View source for Ergodic theory, non-commutative

{{TEX|done}}

A branch of the theory of operator algebras in which one studies automorphisms of [[C*-algebra|$C^*$-algebra]]s 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 [[#References|[1]]]) for approximately-finite factors (cf. [[Factor]]) of type $\mathrm{II}$ and type $\mathrm{III}_\lambda$, $0 < \lambda < 1$ (see [[#References|[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 [[#References|[3]]]). Closely related to this group of problems are investigations on non-commutative generalizations of ergodic theorems (see, for example, [[#References|[4]]], [[#References|[5]]]).

The third group consists of results concerning the entropy theory of automorphisms. For automorphisms of finite $W^*$-algebras (see [[Von Neumann algebra|von Neumann algebra]]) an invariant has been constructed [[#References|[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 [[#References|[7]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Connes,   "Outer conjugacy classes of automorphisms of factors"  ''Ann. Sci. Ecole. Norm. Sup.'' , '''8'''  (1975)  pp. 383–419</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Araki,   "$C^*$-algebras and applications to physics" , ''Lect. notes in math.'' , '''650''' , Springer  (1978)  pp. 66–84</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  E.C. Lance,   "Ergodic theorems for convex sets and operator algebras"  ''Invent. Math.'' , '''37'''  (1976)  pp. 201–214</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  A. Connes,   E. Størmer,   "Entropy for automorphisms of $\mathrm{II}_1$ Von Neumann algebras"  ''Acta Math.'' , '''134'''  (1975)  pp. 289–306</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR>
</table>