论文标题
覆盖tracial $ \ mathrm {w}^*$ - 代数的熵
Covering entropy for types in tracial $\mathrm{W}^*$-algebras
论文作者
论文摘要
我们通过自由概率和模型理论技术的合并来研究奇特$ \ mathrm {w}^*$ - 代数的嵌入到基质代数的超副作用中。 Jung隐含地，海斯通过渐近覆盖的voiculescu的微层面空间明确定义了$ 1 $结合的熵，也就是说，矩阵的空间$（x_1^{（n）}，x_2^{（n）}，x_2^{（n）}，\ dots），与大约相同的$*$*$*$*$*$*$*的生成$（x_1，x_2，\ dots）$ $ \ mathrm {w}^*$ - 代数。我们研究了通过公式定义的微晶格空间的类似熵，这些公式不仅使用$*$ - 代数操作和痕迹，还使用Suprema和Infima，例如在Tracial $ \ Mathrm {W}^*$ - 代数的模型理论中出现，由Farah，Hart，Hart，Hart和Sherman和Sherman和Sherman发起。通过将新理论与原始$ 1 $结合的熵联系起来，我们表明，如果$ h（\ nathcal {n}：\ Mathcal {m}）\ geq 0 $，则存在$ \ nathcal {m Mathcal {m} $的嵌入到Matrix ultrix ultrix ultrix ultraproduct $ \ mathcal中\ Mathcal {u}} m_n（\ Mathbb {c}）$，以至于$ h（\ Mathcal {n}：\ Mathcal {q}）$任意接近$ h（\ Mathcal {n}：\ Mathcal {m}）$。我们推断出$ \ Mathcal {M} $的所有嵌入到$ \ Mathcal {Q} $中都是自动等价的，则$ \ Mathcal {M} $强烈$ 1 $ bunded，实际上具有$ h（\ Mathcal {M}）\ leq 0 $。
We study embeddings of tracial $\mathrm{W}^*$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined $1$-bounded entropy through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples $(X_1^{(N)},X_2^{(N)},\dots)$ having approximately the same $*$-moments as the generators $(X_1,X_2,\dots)$ of a given tracial $\mathrm{W}^*$-algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use not only $*$-algebra operations and the trace, but also suprema and infima, such as arise in the model theory of tracial $\mathrm{W}^*$-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original $1$-bounded entropy, we show that if $h(\mathcal{N}:\mathcal{M}) \geq 0$, then there exists an embedding of $\mathcal{M}$ into a matrix ultraproduct $\mathcal{Q} = \prod_{n \to \mathcal{U}} M_n(\mathbb{C})$ such that $h(\mathcal{N}:\mathcal{Q})$ is arbitrarily close to $h(\mathcal{N}:\mathcal{M})$. We deduce if all embeddings of $\mathcal{M}$ into $\mathcal{Q}$ are automorphically equivalent, then $\mathcal{M}$ is strongly $1$-bounded and in fact has $h(\mathcal{M}) \leq 0$.
