The Connes embedding property for quantum group von Neumann algebras
The Connes embedding property for quantum group von Neumann algebras
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量子群冯诺依曼代数的 Connes 嵌入性质
DOI：
10.1090/tran/6752
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Roland Vergnioux
中科院分区：
文献类型：
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作者：
Michael Brannan;B. Collins;Roland Vergnioux
对于Kac型紧量子群$\mathbb G$，研究了von Neumann代数$L^\infty（\mathbb G）$到超有限II$_1$-因子的超幂的Haar保迹嵌入的存在性（L^\infty（\mathbb G）$的Connes嵌入性质）.建立了L^\infty（\mathbb G）$的Connes嵌入性质与G的某些量子子群结构之间的联系，并以此证明了与自由正交和自由幺正量子群相关联的II $1 $-因子L^\infty（O_N^+）$和L^\infty（U_N^+）$对所有N \ge 4$都具有Connes嵌入性质.作为应用，我们推导出$L^\infty（O_N^+）$的标准生成元的自由熵维数对所有$N \ge 4$都等于$1$。我们还提到了我们的工作的一个应用，$O_N^+$的量子子群的分类问题。
For a compact quantum group $\mathbb G$ of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra $L^\infty(\mathbb G)$ into an ultrapower of the hyperfinite II$_1$-factor (the Connes embedding property for $L^\infty(\mathbb G)$). We establish a connection between the Connes embedding property for $L^\infty(\mathbb G)$ and the structure of certain quantum subgroups of $\mathbb G$, and use this to prove that the II$_1$-factors $L^\infty(O_N^+)$ and $L^\infty(U_N^+)$ associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all $N \ge 4$. As an application, we deduce that the free entropy dimension of the standard generators of $L^\infty(O_N^+)$ equals $1$ for all $N \ge 4$. We also mention an application of our work to the problem of classifying the quantum subgroups of $O_N^+$.