标题： 熵和单位表示的行列式 显示英文标题

标题： Entropy and determinants for unitary representations

摘要： 在可数群的概率保持作用的遍历理论中，已经广泛研究了几种熵的概念。 这些包括基于F{\o }lner序列对可换群的Kolmogorov--Sinai熵的推广，针对一般可数群的渗透熵，以及针对素数群的Bowen熵。 在这项工作中，我们沿着遍历理论和表示论之间的类比来探讨这些概念。 我们得到了与群的酉表示和其他C*-代数的表示相关的新量。 我们的主要结果表明，这些新量通常可以作为Fuglede--Kadison行列式来计算。 由此得到的行列式公式提供了Toeplitz行列式的Szegő极限定理的各种非交换推广。 它们与Arveson的子对角子代数理论有关，也与紧阿贝尔群自同构作用的遍历理论中的一些精确熵计算有关。 显示英文摘要

摘要： Several notions of entropy have been studied widely in the ergodic theory of probability-preserving actions of countable groups. These include the generalization of Kolmogorov--Sinai entropy to amenable groups based on F{\o}lner sequences, percolative entropy for general countable groups, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across an analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can often be evaluated as Fuglede--Kadison determinants. The resulting determinantal formulas offer various non-commutative generalizations of Szeg\H{o}'s limit theorem for Toeplitz determinants. They make contact with Arveson's theory of subdiagonal subalgebras, and also with some exact entropy calculations in the ergodic theory of actions by automorphisms of compact Abelian groups.