C*-代数的1-有界熵
1-Bounded Entropy for $C^*$-Algebras
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中文总结 AI 辅助
本文针对Voiculescu的问题，为C*-代数定义两种1-有界熵h与h^top，证明h^top(𝒜)≤h(𝒜)，引入计算h的变分原理，通过实例计算h验证相关结构结论，并将熵概念推广至算子系统。
中文摘要 AI 辅助
针对Voiculescu提出的一个问题，本文为C*-代数定义了两种1-有界熵概念：h与h^top。其中h涉及所有迹相对于迹的完备化，而h^top则依赖于算子范数微状态。研究证明，对所有C*-代数𝒜，均有h^top(𝒜)不超过h(𝒜)；此外，本文引入了一种可用于计算h的变分原理。本文在多个实例中计算了h，并利用该计算结果验证了结构结论，如C*-素性、交叉积不可分解性及自由不可分解性。这些熵概念还被推广到算子系统，研究证明二者均可在基上计算。
英文摘要
Towards a question of Voiculescu, two notions of $1$-bounded entropy, $h$ and $h^\text{top}$, are defined for $C^*$-algebras. The quantity $h$ involves the tracial completion with respect to all traces while the quantity $h^\text{top}$ depends on operator norm microstates. It is demonstrated that $h^\text{top}(\mathscr{A})$ does not exceed $h(\mathscr{A})$ for all $C^*$-algebras $\mathscr{A}$. Moreover, a variational principle enabling the computation of $h$ is introduced. The quantity $h$ is computed in various examples, and the computation is used to justify structural conclusions such as $C^*$-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.
