Entropy on non-commutative dynamical systems and the structure of operator algebras

1)定义了可数离散可服从群的自同构的熵不变量，并给出了C^*交叉积代数上的自同构的两个熵之间的关系及其对原代数的限制。作为自由积^*的应用，自同构θ的拓扑熵ht（θ）满足ht(θ^*θ^*......^*θ) = ht（θ^*1）2)我们证明了对于所有m= 2,3，…，自由群因子L（Fm）具有GL（n， Z）的连续非共轭外作用族。，并给出了动作中出现的每个自同构的cones - stormer熵的估计。通过将它们限制为具有Kazhdan性质T的子群（例如SL(n， Z)），我们在L(Fm), m>1.3上得到了连续的非环共轭外作用族。设G是算子代数a的自同构群的离散子群。那么，代数A的自同构的熵不会因A与g的叉积的正则扩展而改变。这表明了精确C^*代数A的拓扑熵和AFD冯·诺伊曼代数A的动态熵。

1)An entropical invariant is defined for automorphisms of countable discrete amenable groups, and it is shown relations between two entropies for an automorphism on the C^*-crossed product algebra and for its restriction to the original algebra.As an application to the free product^*, the topological entropy ht(θ) of an automorphism θ satishies that ht(θ^*θ^*......^*θ) = ht(θ^*1).2)We showed that the free group factor L(Fm) has a continuous family of non conjugate outer actions of GL(n, Z) for all m=2, 3,..., and give an estimation of the Connes-Stormer entropy for each automorphism appearing in the actions.By restrictng them to subgroups with Kazhdan's property T (for an example SL(n, Z), we have a continuous family of non cocycle conjugate outer actions on L(Fm), m>1.3)Let G be a discrete subgroup of the automorphism group of an operator algebra A. Assume that G is exact, that is, it admits an amenable action on some compact space.Then the entropy of an automorphism of the algebra A does not change by the canonical extension to the crossed product of A by G.This is shown for the topological entropy of an exact C^* algebra A and for the dynamical entropy of an AFD von Neumann algebra A.