论文标题
自由概率和模型理论$ \ mathrm {w}^*$ - 代数
Free probability and model theory of tracial $\mathrm{W}^*$-algebras
论文作者
论文摘要
$*$ - 法律或$*$ - 免费概率分布的概念也被称为Farah，Hart和Sherman的模型理论框架的无量词类型。但是，整个类型也可以被认为是经典概率分布的类似物（实际上，本·亚科夫（Ben Yaacov）表明，在经典的环境中，原子概率空间可以消除量词，因此全类型和无量词类型之间没有差异）。因此，我们为完整类型制定了Voiculescu的免费微晶格熵的概念，并且我们表明，如果$ \ Mathbf {X} $是$ \ Mathcal {m Mathcal {M} $的$ d $ -tuple，则具有$χ^{\ Mathcal {\ Mathcal {u}}}（U}}（\ mathbf）（\ Mathbf） Ultrafilter $ \ MATHCAL {U} $，然后存在$ \ Mathcal {m} $的嵌入$ \ Mathcal {m} $中的$ \ Mathcal {q} = \ prod_ {n \ to \ to \ nathcal {u}}}} m_n（\ mathbb {c}） \ Mathcal {Q}）=χ（\ MathBf {X}：\ Mathcal {M}）$;特别是，通过VoiculeScu的结果，这种嵌入将满足$ b（\ Mathbf {x}）'\ cap \ mathcal {q} = \ mathbb {c} $。此外，我们草拟了一些开发自由独立性和自由吉布斯法律的模型理论版本的开放问题和挑战。
The notion of a $*$-law or $*$-distribution in free probability is also known as the quantifier-free type in Farah, Hart, and Sherman's model theoretic framework for tracial von Neumann algebras. However, the full type can also be considered an analog of a classical probability distribution (indeed, Ben Yaacov showed that in the classical setting, atomless probability spaces admit quantifier elimination and hence there is no difference between the full type and the quantifier-free type). We therefore develop a notion of Voiculescu's free microstates entropy for a full type, and we show that if $\mathbf{X}$ is a $d$-tuple in $\mathcal{M}$ with $χ^{\mathcal{U}}(\mathbf{X}:\mathcal{M}) > -\infty$ for a given ultrafilter $\mathcal{U}$, then there exists an embedding $ι$ of $\mathcal{M}$ into $\mathcal{Q} = \prod_{n \to \mathcal{U}} M_n(\mathbb{C})$ with $χ(ι(\mathbf{X}): \mathcal{Q}) = χ(\mathbf{X}:\mathcal{M})$; in particular, such an embedding will satisfy $ι(\mathbf{X})' \cap \mathcal{Q} = \mathbb{C}$ by the results of Voiculescu. Furthermore, we sketch some open problems and challenges for developing model-theoretic versions of free independence and free Gibbs laws.
