}_{{\mathcal{F}}_n}$. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms ${\lambda }_u$ for which the relative commutant ${\lambda }_u({\mathcal{F}}_n{)}^{\prime }\cap {\mathcal{F}}_n$ is finite dimensional.

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