##
**Intermediate subfactors with no extra structure.**
*(English)*
Zbl 1131.46041

Let \(N \subset M\) be II\(_{1}\) von Neumann factors with index \([M:N] < \infty\) [see V.F.R.Jones, Invent.Math.72, 1–25 (1983; Zbl 0508.46040)]. We assume almost throughout that \(N\) is irreducible, i.e., the relative commutant \(N' \cap M = \mathbb{C}\). The standard invariant of V.F.R.Jones [loc.cit.] is a certain double sequence of multi-matrix algebras (direct sums of matrix algebras); he used a fundamental construction of \(M_{1}\) from \(M\) to build a tower of multi-matrix algebras \(\{ M_{k} \}\) over \(M_{0} = M\) together with a corresponding sequence of projections \(\{ e_{k} \}\). Extending this to von Neumann factors, the fundamental construction generates a von Neumann factor \(M_{1}\) from \(M\) and \(e_{1}\). This \(e_{1}\) is a projection whose restriction, for the Hilbert space structures given by the trace as inner product, is the conditional expectation of \(M\) onto \(N\). Subsequent projections \(\{ e_{i}\}\) are constructed by repeating the fundamental construction to build a tower of factors \(\{ M_{k} \}\). The vector space underlying the relative commutant \(N' \cap M_{k-1}\) is denoted by \(P_{k}\). For details, see [V.F.R.Jones, J. Funct.Anal.122, No.1, 84–90 (1994; Zbl 0821.46076)] and [F.M.Goodman, P.de la Harpe and V.F.R.Jones, “Coxeter graphs and towers of algebras” (MSRI Publ.14, New York etc.:Springer–Verlag) (1989; Zbl 0698.46050)]. D.Bisch [Pac.J.Math.163, No.2, 201–216 (1994; Zbl 0814.46053)] has given an abstract characterisation of the standard invariant in terms of relative commutants. There are twin towers of relative commutants, one the \(\{ N' \cap M_{i}\}\) and the other the \(\{ M' \cap M_{i} \}\).

The standard invariant is sufficient to characterise the subfactor structure of certain hyperfinite II\(_{1}\) factors [see S.Popa, Invent.Math.120, No.3, 427–445 (1995; Zbl 0831.46069)]. All II\(_{1}\) factors are isomorphic as von Neumann algebras, but the subfactor structure can vary. For general II\(_{1}\) factors, one needs further invariants. An intermediate subfactor is of the form \(N \subset P \subset M\) and the inclusions can be described using conditional expectations. An intermediate factor provides additional symmetries. Invariants can be found by considering the lattice structure of the inclusions (dating back to John von Neumann’s lattice structure for type I factors and his continuous geometry) or by using A.Ocneanu’s concept of crossed products by paragroups approach; see [Lond.Math.Soc.Lect.Note 136, 119–172 (1987; Zbl 0696.46048)], for which an intermediate subfactor can be considered as a quantization of a finite group. S.Popa’s \(\lambda\)- lattices [loc.cit.] axiomatize the standard invariant.

A pair \([M,N]\) is said to have no extra structure in the sense that a tower has been built where there is no intermediate subfactor between \(M\) and \(N\). When \([M:N] < 4\) and there is no extra structure, the algebras obtained in the tower of multimatrix algebras are called Temperley–Lieb algebras (whose dimensions are Catalan numbers) leading by induction to the II\(_{1}\) Jones–Temperley–Lieb factors. The \(\{ e_{i} \}\) satisfy the Temperley–Lieb relations with a parameter \([M:N]^{-1}\) [cf.H.V.N.Temperley and E.H.Lieb, Proc.R.Soc.Lond., Ser.A 322, 251–280 (1971; Zbl 0211.56703)]; these were used for defining solvable models in statistical mechanics, solvable in the vague sense that the asymptotic growth rate of the partition function can be reasonably determined – examples are the spin models, say of the ice-type (oxygen atoms with hydrogen bonds), where the states are functions from the vertices to to a finite set of spins and the energy is in terms of the bonds. Vertex models in statistical mechanics have states which are functions of the edges to the set of spins and the energy is associated with the vertices. Quantum groups give rise to vertex models.

When there is a single intermediate factor, so providing an extra symmetry, D.Bisch and V.F.R.Jones [Invent.Math.128, No.1, 89–157 (1997; Zbl 0891.46035)], and when there is a chain of intermediate factors, Z.Landau [Pac.J.Math.197, No.2, 325–367 (2001; Zbl 1055.46511)] constructed the so-called Fuss–Catalan subfactors with colours which are represented by positive integers \(n\). For \(n = 1\), the algebras are Temperley–Lieb and are colourless; see N.I.Fuss [“Solutio quaestionis, quot modis polygonum \(n\) laterum in polygona \(m\) laterum, per diagonales resolui quaeat”, Nova Acta Acadamiae Scientiarum Petropolitanae 9, 243–251 (1791)] who calculated the number of partitions, by noncrossing diagonals, of convex polygons, \(n\)-gons into \(m\)-gons, and E.G.Catalan [J. Reine Angew.Math.27, 192 (1844; ERAM 027.0790cj)] who did the simpler calculation of the numbers of partitions into triangles.

A graphical approach is to use graphs whose vertices correspond to the multi-matrix algebras and to construct Bratteli diagrams [O.Bratteli, Trans.Am.Math.Soc.171, 195–234 (1972; Zbl 0264.46057)] which are a graphical way of describing inclusions. The Bratteli diagram records the ranks of minimal projections in the simple components of the bigger one. In some cases, stacking successive Bratelli diagams on top of each other leads to II\(_{1} \)-factors as inductive limits of finite-dimensional ones (e.g., Jones–Temperley–Lieb). Principal graphs were constructed from Bratteli diagrams by A.Ocneanu. The mirror image of the previous subfactor inclusion is always part of the next inclusion and, by throwing away reflections in the diagrams, one gets the principal graph, and this provides the same information. If \([M:N] < 4\), the principal graph must be an indecomposable affine A, D, or E Coxeter graph [see F.M.Goodman, P.de la Harpe and V.F.R.Jones, loc.cit.].

The planar algebras of V.F.R.Jones [in:Knots in Hellas ’98, World Scientific Ser.Knots Everything 24, 94–117 (2000; Zbl 1021.46047)] are algebras over the vector space generated by the \(P_{k}\) and lead to a more comprehensive kind of invariant for the subfactor structure. They present an algebraic-combinatorial interpretation of the standard invariant and can be considered as a geometric topological axiomatisation of Popa’s \(\lambda\)-lattice structure. The algebra may be represented graphically as an algebra over an operad of tangles. It forms a pivotal category [see, e.g., S.Abramsky, in: “Mathematics of quantum computation and quantum technology” CRC applied mathematics and nonlinear science series 14, 515–558 (2008; Zbl 1135.81006)]. The algebras are graphically represented like multi-coloured discs and subdiscs in the plane with strings cutting an even number of points on the discs, and of course an operation of composition. Planar signifies that the strings are to be non-crossing. From the viewpoint of planar algebras, the simplest of structures for inclusions of intermediate factors are the Fuss–Catalan algebras.

A quadrilateral of subfactors consists of \(N \subset P \subset M\), \(N \subset Q \subset M\) where \(M\) is generated by \(P,Q\) and \(N = P \cap Q\), and will be assumed to be irreducible. The quadrilateral is called a commuting square if the conditional expectations of \(P\) and \(Q\) commute. The quadrilateral can be considered as a dual object, under co-multiplication, to a commuting square. The quadrilateral is called co-commutative if the commutants of the factors form a commuting square so it can be considered as a dual object to a commuting square.

K.Gustafson [Bull.Am.Math.Soc.74, 488–492 (1968; Zbl 0172.40702)], dealing with positive non-commutative selfadjoint operators on a Hilbert space, has defined an angle of an operator as the least upper bound on the angle by which the operator can rotate any unit vector. T.Sano and Y.Watatani [Proceedings of ICM satellite conference, Nara, 1990, 72–82 (1991; Zbl 0821.46079)] considered the relative positions of four subspaces of a Hilbert space [cf.P.R.Halmos, Trans.Am.Math.Soc.144, 381–389 (1969; Zbl 0187.05503)]. They used Gustafson’s idea to define the angle at a vertex between two subfactors of \(M\) as the spectrum of the angle operator at the vertex. The trigonometric functions are expressed in terms of the conditional expectations. A quadrilateral is a commuting square if and only if the angles are all right-angles.

The angle operators between two subspaces in a Hilbert space can have any closed subset of \((0,{\pi \over 2}]\) as spectrum. A major problem is to quantize the angle operators for intermediate subfactors, in the sense that they take a countable discrete set of values in the interval \((0, {\pi \over 2}] \). The main theorem of this paper is an initial attempt at quantization.

Let \(G\) be a Coxeter group. For finite von Neumann factors \(A_{0} \subset A_{1}\), one constructs a tower over \(A_{0}\) where the \(B_{i}\) are subalgebras of \(A_{i}\) generated by Jones’ projections \(e_{1} \dots e_{i-1}\). Let \(r\) denote the selected vertex for the Bratteli diagram. The subfactor \(rB \subset rAr\) is called the GHJ subfactor for \(G\). The diagrams are bipartite graphs in the sense that there are two colours; black (odd) and white (even) vertices occur in alternating rows and no edges join a black to a white (apartheid?) or a white to a white. In their main theorem, the authors consider the GHQ subfactors for two Coxeter groups, the symmetric group \(S_{3}\), and the group \(D_{5}\) with selected vertex the trivalent vertex, incident to three edges.

For their main theorem, they assume that there is no extra structure and classify three cases where the angle operator takes a finite number of values which can be determined. When \([M:N] = 6\), \(N\) is the fixed point algebra for the outer action of \(S_{3}\) on \(M\) and the angle is \(\pi \over 3\). When \([M:N] = 6 + 4 \sqrt{2}\), the quadrilateral can be associated with a GHQ subfactor associated to the Coxeter group \(D_{5}\); here, the angle is \(\arccos (\sqrt{2} -1\)). Otherwise, the expectations onto \(P\) and \(Q\) commute and the quadrilateral is a commutative square, thus both commutative and co-commutative.

The standard invariant is sufficient to characterise the subfactor structure of certain hyperfinite II\(_{1}\) factors [see S.Popa, Invent.Math.120, No.3, 427–445 (1995; Zbl 0831.46069)]. All II\(_{1}\) factors are isomorphic as von Neumann algebras, but the subfactor structure can vary. For general II\(_{1}\) factors, one needs further invariants. An intermediate subfactor is of the form \(N \subset P \subset M\) and the inclusions can be described using conditional expectations. An intermediate factor provides additional symmetries. Invariants can be found by considering the lattice structure of the inclusions (dating back to John von Neumann’s lattice structure for type I factors and his continuous geometry) or by using A.Ocneanu’s concept of crossed products by paragroups approach; see [Lond.Math.Soc.Lect.Note 136, 119–172 (1987; Zbl 0696.46048)], for which an intermediate subfactor can be considered as a quantization of a finite group. S.Popa’s \(\lambda\)- lattices [loc.cit.] axiomatize the standard invariant.

A pair \([M,N]\) is said to have no extra structure in the sense that a tower has been built where there is no intermediate subfactor between \(M\) and \(N\). When \([M:N] < 4\) and there is no extra structure, the algebras obtained in the tower of multimatrix algebras are called Temperley–Lieb algebras (whose dimensions are Catalan numbers) leading by induction to the II\(_{1}\) Jones–Temperley–Lieb factors. The \(\{ e_{i} \}\) satisfy the Temperley–Lieb relations with a parameter \([M:N]^{-1}\) [cf.H.V.N.Temperley and E.H.Lieb, Proc.R.Soc.Lond., Ser.A 322, 251–280 (1971; Zbl 0211.56703)]; these were used for defining solvable models in statistical mechanics, solvable in the vague sense that the asymptotic growth rate of the partition function can be reasonably determined – examples are the spin models, say of the ice-type (oxygen atoms with hydrogen bonds), where the states are functions from the vertices to to a finite set of spins and the energy is in terms of the bonds. Vertex models in statistical mechanics have states which are functions of the edges to the set of spins and the energy is associated with the vertices. Quantum groups give rise to vertex models.

When there is a single intermediate factor, so providing an extra symmetry, D.Bisch and V.F.R.Jones [Invent.Math.128, No.1, 89–157 (1997; Zbl 0891.46035)], and when there is a chain of intermediate factors, Z.Landau [Pac.J.Math.197, No.2, 325–367 (2001; Zbl 1055.46511)] constructed the so-called Fuss–Catalan subfactors with colours which are represented by positive integers \(n\). For \(n = 1\), the algebras are Temperley–Lieb and are colourless; see N.I.Fuss [“Solutio quaestionis, quot modis polygonum \(n\) laterum in polygona \(m\) laterum, per diagonales resolui quaeat”, Nova Acta Acadamiae Scientiarum Petropolitanae 9, 243–251 (1791)] who calculated the number of partitions, by noncrossing diagonals, of convex polygons, \(n\)-gons into \(m\)-gons, and E.G.Catalan [J. Reine Angew.Math.27, 192 (1844; ERAM 027.0790cj)] who did the simpler calculation of the numbers of partitions into triangles.

A graphical approach is to use graphs whose vertices correspond to the multi-matrix algebras and to construct Bratteli diagrams [O.Bratteli, Trans.Am.Math.Soc.171, 195–234 (1972; Zbl 0264.46057)] which are a graphical way of describing inclusions. The Bratteli diagram records the ranks of minimal projections in the simple components of the bigger one. In some cases, stacking successive Bratelli diagams on top of each other leads to II\(_{1} \)-factors as inductive limits of finite-dimensional ones (e.g., Jones–Temperley–Lieb). Principal graphs were constructed from Bratteli diagrams by A.Ocneanu. The mirror image of the previous subfactor inclusion is always part of the next inclusion and, by throwing away reflections in the diagrams, one gets the principal graph, and this provides the same information. If \([M:N] < 4\), the principal graph must be an indecomposable affine A, D, or E Coxeter graph [see F.M.Goodman, P.de la Harpe and V.F.R.Jones, loc.cit.].

The planar algebras of V.F.R.Jones [in:Knots in Hellas ’98, World Scientific Ser.Knots Everything 24, 94–117 (2000; Zbl 1021.46047)] are algebras over the vector space generated by the \(P_{k}\) and lead to a more comprehensive kind of invariant for the subfactor structure. They present an algebraic-combinatorial interpretation of the standard invariant and can be considered as a geometric topological axiomatisation of Popa’s \(\lambda\)-lattice structure. The algebra may be represented graphically as an algebra over an operad of tangles. It forms a pivotal category [see, e.g., S.Abramsky, in: “Mathematics of quantum computation and quantum technology” CRC applied mathematics and nonlinear science series 14, 515–558 (2008; Zbl 1135.81006)]. The algebras are graphically represented like multi-coloured discs and subdiscs in the plane with strings cutting an even number of points on the discs, and of course an operation of composition. Planar signifies that the strings are to be non-crossing. From the viewpoint of planar algebras, the simplest of structures for inclusions of intermediate factors are the Fuss–Catalan algebras.

A quadrilateral of subfactors consists of \(N \subset P \subset M\), \(N \subset Q \subset M\) where \(M\) is generated by \(P,Q\) and \(N = P \cap Q\), and will be assumed to be irreducible. The quadrilateral is called a commuting square if the conditional expectations of \(P\) and \(Q\) commute. The quadrilateral can be considered as a dual object, under co-multiplication, to a commuting square. The quadrilateral is called co-commutative if the commutants of the factors form a commuting square so it can be considered as a dual object to a commuting square.

K.Gustafson [Bull.Am.Math.Soc.74, 488–492 (1968; Zbl 0172.40702)], dealing with positive non-commutative selfadjoint operators on a Hilbert space, has defined an angle of an operator as the least upper bound on the angle by which the operator can rotate any unit vector. T.Sano and Y.Watatani [Proceedings of ICM satellite conference, Nara, 1990, 72–82 (1991; Zbl 0821.46079)] considered the relative positions of four subspaces of a Hilbert space [cf.P.R.Halmos, Trans.Am.Math.Soc.144, 381–389 (1969; Zbl 0187.05503)]. They used Gustafson’s idea to define the angle at a vertex between two subfactors of \(M\) as the spectrum of the angle operator at the vertex. The trigonometric functions are expressed in terms of the conditional expectations. A quadrilateral is a commuting square if and only if the angles are all right-angles.

The angle operators between two subspaces in a Hilbert space can have any closed subset of \((0,{\pi \over 2}]\) as spectrum. A major problem is to quantize the angle operators for intermediate subfactors, in the sense that they take a countable discrete set of values in the interval \((0, {\pi \over 2}] \). The main theorem of this paper is an initial attempt at quantization.

Let \(G\) be a Coxeter group. For finite von Neumann factors \(A_{0} \subset A_{1}\), one constructs a tower over \(A_{0}\) where the \(B_{i}\) are subalgebras of \(A_{i}\) generated by Jones’ projections \(e_{1} \dots e_{i-1}\). Let \(r\) denote the selected vertex for the Bratteli diagram. The subfactor \(rB \subset rAr\) is called the GHJ subfactor for \(G\). The diagrams are bipartite graphs in the sense that there are two colours; black (odd) and white (even) vertices occur in alternating rows and no edges join a black to a white (apartheid?) or a white to a white. In their main theorem, the authors consider the GHQ subfactors for two Coxeter groups, the symmetric group \(S_{3}\), and the group \(D_{5}\) with selected vertex the trivalent vertex, incident to three edges.

For their main theorem, they assume that there is no extra structure and classify three cases where the angle operator takes a finite number of values which can be determined. When \([M:N] = 6\), \(N\) is the fixed point algebra for the outer action of \(S_{3}\) on \(M\) and the angle is \(\pi \over 3\). When \([M:N] = 6 + 4 \sqrt{2}\), the quadrilateral can be associated with a GHQ subfactor associated to the Coxeter group \(D_{5}\); here, the angle is \(\arccos (\sqrt{2} -1\)). Otherwise, the expectations onto \(P\) and \(Q\) commute and the quadrilateral is a commutative square, thus both commutative and co-commutative.

Reviewer: Aubrey Wulfsohn (Coventry)

### MSC:

46L37 | Subfactors and their classification |

### Keywords:

von Neumann II\(_1\)-factors; intermediate subfactors; index; standard invariant; commuting square; quadrilateral; angle operator; Bratelli diagram; planar algebra; Coxeter group; Coxeter graph; Coxeter-Dynkin diagram### Citations:

Zbl 0508.46040; Zbl 0698.46050; Zbl 0814.46053; Zbl 0821.46076; Zbl 0831.46069; Zbl 0211.56703; Zbl 0891.46035; Zbl 1055.46511; Zbl 0264.46057; Zbl 0172.40702; Zbl 0821.46079; Zbl 0187.05503; Zbl 1021.46047; Zbl 0696.46048; Zbl 1135.81006; ERAM 027.0790cj
PDFBibTeX
XMLCite

\textit{P. Grossman} and \textit{V. F. R. Jones}, J. Am. Math. Soc. 20, No. 1, 219--265 (2007; Zbl 1131.46041)

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