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 m≥1$m\ge 1$ and aab" title="Click to view the MathML source">M₁,…,M_m$M_1,\dots ,M_m$ are nonamenable factors in a large class of von Neumann algebras that we call C_(AO)${\mathcal{C}}_{(\text{AO})}$ and which contains all free Araki–Woods factors, the tensor product factor $M_1\bar{\otimes }\cdots \bar{\otimes }{M}_m$ retains the integer m   and each factor M_i$M_i$ 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 aab" title="Click to view the MathML source">M₁,…,M_m$M_1,\dots ,M_m$ 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 III₁$\mathrm{I}\mathrm{I}{\mathrm{I}}_{\mathrm{1}}$ factors in the class C_(AO)${\mathcal{C}}_{(\text{AO})}$.