The fundamental results of A. Connes which determine a complete set of isomorphism classes for most injectlve factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines of the investigation in the classification of factors is presented, culminating in the Connes-Takesakl structure theory of type III factors. After a discussion of inJectlvity for finite factors, the main result of the paper, the uniqueness of the injectlve II 1 factor, is deduced, and the structure of II. and type III injectlve factors is then obtained as corollaries of the main result. 1980 AMS (MOS) SUBJECT CLASSIFICATION: PRIMARY 46L35, 46LI0; SECONDARY- 46L50, 46L05 KEY WORI)S AND PHRASES: jiVe yon Nann o.%gebra, fi facr, dcre

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Central Sequences and the Hyperfinite Factor

Classification of injective factors

Soit M un facteur de type II 1 de trace normalise τ et N⊂M un sous-facteur. On demontre que l'indice [M:N] est fini si et seulement si M est un module projectif finiment genere sur N et que si c'est le cas ou M a une decomposition orthonormale sur N. On demontre la formule pour l'indice

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Entropy and index for subfactors

background

Conditional Expectations in von Neumann Algebras

We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A io M, the Betti numbers of the standard equivalence relation associated with A io M ([G2]), are in fact isomorphism invariants for the factors M, ¥â HT n (M), n iA 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying ¥â HT n (Mt) = ¥â HT n (M)/t, ¢£t > 0, and a Ki§unneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2  SL(2,Z)), for which ¥â HT 1 (M) = ¥â1(SL(2,Z)) = 1/12. Thus, Mt  M,¢£t = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R. V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.

http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n3-p03.pdf

On a class of type $II_1$ factors with Betti numbers invariants

We consider crossed product II1 factors $M = N\rtimes_{\sigma}G$
 , with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are “malleable” and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical) and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, $\mathcal{F}(M)$
 , in terms of the weights of the shift σ, for $G=\mathbb{Z}^2\rtimes SL(2,\mathbb{Z})$
 and other special arithmetic groups. We deduce that for any subgroup S⊂ℝ+
 * there exist II1 factors M (separable if S is countable or S=ℝ+
 *) with $\mathcal{F}(M)=S$
 . This brings new light to a long standing open problem of Murray and von Neumann.

Central Sequences and the Hyperfinite Factor

Citations

Classification of injective factors

Entropy and index for subfactors

Conditional Expectations in von Neumann Algebras

On a class of type $II_1$ factors with Betti numbers invariants

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

Related Papers (5)

Classification of Injective Factors Cases II 1 , II ∞ , III λ , λ 1

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Index for subfactors