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arXiv math/0402109math.OA
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自由熵的一个估计及其应用

An estimate of free entropy and applications

Marius Stefan

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摘要：

我们得到一个估计的自由熵的生成器在一个类型${II}_1$ -因子$\mc{M}$它有一个子因子$\mc{N}$的有限指数与一个子代数$\mc{P}=\mc{P}_1\vee\mc{P}_2\subset\mc{N}$其中$\mc{P}_1=\mc{R}_1'\cap\mc{P}$，$\mc{P}_2=\mc{R}_2'\cap\mc{P}$是扩散的，$\mc{R}_1,\mc{R}_2\subset\mc{P}$是相互交换的超有限子因子，和一个阿贝尔子代数$\mc{A}\subset\mc{N}$，使对应$_\mc{P}L^2(\mc{N},τ)_\mc{A}$是$\mc{M}$ -弱包含在一个子对应$_\mc{P}H_\mc{A}$的$_\mc{P}L^2(\mc{M},τ)_\mc{A}$，由$v$向量生成。$\mc{M}$的任何生成集的（修改的）自由熵维数是$\leq 2r+2v+4$，其中$r$是索引的整数部分。因此，有限指标的内插自由群子因子不具有正则非素子因子或正则扩散超有限子代数。

英文摘要：

We obtain an estimate of free entropy of generators in a type ${II}_1$-factor $\mc{M}$ which has a subfactor $\mc{N}$ of finite index with a subalgebra $\mc{P}=\mc{P}_1\vee\mc{P}_2\subset\mc{N}$ where $\mc{P}_1=\mc{R}_1'\cap\mc{P}$, $\mc{P}_2=\mc{R}_2'\cap\mc{P}$ are diffuse, $\mc{R}_1,\mc{R}_2\subset\mc{P}$ are mutually commuting hyperfinite subfactors, and an abelian subalgebra $\mc{A}\subset\mc{N}$ such that the correspondence $_\mc{P}L^2(\mc{N},τ)_\mc{A}$ is $\mc{M}$-weakly contained in a subcorrespondence $_\mc{P}H_\mc{A}$ of $_\mc{P}L^2(\mc{M},τ)_\mc{A}$, generated by $v$ vectors. The (modified) free entropy dimension of any generating set of $\mc{M}$ is $\leq 2r+2v+4$, where $r$ is the integer part of the index. As a consequence, the interpolated free group subfactors of finite index do not have regular non-prime subfactors or regular diffuse hyperfinite subalgebras.

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