On endomorphisms of the Cuntz algebra which preserve the canonical UHF-subalgebra, II

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• 作者：Tomohiro Hayashi^a ; ^{hayashi.tomohiro@nitech.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Jeong Hee Hong^b ; ^{hongjh@kmou.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Wojciech Szymański^c ; ^{szymanski@imada.sdu.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cuntz algebra ; Endomorphism ; UHF-subalgebra
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：759-775
• 全文大小：388 K

文摘

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