Action of discrete amenable groups on von Neumann algebras.

*(English)*Zbl 0608.46035
Lecture Notes in Mathematics, 1138. Berlin etc.: Springer-Verlag. V, 115 p. DM 21.50 (1985).

In the new stage of the theory of operator algebras established by the work of Alain Connes an objective of great interest was the study of automorphic actions of groups on von Neumann algebras and their classification up to outer conjugacy. In his pioneering works [see Acta Sci. Math. 39, 39-66 (1977; Zbl 0382.46027); Ann. Sci. Ec. Norm. Sup., IV. Ser. 8, 383-420 (1975; Zbl 0342.46052)] A. Connes classifies the actions on the hyperfinite type \(II_ 1\) factor R of the groups \({\mathbb{Z}}_ n\) (up to conjugacy) and \({\mathbb{Z}}\) (up to outer conjugacy). Subsequently, V. F. R.Jones [see Lecture Notes Math. 725, 237-253 (1979; Zbl 0497.46044)] studied the cohomological invariants and extended the characteristic invariant of A. Connes for arbitrary discrete group actions, and [see Mem. Am. Math. Soc. 237, 70 p. (1980; Zbl 0454.46045)] classified (up to conjugacy) the actions of finite groups on R.

In [C. R. Acad. Sci., Paris, Ser. A 291, 399-401 (1980; Zbl 0471.46045)] the author announced the classification up to outer conjugacy of the actions of amenable discrete groups on R. The book under review contains the complete proof of this classification whose main result is the uniqueness up to outer conjugacy of the free action of an amenable discrete group on R (actually, the outer conjugacy is proved for any centrally trivial and approximately inner actions of amenable discrete groups on McDuff factors M, i.e. \(M\approx M\times R)\). For not ncessarily free actions of a discrete amenable group on R, the characteristic invariant is proved to be complete for their outer conjugacy. A parallel study is done for G-kernels (homomorphisms \(G\to Aut R=Aut R/Int R)\) where the Eilenberg-MacLane \(H^ 3\) obstruction is proved to be a complete conjugacy invariant.

An important ingredient of the proof and a main result of independent interest is the vanishing of the \({\mathbb{Z}}\)-cohomology for centrally free actions of amenable discrete groups on von Neumann algebras with separable predual (while, usually, the 1-cohomology does not vanish, i.e. outer conjugacy does not imply conjugacy).

The author begins the proof with a detailed analysis of an amenable group G: using the Fölner condition and reproving the paring theorem of Ornstein and Weiss he obtains a paring structure for G, that is a projective system of finite subsets of G endowed with an approximate G- action, which gives an approximation of the behaviour of left G-spaces. The paring structure is then used to construct a model for a free action of G on R which is similar to the V. Jones model for G finite, but different to A. Connes model for \(G={\mathbb{Z}}\) (the author had previously extended the A. Connes model to \(G={\mathbb{Z}}\times {\mathbb{Z}})\). Further, the author makes a systematic study of the ultraproduct techniques introduced by A. Connes which is used in the proof of the main technical result - the Rohlin theorem - providing, for centraly free actions of amenable groups, an equivariant partition of the identity. The method here is based on a local phenomenon discovered by S. Popa and gives new proofs of the Rohlin theorem both for amenable group actions on measure spaces and for centrally free actions of \({\mathbb{Z}}\) on von Neumann algebras. The Rohlin theorem is then used to obtain stability properties for centrally free actions of amenable groups, including the exact vanishing of the second cohomology. The final part of the book is devoted to the recovery of the model action inside given actions and to the proof of the isomorphism theorem; here also, the method differs of the original method of A. Connes and avoids the use of spectral techniques.

This work of Adrian Ocneanu is one of the most important recent achievements in the theory of operator algebras.

In [C. R. Acad. Sci., Paris, Ser. A 291, 399-401 (1980; Zbl 0471.46045)] the author announced the classification up to outer conjugacy of the actions of amenable discrete groups on R. The book under review contains the complete proof of this classification whose main result is the uniqueness up to outer conjugacy of the free action of an amenable discrete group on R (actually, the outer conjugacy is proved for any centrally trivial and approximately inner actions of amenable discrete groups on McDuff factors M, i.e. \(M\approx M\times R)\). For not ncessarily free actions of a discrete amenable group on R, the characteristic invariant is proved to be complete for their outer conjugacy. A parallel study is done for G-kernels (homomorphisms \(G\to Aut R=Aut R/Int R)\) where the Eilenberg-MacLane \(H^ 3\) obstruction is proved to be a complete conjugacy invariant.

An important ingredient of the proof and a main result of independent interest is the vanishing of the \({\mathbb{Z}}\)-cohomology for centrally free actions of amenable discrete groups on von Neumann algebras with separable predual (while, usually, the 1-cohomology does not vanish, i.e. outer conjugacy does not imply conjugacy).

The author begins the proof with a detailed analysis of an amenable group G: using the Fölner condition and reproving the paring theorem of Ornstein and Weiss he obtains a paring structure for G, that is a projective system of finite subsets of G endowed with an approximate G- action, which gives an approximation of the behaviour of left G-spaces. The paring structure is then used to construct a model for a free action of G on R which is similar to the V. Jones model for G finite, but different to A. Connes model for \(G={\mathbb{Z}}\) (the author had previously extended the A. Connes model to \(G={\mathbb{Z}}\times {\mathbb{Z}})\). Further, the author makes a systematic study of the ultraproduct techniques introduced by A. Connes which is used in the proof of the main technical result - the Rohlin theorem - providing, for centraly free actions of amenable groups, an equivariant partition of the identity. The method here is based on a local phenomenon discovered by S. Popa and gives new proofs of the Rohlin theorem both for amenable group actions on measure spaces and for centrally free actions of \({\mathbb{Z}}\) on von Neumann algebras. The Rohlin theorem is then used to obtain stability properties for centrally free actions of amenable groups, including the exact vanishing of the second cohomology. The final part of the book is devoted to the recovery of the model action inside given actions and to the proof of the isomorphism theorem; here also, the method differs of the original method of A. Connes and avoids the use of spectral techniques.

This work of Adrian Ocneanu is one of the most important recent achievements in the theory of operator algebras.

Reviewer: Ş.Strátilá

##### MSC:

46L55 | Noncommutative dynamical systems |

20F29 | Representations of groups as automorphism groups of algebraic systems |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L40 | Automorphisms of selfadjoint operator algebras |

46L35 | Classifications of \(C^*\)-algebras |