View source for Ergodic theory, non-commutative

A branch of the theory of operator algebras in which one studies automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361601.png" />-algebras (cf. [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361602.png" />-algebra]]) from the point of view of [[Ergodic theory|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361604.png" /> are called outer conjugate if there exists an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361605.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361606.png" /> is an inner automorphism.) The corresponding classification problem has been solved (see [[#References|[1]]]) for approximately-finite factors (cf. [[Factor|Factor]]) of type II and type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e0361608.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e03616010.png" />-algebras (see [[Von Neumann algebra|von Neumann algebra]]) an invariant has been constructed [[#References|[6]]] that generalizes the [[Entropy|entropy]] of a metric dynamical system. The entropy of automorphisms of an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e03616011.png" />-algebra with respect to a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e03616012.png" /> 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,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e03616013.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e03616014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036160/e03616015.png" />-algebras"  ''Soviet Math. (Vuz)'' , '''26''' :  8  (1982)  pp. 61–71  ''Izv. Vuzov. Mat.'' , '''8'''  (1982)  pp. 52–60</TD></TR></table>