# Topics on von Neumann algebras (06w5086)

Arriving in Banff, Alberta Saturday, September 16 and departing Thursday September 21, 2006

## Organizers

Juliana Erlijman (University of Regina)

Hans Wenzl (University of California, San Diego)

## Objectives

Von Neumann algebras are algebras of bounded linear operators on a Hilbert space which are closed under the topology of pointwise convergence. If their center only consists of multiples of the identity, they are called factors. Von Neumann algebras were first studied in a series of papers by Murray and von Neumann in the 1930's. Their motivation was to have a tool for studying quantum mechanics and representations of infinite groups. As will be seen below, these are still some of the major driving forces in research related to von Neumann algebras with exciting recent developments. Moreover, von Neumann algebras play an important role in additional areas such as topology and quantum computing.

In spite of these activities, there have not been many instances of a general conference on von Neumann algebras with experts from interacting areas. In fact, since the special year on operator algebras at MSRI in 2000/2001, there was only one such meeting earlier this year at CIRM, Marseille, France in which specialists in von Neumann algebras met with experts in group theory and topology. The proposed conference would try to fill this gap. It would also be at a more convenient location for researchers in North America.

Timeliness of this workshop: There are fairly new results that involve young mathematicians with fresh ideas, which will be mentioned below. It would be optimal to hold a workshop of this nature within the next two years in order to maximize the exposure and discussion of these ideas and stimulate further progress.

The organizers have contacted some of the leading researchers in von Neumann algebras, such as V.Jones, D.Voiculescu, S.Popa and A.Wassermann, who have expressed their interest and support for this project.

It will be convenient for our exposition to roughly divide the recent activities in von Neumann algebras into the study of subfactors and the study of factors.

I) The study of subfactors was initiated by Vaughan Jones in the 1980's by introducing an important invariant for them called the index. Moreover, he proved a surprising and fundamental theorem on the set of possible index values and he produced an important class of examples called the Jones subfactors. This class of examples carried a representation of braid groups, and was later used to define link invariants. The Jones subfactors are related to the Lie group SU(2); additional examples for all other Lie types were subsequently constructed by Wassermann (via loop groups) and by H.Wenzl and by F.Xu via quantum groups. Wassermann's construction was in part inspired by results in algebraic quantum field theory, where von Neumann algebras have played an important role for a long time. His crucial result in this context was the definition and explicit computation of a highly nontrivial tensor product between two representations of loop groups of the same level, which is usually called the fusion tensor product.

a) Wassermann's work established a useful link between conformal field theory and von Neumann algebras: It follows from work of Wassermann and Xu that special inclusions of a loop group in another one, so-called conformal inclusions, produce additional examples of subfactors. This explains, in particular, the occurrence of certain exceptional subfactors in the classification of subfactors of index less than 4. The latter subfactors were constructed before by different methods and can be understood as coming from some suitable 'quantum subgroup' of a quantum version of SU(2). A.Ocneanu has continued these studies to classify analogous quantum subgroups of higher rank groups. An important tool in his studies are certain functions called modular invariants. These have been studied in depth by T.Gannon from a completely different point of view. It would be interesting to see how these different approaches complement each other.

There have been further interesting developments in this area by recent work by V.Kac, R.Longo and F.Xu involving actions of finite groups and solitons in connection with the loop group approach.

b) The fact that one obtains essentially the same subfactors from the loop group approach as well as from the quantum group approach can probably easiest be expressed via the notion of tensor categories: quantum groups and loop groups (with the fusion tensor product) yield equivalent tensor categories. Not surprisingly, similar results appear in different contexts. E.g. categorical analogs of the already mentioned quantum subgroups and of the ADE classification have been found by Wassermann and by Kirillov jr and Ostrik. There seems to be more potential to use this formalism to translate between conformal field theory and subfactors.

c) An important classification result for amenable subfactors of the hyperfinite II_1 factor was proved by Popa. He showed that they can be reconstructed by what he calls the standard invariant; it is, however, still a wide open problem what values this standard invariant can take. On the other hand, he did show that whenever a system of finite dimensional algebras satisfies certain axioms, there always exists a subfactor of a (not necessarily hyperfinite) II_1 factor for which the given system appears as standard invariant. This inspired Jones to describe these systems in a graphical way by planar algebras which is more suitable for a more explicit classification Several important results about planar algebras were obtained by him and D.Bisch. Moreover, planar algebras also appear in free probability theory which will be explained below. In a recent development, new results have been obtained by Jones and Xu regarding the relative position of several subfactors.

d) The above mentioned fusion tensor categories also play an important role in Michael Freedman's approach towards building a quantum computer. This is done in the context of topological quantum field theory, with positivity and unitarity questions playing a crucial role. Von Neumann algebras provide a perfect framework in which to study this questions. Indeed there has already been some interaction, where subfactor related results played a useful role.

Von Neumann algebras also appear as a tool in the work of S.Stolz and P.Teichner. Their project is to find a geometric interpretation of elliptic cohomology in the context of conformal field theory. In their approach they use bimodules of type III_1 factors. Not only do they need results by Wassermann; their approach also opens new question within von Neumann algebras, some of which have been solved by

Wassermann.

II) Factors: One of the big problems in von Neumann algebras is the classification of II_1 factors. One can define an important class of examples of such factors from the group von Neumann algebras of infinite discrete groups for which all nontrivial conjugate classes are infinite. However, it is very difficult to decide when these factors are isomorphic. Various invariants for II_1 factors have been introduced by A.Connes and several deep results have been proved by him. The most spectacular result was that all the factors obtained from amenable groups are isomorphic to the hyperfinite II_1 factor. It is known that this factor is not isomorphic to the one obtained from a free group with n generators.

(a) It has been a longstanding unsolved problem to decide whether the factors obtained from the free groups with n and m generators respectively are isomorphic if n is not equal to m with both n,m >1. This problem was one of the inspirations for Voiculescu's theory of free probability. While this has not led to a solution of the original problem yet, it produced many interesting results in its right, such as e.g. Voiculescu's proof of the absence of Cartan subalgebras for free group factors. There has been a recent workshop on free probability theory also at the BIRS. So it is planned to limit free probability in this workshop to pure applications to von Neumann algebras. In particular we would like to mention results by J.Mingo, A.Nica and R.Speicher on noncrossing permutations and second order asymptotics for random matrices. The combinatorics in their study is very similar to the one in Jones' planar algebras. It would be interesting to see whether there are any deeper connections.

b) The most exciting developments in the theory of von Neumann algebras in the last few years undoubtedly took place in connection with group theory. D.Gaboriau defined a notion of ell^2 Betti numbers for countable measure preserving equivalence relations in a Borel space. This proved a crucial tool in Popa's recent work: One can define for any II_1 factor a multiplicative subgroup of the positive real numbers called the fundamental group. It has been a longstanding problem whether there exists a II_1 factor with trivial fundamental group. The existence of such factors has now been established by Popa, using a version of these ell^2 Betti numbers in the von Neumann algebra context.

Additional interesting results were recently obtained by N.Ozawa: based on his notion of solid von Neumann algebras he obtained many examples of prime factors (i.e. II_1 factors which are not the tensor product of two II_1 factors); moreover, in collaboration with Popa, they prove unique prime factorization results for tensor products of factors coming from subgroups of hyperbolic groups.

Another important work inspired by D.Gaboriau's paper is the definition of an L^2-homology for von Neumann algebras by Connes and D.Shlyakhtenko. There is a well-known analogy between certain properties of groups (such as `amenability' or `property T') and II_1 factors. The idea is to replace modules of groups by bimodules of factors. There is some evidence, supplied in part by results in free probability that this new homology theory provides the right framework to study which properties of groups carry over to their corresponding group von Neumann algebras. Such tools would be all the more relevant in view of exciting new results obtained by N.Monod and Y.Shalom in the group context. While there has been contact between these different research groups, it would seem that the potential for future exciting results is still far from being exhausted. It is hoped that the proposed conference will contribute to further progress in these areas.

In spite of these activities, there have not been many instances of a general conference on von Neumann algebras with experts from interacting areas. In fact, since the special year on operator algebras at MSRI in 2000/2001, there was only one such meeting earlier this year at CIRM, Marseille, France in which specialists in von Neumann algebras met with experts in group theory and topology. The proposed conference would try to fill this gap. It would also be at a more convenient location for researchers in North America.

Timeliness of this workshop: There are fairly new results that involve young mathematicians with fresh ideas, which will be mentioned below. It would be optimal to hold a workshop of this nature within the next two years in order to maximize the exposure and discussion of these ideas and stimulate further progress.

The organizers have contacted some of the leading researchers in von Neumann algebras, such as V.Jones, D.Voiculescu, S.Popa and A.Wassermann, who have expressed their interest and support for this project.

It will be convenient for our exposition to roughly divide the recent activities in von Neumann algebras into the study of subfactors and the study of factors.

I) The study of subfactors was initiated by Vaughan Jones in the 1980's by introducing an important invariant for them called the index. Moreover, he proved a surprising and fundamental theorem on the set of possible index values and he produced an important class of examples called the Jones subfactors. This class of examples carried a representation of braid groups, and was later used to define link invariants. The Jones subfactors are related to the Lie group SU(2); additional examples for all other Lie types were subsequently constructed by Wassermann (via loop groups) and by H.Wenzl and by F.Xu via quantum groups. Wassermann's construction was in part inspired by results in algebraic quantum field theory, where von Neumann algebras have played an important role for a long time. His crucial result in this context was the definition and explicit computation of a highly nontrivial tensor product between two representations of loop groups of the same level, which is usually called the fusion tensor product.

a) Wassermann's work established a useful link between conformal field theory and von Neumann algebras: It follows from work of Wassermann and Xu that special inclusions of a loop group in another one, so-called conformal inclusions, produce additional examples of subfactors. This explains, in particular, the occurrence of certain exceptional subfactors in the classification of subfactors of index less than 4. The latter subfactors were constructed before by different methods and can be understood as coming from some suitable 'quantum subgroup' of a quantum version of SU(2). A.Ocneanu has continued these studies to classify analogous quantum subgroups of higher rank groups. An important tool in his studies are certain functions called modular invariants. These have been studied in depth by T.Gannon from a completely different point of view. It would be interesting to see how these different approaches complement each other.

There have been further interesting developments in this area by recent work by V.Kac, R.Longo and F.Xu involving actions of finite groups and solitons in connection with the loop group approach.

b) The fact that one obtains essentially the same subfactors from the loop group approach as well as from the quantum group approach can probably easiest be expressed via the notion of tensor categories: quantum groups and loop groups (with the fusion tensor product) yield equivalent tensor categories. Not surprisingly, similar results appear in different contexts. E.g. categorical analogs of the already mentioned quantum subgroups and of the ADE classification have been found by Wassermann and by Kirillov jr and Ostrik. There seems to be more potential to use this formalism to translate between conformal field theory and subfactors.

c) An important classification result for amenable subfactors of the hyperfinite II_1 factor was proved by Popa. He showed that they can be reconstructed by what he calls the standard invariant; it is, however, still a wide open problem what values this standard invariant can take. On the other hand, he did show that whenever a system of finite dimensional algebras satisfies certain axioms, there always exists a subfactor of a (not necessarily hyperfinite) II_1 factor for which the given system appears as standard invariant. This inspired Jones to describe these systems in a graphical way by planar algebras which is more suitable for a more explicit classification Several important results about planar algebras were obtained by him and D.Bisch. Moreover, planar algebras also appear in free probability theory which will be explained below. In a recent development, new results have been obtained by Jones and Xu regarding the relative position of several subfactors.

d) The above mentioned fusion tensor categories also play an important role in Michael Freedman's approach towards building a quantum computer. This is done in the context of topological quantum field theory, with positivity and unitarity questions playing a crucial role. Von Neumann algebras provide a perfect framework in which to study this questions. Indeed there has already been some interaction, where subfactor related results played a useful role.

Von Neumann algebras also appear as a tool in the work of S.Stolz and P.Teichner. Their project is to find a geometric interpretation of elliptic cohomology in the context of conformal field theory. In their approach they use bimodules of type III_1 factors. Not only do they need results by Wassermann; their approach also opens new question within von Neumann algebras, some of which have been solved by

Wassermann.

II) Factors: One of the big problems in von Neumann algebras is the classification of II_1 factors. One can define an important class of examples of such factors from the group von Neumann algebras of infinite discrete groups for which all nontrivial conjugate classes are infinite. However, it is very difficult to decide when these factors are isomorphic. Various invariants for II_1 factors have been introduced by A.Connes and several deep results have been proved by him. The most spectacular result was that all the factors obtained from amenable groups are isomorphic to the hyperfinite II_1 factor. It is known that this factor is not isomorphic to the one obtained from a free group with n generators.

(a) It has been a longstanding unsolved problem to decide whether the factors obtained from the free groups with n and m generators respectively are isomorphic if n is not equal to m with both n,m >1. This problem was one of the inspirations for Voiculescu's theory of free probability. While this has not led to a solution of the original problem yet, it produced many interesting results in its right, such as e.g. Voiculescu's proof of the absence of Cartan subalgebras for free group factors. There has been a recent workshop on free probability theory also at the BIRS. So it is planned to limit free probability in this workshop to pure applications to von Neumann algebras. In particular we would like to mention results by J.Mingo, A.Nica and R.Speicher on noncrossing permutations and second order asymptotics for random matrices. The combinatorics in their study is very similar to the one in Jones' planar algebras. It would be interesting to see whether there are any deeper connections.

b) The most exciting developments in the theory of von Neumann algebras in the last few years undoubtedly took place in connection with group theory. D.Gaboriau defined a notion of ell^2 Betti numbers for countable measure preserving equivalence relations in a Borel space. This proved a crucial tool in Popa's recent work: One can define for any II_1 factor a multiplicative subgroup of the positive real numbers called the fundamental group. It has been a longstanding problem whether there exists a II_1 factor with trivial fundamental group. The existence of such factors has now been established by Popa, using a version of these ell^2 Betti numbers in the von Neumann algebra context.

Additional interesting results were recently obtained by N.Ozawa: based on his notion of solid von Neumann algebras he obtained many examples of prime factors (i.e. II_1 factors which are not the tensor product of two II_1 factors); moreover, in collaboration with Popa, they prove unique prime factorization results for tensor products of factors coming from subgroups of hyperbolic groups.

Another important work inspired by D.Gaboriau's paper is the definition of an L^2-homology for von Neumann algebras by Connes and D.Shlyakhtenko. There is a well-known analogy between certain properties of groups (such as `amenability' or `property T') and II_1 factors. The idea is to replace modules of groups by bimodules of factors. There is some evidence, supplied in part by results in free probability that this new homology theory provides the right framework to study which properties of groups carry over to their corresponding group von Neumann algebras. Such tools would be all the more relevant in view of exciting new results obtained by N.Monod and Y.Shalom in the group context. While there has been contact between these different research groups, it would seem that the potential for future exciting results is still far from being exhausted. It is hoped that the proposed conference will contribute to further progress in these areas.