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

Operator Algebras and Applications: Coactions and Yang-Baxter equations for ergodic actions and subfactors

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

We identify the statistical dimension of a superselection sector in a local quantum field theory with the square root of the index of a localized endomorphism of the quasi-local C*-algebra that represents the sector. As a consequence in a two-dimensional theory the possible values of the statistical dimension below 2 are restricted to a given discrete set.

/pdf/index-of-subfactors-and-statistics-of-quantum-fields-i-s7dk0ecmu4.pdf

Index of subfactors and statistics of quantum fields. I

0. Introduction 1. Basics of the theory of subfactors 1.1. General inclusions 1.2. Subfactors of finite index 1.3. The s tandard invariant (the paragroup) 1.4. Core and model inclusions 2. A representation theory for subfactors 2.1. Definition and motivation 2.2. Basic construction for representations 2.3. Smooth representations 2.4. Standard representations 3. Amenabli ty for inclusions of type II1 factors 3.1. Definitions and motivations 3.2. Basic properties 4. Approximation of amenable inclusions by higher relative commutants 4.1. Statement of results 4.2. A F01ner-type condition 4.3. Local approximation 4.4. Global approximation 4.5. Bicommutant condition and existence of hypertraces 5. Classification by s tandard invariants 5.1. The main result 5.2. A list of subfactors of index 4 4 5.3. Strong amenabili ty for paragroups and s tandard invariants 5.4. ~ r t h e r characterizations of amenabili ty for inclusions Appendix A.1. A local quantization principle A.2. A per turbat ion result

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Classification of amenable subfactors of type II

We prove that a type II$_1$ 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 $\Cal H \Cal T$ of factors $M$ having such Cartan subalgebras $A \subset M$, the Betti numbers of the standard equivalence relation associated with $A \subset M$ ([G2]), are in fact isomorphism invariants for the factors $M$, $\beta^{^{HT}}_n(M), n\geq 0$. The class $\Cal H\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $\beta^{^{HT}}_n(M^t)= \beta^{^{HT}}_n(M)/t, \forall t>0$, and a K{\"u}nneth type formula. An example of a factor in the class $\Cal H\Cal T$ is given by the group von Neumann factor $M=L(\Bbb Z^2 \rtimes SL(2, \Bbb Z))$, for which $\beta^{^{HT}}_1(M) = \beta_1(SL(2, \Bbb Z)) = 1/12$. Thus, $M^t \not\simeq M, \forall t \neq 1$, 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.

On a Class of Type II$_1$ Factors with Betti Numbers Invariants

The main object of this paper is the study of a sequence of finite dimensional algebras, depending on 2 parameters, which appear in connection with the Kauffman link invariant and with Drinfeld's and Jimbo'sq deformation of Lie algebras of typesB, C andD We determine for which parameters these algebras are semisimple Moreover, we classify all unitary representations of the infinite braid groupB∞ factoring through the inductive limit of these algebras This yields new examples of irreducible subfactors of finite depth, whose indices are squares ofq dimensions of irreducible representations of sympletic and orthogonal groups In the combinatorial description of these subfactors one naturally obtains truncated Weyl chambers (as for loop groups for a given level) and multiplicity coefficients of fusion rules for Wess-Zumino-Witten models

/pdf/quantum-groups-and-subfactors-of-type-b-c-and-d-1oaiexp82j.pdf

Operator Algebras and Applications: Coactions and Yang-Baxter equations for ergodic actions and subfactors

Citations

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

Index of subfactors and statistics of quantum fields. I

Classification of amenable subfactors of type II

On a Class of Type II$_1$ Factors with Betti Numbers Invariants

Quantum groups and subfactors of type $B$, $C$, and $D$

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