Ergodic theory, non-commutative

A branch of the theory of operator algebras in which one studies automorphisms of $C^{\ast }$-algebras from the point of view of ergodic theory.

The range of questions considered in non-commutative ergodic theory and the results obtained so far (1984) can be basically divided into three groups. To the first group belong results connected with the construction of a complete system of invariants for outer conjugacy. (Two automorphisms ${\theta }_1$ and ${\theta }_2$ are called outer conjugate if there exists an automorphism $\sigma$ such that ${\theta }_1\sigma {\theta }_2^{-1}{\sigma }^{-1}$ is an inner automorphism.) The corresponding classification problem has been solved (see [1]) for approximately-finite factors (cf. Factor) of type $\mathrm{II}$ and type ${\mathrm{III}}_{\lambda }$, $0<\lambda <1$ (see [2]).

To the second group belong articles devoted to the study of properties of equilibrium states (by a state in an algebra one means a positive linear normalized functional on the algebra) which are invariant under a one-parameter group of automorphisms. In particular, one considers questions of existence and uniqueness of Gibbs states (see [3]). Closely related to this group of problems are investigations on non-commutative generalizations of ergodic theorems (see, for example, [4], [5]).

The third group consists of results concerning the entropy theory of automorphisms. For automorphisms of finite $W^{\ast }$-algebras (see von Neumann algebra) an invariant has been constructed [6] that generalizes the entropy of a metric dynamical system. The entropy of automorphisms of an arbitrary $W^{\ast }$-algebra with respect to a state $\varphi$ has been investigated [7].

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