Dietmar Bisch

Singly generated planar algebras of small dimension

0. Introduction. A subfactorN ⊂M gives rise to a powerful set of invariants that can be approached successfully in several ways. (See, for instance, [B2], [EK], [FRS], [GHaJ], [H], [Iz], [JSu], [Lo1], [Lo2], [Oc1], [Oc2], [Po1], [Po2], [Po3], [Po4], [Wa], [We1], and [We2]). A particular approach suggests a particular kind of subfactor as the “simplest.” For instance, in Haagerup’s approach [H], subfactors of small index are the simplest. In [J2], a pictorial language is developed in which the invariants appear as a graded vector space V = (Vn)n≥0 whose elements can be combined in planar, but otherwise quite arbitrary, ways. Thus, for instance, in the diagram

On the existence of central sequences in subfactors

We prove a relative version of [Co 1, Theorem 2.1 ] for a pair of type III-factors N c M. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of M contained in the subfactor N. As an immediate application we obtain a result by Bedos [Be, Theorem A], showing that if N has property F and G is an amenable group acting freely on N via some action a, then the crossed product N x . G has property F. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems 1, 2 and 3]), which gives necessary and sufficient conditions implying that the pair N c M is stable.

Entropy of groups and subfactors

We show that the classical notion of entropy of a finitely generated group G as introduced by A. Avez (C. R. Acad. Sci. Paris 275A, 1972, 1363–1366) is related by an explicit formula to the entropy of A. Connes and E. Stormer (Acta Math. 134, 1975, 288–306) and the index of V. F. R. Jones (Invent. Math. 72, 1983, 1–25) of the associated pair of finite von Neumann algebras as considered by S. Popa (C. R. Acad. Sci. Paris Ser. I Math. 309, 1989, 771–776). This construction is discussed in detail. We prove that the entropy of G is maximal if and only if G is the free group and compute its value. Then we show how the entropy of groups is connected to random walks on groups and information theory. Finally we give a brief discussion of the group entropy as a growth invariant and compare it to the Grigorchuk-Cohen cogrowth and Kestens invariant λ(G).

CONTINUOUS FAMILIES OF HYPERFINITE SUBFACTORS WITH THE SAME STANDARD INVARIANT

We construct numerous continuous families of irreducible subfactors of the hyperfinite II1 factor which are non-isomorphic, but have all the same standard invariant. In particular, we obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II1 factor with Jones index 6, which have all the same standard invariant with property (T). We exploit the fact that property (T) groups have uncountably many non-cocycle conjugate cocycle actions on the hyperfinite II1 factor.

Singly generated planar algebras of small dimension, Part II

Abstract We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C ∗ -planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N ′∩ M 2 is ⩽12. We showed that they are given by the Fuss–Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C ∗ -planar algebra generated by a single non-trivial 2-box if the dimension of N ′∩ M 2 is 13. It is given by the standard invariant of the crossed product subfactor R⋊ Z 2 ⊂R⋊D 5 , where D 5 denotes the dihedral group with 10 elements.

On the Structure of Finite Depth Subfactors

We study the basic construction of a pair of reduced subfactors N p ⊂ pM p, p ∈ N 1 ⋂ M a projection and provide a method of computing the graphs of the reduced subfactors using the Bratteli diagrams of the original inclusion N ⊂ M. Then we describe what we mean by a fusion algebra associated to a subfactor and explain how these fusion algebras can be computed explicitly. In particular this provides an explicit method of calculating the graphs of the reduced subfactors. We present the results of these computations for various subfactors such as for the 3+√3 subfactor and for Haagerup’s smallest finite depth subfactor of the hyperfinite II1 factor with index above 4.

Tube algebra of group-type subfactors

We describe the tube algebra and its representations in the cases of diagonal and Bisch–Haagerup subfactors possibly with a scalar 3-cocycle obstruction. We show that these categories are additivel...

Von Neumann Algebras and Ergodic Theory of Group Actions

The theory of von Neumann algebras has seen some dramatic advances in the last few years. Von Neumann algebras are objects which can capture and analyze symmetries of mathematical or physical situations whenever these symmetries can be cast in terms of generalized morphisms of the algebra (Hilbert bimodules, or correspondences). Analyzing these symmetries led to an amazing wealth of new mathematics and the solution of several long-standing problems in the theory. Popa’s new deformation and rigidity theory has culminated in the discovery of new cocycle superrigidity results à la Zimmer, thus establishing a new link to orbit equivalence ergodic theory. The workshop brought together world-class researchers in von Neumann algebras and ergodic theory to focus on these recent developments. Mathematics Subject Classification (2000): 46L10. Introduction by the Organisers The workshop Von Neumann Algebras and Ergodic Theory of Group Actions was organized by Dietmar Bisch (Vanderbilt University, Nashville), Damien Gaboriau (ENS Lyon), Vaughan Jones (UC Berkeley) and Sorin Popa (UC Los Angeles). It was held in Oberwolfach from October 26 to November 1, 2008. This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory. The organizers took special care to invite many young mathematicians and more than half of the 28 talks were given by them. The meeting was very well attended by over 40 participants, leading senior researchers and junior mathematicians in the field alike. Participants came from 2764 Oberwolfach Report 49/2008 about a dozen different countries including Belgium, Canada, Denmark, France, Germany, Great Britain, Japan, Poland, Switzerland and the USA. The first day of the workshop featured beautiful introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa’s deformation/rigidity techniques and applications to rigidity in II1 factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa). There were many excellent lectures during the subsequent days of the conference and many new results were presented, some for the first time during this meeting. A few of the highlights of the workshop were Vaes’ report on a new cocycle superrigidity result for non-singular actions of lattices in SL(n,R) on Rn and on other homogeneous spaces (joint with Popa), Ioana’s result showing that every sub-equivalence relation of the equivalence relation arising from the standard SL(2,Z)-action on the 2-torus T is either hyperfinite, or has relative property (T), and Epstein’s report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space. Other talks discussed new results on fundamental groups of II1 factors, L -rigidity in von Neumann algebras, II1 factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra and new results on topological rigidity and the Atiyah conjecture. Many interactions and stimulating discussions took place at this workshop, which is of course exactly what the organizers had intended. The organizers would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing the splendid environnment for holding this conference. Special thanks go to the very helpful and competent staff of the institute.