Unique prime factorization and bicentralizer problem for a class of type III factors

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• 作者：Cyril Houdayer^a ; ¹ ; ^{cyril.houdayer@math.u-psud.fr" class="auth_mail" title="E-mail the corresponding author} ; Yusuke Isono^b ; ² ; ^{isono@kurims.kyoto-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：46L10 ; 46L36
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：402-455
• 全文大小：921 K

文摘

We show that whenever class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si1.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=927f7b1b53e088683d8b955858b3a6c3" title="Click to view the MathML source">m≥1class="mathContainer hidden">class="mathCode">$m\ge 1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si2.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=093d783ddf86d6c398e28f8d61c6daab" title="Click to view the MathML source">M₁,…,M_mclass="mathContainer hidden">class="mathCode">$M_1,\dots ,M_m$ are nonamenable factors in a large class of von Neumann algebras that we call class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si22.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=e5c944fc4fdbe12ee1bebdb6ffe5395b" title="Click to view the MathML source">C_(AO)class="mathContainer hidden">class="mathCode">${\mathcal{C}}_{(\text{AO})}$ and which contains all free Araki–Woods factors, the tensor product factor class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si12.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=08c7cf81f40bcb2e18a6b51d53fbda49">class="imgLazyJSB inlineImage" height="15" width="110" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816303383-si12.gif">class="mathContainer hidden">class="mathCode">$M_1\bar{\otimes }\cdots \bar{\otimes }{M}_m$ retains the integer m   and each factor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si5.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=8517fd7d5cd04af2cf7eb29ea02b07d0" title="Click to view the MathML source">M_iclass="mathContainer hidden">class="mathCode">$M_i$ up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from  and  and moreover provides new UPF results in the case when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si2.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=093d783ddf86d6c398e28f8d61c6daab" title="Click to view the MathML source">M₁,…,M_mclass="mathContainer hidden">class="mathCode">$M_1,\dots ,M_m$ are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si33.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=5afb47bb2889f644d6bfcbdd64dd20ea" title="Click to view the MathML source">III₁class="mathContainer hidden">class="mathCode">$\mathrm{I}\mathrm{I}{\mathrm{I}}_{\mathrm{1}}$ factors in the class class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816303383&_mathId=si22.gif&_user=111111111&_pii=S0001870816303383&_rdoc=1&_issn=00018708&md5=e5c944fc4fdbe12ee1bebdb6ffe5395b" title="Click to view the MathML source">C_(AO)class="mathContainer hidden">class="mathCode">${\mathcal{C}}_{(\text{AO})}$.