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

1)对可列离散自同构定义了一个熵不变量，并证明了C^*-交叉积代数上自同构的两个熵及其对原代数的限制之间的关系。作为对自由积^*的应用，自同构θ的拓扑熵ht(θ)满足ht(θ^*θ^*......^*θ)=ht(θ^*1)。2)我们证明了当m=2，3，……时，自由群L(Fm)有一个连续的非共轭外作用族GL(n，Z)。通过把它们限制在具有Kazhdan性质T的子群上(例如SL(n，Z)，我们得到了L(Fm)，m&GT上的一个连续的非上循环共轭外作用族；1.3)设G是算子代数A的自同构群的离散子群，假设G是精确的，即它允许在某个紧空间上有一个从属作用，则代数A的自同构的熵不随G对A的交叉积的正则扩张而改变。本文证明了正则C^*代数A的拓扑熵和AFD von Neumann代数A的动力熵。

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.