Celebratio Mathematica

D. Bisch and V. Jones: “A note on free composition of subfactors,” pp. 339–361 in Geometry and physics (Aarhus, Denmark, 18–27 July 1995). Edited by J. E. Andersen, J. Dupont, H. Pedersen, and A. Swann. Lecture Notes in Pure and Applied Mathematics 184. Dekker (New York), 1997. MR 1423180 Zbl 0968.46045 incollection

Johan L. Dupont	Related
Jørgen Ellegaard Andersen	Related
Henrik Pedersen	Related
Dietmar Herbert Bisch	Related
Andrew Swann	Related

D. Bisch and V. Jones: “Algebras associated to intermediate subfactors,” Invent. Math. 128 : 1 (1997), pp. 89–157. MR 1437496 Zbl 0891.46035 article

The Temperley–Lieb algebras are the fundamental symmetry associated to any inclusion of \( \mathrm{II}_1 \) factors \( N\subset M \) with finite index. We analyze in this paper the situation when there is an intermediate subfactor \( P \) of \( N\subset M \). The additional symmetry is captured by a tower of certain algebras \( \mathrm{IA}_n \) associated to \( N\subset \) \( P\subset M \). These algebras form a Popa system (or standard lattice) and thus, by a theorem of Popa, arise as higher relative commutants of a subfactor. This subfactor gives a free composition (or minimal product) of an \( A_n \) and an \( A_m \) subfactor. We determine the Bratteli diagram describing their inclusions. This is done by studying a hierarchy \[ (FC_{m,n})_{n\in \mathbb{N}} \] of colored generalizations of the Temperley–Lieb algebras, using a diagrammatic approach, à la Kauffman, that is independent of the subfactor context. The Fuss–Catalan numbers \[ \frac{1}{(m+1)n+1}{(m+2)n\choose n} \] appear as the dimensions of our algebras. We give a presentation of the \( FC_{1,n} \) and calculate their structure in the semisimple case employing a diagrammatic method. The principal part of the Bratteli diagram describing the inclusions of the algebras \( FC_{1,n} \) is the Fibonacci graph. Our algebras have a natural trace and we compute the trace weights explicitly as products of Temperley–Lieb traces. If all indices are \( \geq 4 \), we prove that the algebras \( \mathrm{IA}_n \) and \( FC_{1,n} \) coincide. If one of the indices is \( < 4 \), \( \mathrm{IA}_n \) is a quotient of \( FC_{1,n} \) and we compute the Bratteli diagram of the tower \( (\mathrm{IA}_k)_{k\in\mathbb{N}} \). Our results generalize to a chain of \( m \) intermediate subfactors.

D. Bisch and V. Jones: “Singly generated planar algebras of small dimension,” Duke Math. J. 101 : 1 (2000), pp. 41–75. Part II was published in Adv. Math. 175:2 (2003). Part III was published in Trans. Am. Math. Soc. 369:14 (2017). MR 1733737 Zbl 1075.46053 article

D. Bisch and V. Jones: “Singly generated planar algebras of small dimension, II,” Adv. Math. 175 : 2 (May 2003), pp. 297–318. Part I was published in Duke Math. J. 101:1 (2000). Part III was published in Trans. Am. Math. Soc. 369:14 (2017). MR 1972635 Zbl 1041.46048 article

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 \( N^{\prime}\cap M_2 \) is \( \leq 12 \). 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 \( N^{\prime}\cap M_2 \) is 13. It is given by the standard invariant of the crossed product subfactor \[ R \rtimes \mathbb{Z}_2 \subset R \rtimes D_5 ,\] where \( D_5 \) denotes the dihedral group with 10 elements.

D. Bisch, D. Gaboriau, V. F. R. Jones, and S. Popa: “Von Neumann algebras and ergodic theory of group actions,” Oberwolfach Rep. 5 : 4 (2008), pp. 2763–2814. MR 2568656 Zbl 1178.46003 article

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 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 \( \mathrm{II}_1 \) 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 \( SL(n,\mathbb{R}) \) on \( \mathbb{R}^n \) 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 \( SL(2,\mathbb{Z}) \)-action on the 2-torus \( \mathbb{T}^2 \) 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 \( \mathrm{II}_1 \) factors, \( L^2 \)-rigidity in von Neumann algebras, \( \mathrm{II}_1 \) 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.

Sorin Teodor Popa	Related
Dietmar Herbert Bisch	Related
Damien Gaboriau	Related

D. Bisch, T. J. Gannon, V. F. R. Jones, and Y. Kawahigashi: “Subfactors and conformal field theory,” Oberwolfach Rep. 12 : 2 (2015), pp. 849–926. MR 3443131 Zbl 1380.00034 article

Dietmar Herbert Bisch	Related
Yasuyuki Kawahigashi	Related
Terry J. Gannon	Related

D. Bisch, V. F. R. Jones, and Z. Liu: “Singly generated planar algebras of small dimension, III,” Trans. Am. Math. Soc. 369 : 4 (2017), pp. 2461–2476. Part I was published in Duke Math. J. 101:1 (2000). Part II was published in Adv. Math. 175:2 (2003). MR 3592517 Zbl 1370.46036 ArXiv 1410.2876 article

Zhengwei Liu	Related
Dietmar Herbert Bisch	Related

D. Bisch, D. E. Evans, R. Kirby, and S. Popa: “Memories of Vaughan Jones,” Notices Amer. Math. Soc. 68 : 9 (2021), pp. 1540–1563. MR 4323829 Zbl 1480.01028 article

David E. Evans	Related
Dietmar Herbert Bisch	Related
Robion C. Kirby	Related	Volume

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