Ergodic theory, non-commutative

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A branch of the theory of operator algebras in which one studies automorphisms of $C^*$-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 $W^*$-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 $W^*$-algebra with respect to a state $\phi$ has been investigated [7].


[1] A. Connes, "Outer conjugacy classes of automorphisms of factors" Ann. Sci. Ecole. Norm. Sup. , 8 (1975) pp. 383–419
[2] V.Ya. Golodets, "Modular operators and asymptotic commutativity in Von Neumann algebras" Russian Math. Surveys , 33 : 1 (1978) pp. 47–106 Uspekhi Mat. Nauk , 33 : 1 (1978) pp. 43–94
[3] H. Araki, "$C^*$-algebras and applications to physics" , Lect. notes in math. , 650 , Springer (1978) pp. 66–84
[4] Ya.G. Sinai, V.V. Anshelevich, "Some problems of non-commutative ergodic theory" Russian Math. Surveys , 31 : 4 (1976) pp. 157–174 Uspekhi Mat. Nauk , 31 : 4 (1976) pp. 151–167
[5] E.C. Lance, "Ergodic theorems for convex sets and operator algebras" Invent. Math. , 37 (1976) pp. 201–214
[6] A. Connes, E. Størmer, "Entropy for automorphisms of $\mathrm{II}_1$ Von Neumann algebras" Acta Math. , 134 (1975) pp. 289–306
[7] A.M. Stepin, A.G. Shukhov, "The centralizer of diagonable states and entropies of automorphisms of $W^*$-algebras" Soviet Math. (Vuz) , 26 : 8 (1982) pp. 61–71 Izv. Vuzov. Mat. , 8 (1982) pp. 52–60
How to Cite This Entry:
Ergodic theory, non-commutative. Encyclopedia of Mathematics. URL:,_non-commutative&oldid=35673
This article was adapted from an original article by A.M. Stepin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article