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

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• 作者：Tomohiro Hayashiup>aup>up> ; up>up>hayashi.tomohiro@nitech.ac.jp" class="auth_mail" title="E-mail the corresponding authorup> ; Jeong Hee Hongup>bup>up> ; up>up>hongjh@kmou.ac.kr" class="auth_mail" title="E-mail the corresponding authorup> ; Wojciech Szymańskiup>cup>up> ; up>up>szymanski@imada.sdu.dk" class="auth_mail" title="E-mail the corresponding authorup>
• 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 ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si1.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=5a9d11d6d965b406a5956f4ad053d26f" title="Click to view the MathML source">λub>uub> of the Cuntz algebra ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si2.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=aca49272aa3088c7f945ffcb120c431f" title="Click to view the MathML source">Oub>nub> such that ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si3.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=e9354c2a92d55bce9025ad0f7e3258be" title="Click to view the MathML source">λub>uub>(Fub>nub>)⊆Fub>nub> but ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si4.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=7a2e9c6326526cd2d4c8e81e39d07958" title="Click to view the MathML source">u∉Fub>nub>, and a question was raised if for such a u   there must always exist a unitary ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si202.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=adeb6c6193360672f98ec12c80b8ec0f" title="Click to view the MathML source">v∈Fub>nub> with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si203.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=4c7bfa33c4962aba98ffbadf6381fa6d" title="Click to view the MathML source">λub>uub>|ub>Fub>nub>ub>=λub>vub>|ub>Fub>nub>ub>. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si1.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=5a9d11d6d965b406a5956f4ad053d26f" title="Click to view the MathML source">λub>uub> for which the relative commutant ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301926&_mathId=si108.gif&_user=111111111&_pii=S0022123616301926&_rdoc=1&_issn=00221236&md5=39f15958d000c9489b14fb1d4b744155" title="Click to view the MathML source">λub>uub>(Fub>nub>)up>′up>∩Fub>nub> is finite dimensional.