Existence of an infinite-index II_1-subfactor with the GICAR (planar rook) standard invariant

For a finite-index subfactor of II_1 factors, the standard invariant always contains the Temperley-Lieb tower, and Popa (1993) showed this minimal invariant is realized at every index greater than 4. Jones and Penneys (2015) asked the infinite-index analogue — "what is the simplest possible standard invariant?" — and proved that the centralizer algebras $Q_n$ of an arbitrary-index II_1-subfactor (or II_1-factor bimodule) always contain a faithful copy of the GICAR algebras (the fermion, or planar rook, algebras, of dimension $\sum_k\binom{n}{k}^2$), so the GICAR tower is a universal lower bound. They constructed a II_1-factor bimodule attaining this bound exactly, with $Q_n$ equal to the planar rook algebra $G_n$ and one-dimensional spaces of central $L^2$-vectors, but it does not come from a subfactor. The open problem: does there exist an infinite-index II_1-subfactor $A\subset B$ whose standard invariant is exactly the GICAR (planar rook) tower — equivalently, such a bimodule of the form $L^2(B)\ominus L^2(A)$ — or does an obstruction forbid it? The authors also ask whether such a bimodule can be chosen symmetric. The authors state they will "continue to search for an example ... or to attempt to show no such example exists"; no construction or nonexistence theorem is known to date.

Background

A factor is a von Neumann algebra (a weakly closed $*$-algebra of bounded operators on a Hilbert space) with trivial center, and a $II_1$ factor is an infinite-dimensional factor carrying a faithful finite normal trace. A subfactor is an inclusion $A\subset B$ of $II_1$ factors. Its Jones index $[B:A]$ measures the size of $B$ as a module over $A$; Jones proved that the index can only take the values $4\cos^2(\pi/n)$ for integers $n\ge 3$, or values in the interval $[4,\infty]$, with $[B:A]=\infty$ being the infinite index case.

For finite index, the basic construction adjoins the orthogonal projection $e_1$ from $L^2(B)$ onto $L^2(A)$ and iterates to give the Jones tower $A=A_0\subset A_1=B\subset A_2\subset\cdots$ of $II_1$ factors; the resulting Jones projections $(e_n)$ satisfy the defining relations of the Temperley-Lieb algebra $TL_n$, the diagram algebra spanned by non-crossing pairings of $2n$ boundary points. The standard invariant — the two towers of finite-dimensional centralizer algebras (higher relative commutants) $A_i'\cap A_j$ with their inclusion and trace structure — is the central object of finite-index subfactor theory and has three classical equivalent axiomatizations (paragroups, $\lambda$-lattices, and planar algebras, the last a diagrammatic algebra acted on by shaded planar tangles). Because the Jones projections always lie in the centralizer algebras, every finite-index standard invariant contains a canonical copy of the Temperley-Lieb tower: the index-at-most-4 subfactors constructed by Jones have exactly this invariant and no more, and Popa proved that for every index greater than 4 there is a (non-hyperfinite) subfactor whose standard invariant is Temperley-Lieb only. "Temperley-Lieb only" is thus the simplest possible finite-index standard invariant, and it is realizable at every allowed index; very recently Bisch-Cáceres produced the first hyperfinite such examples with integer index greater than 4.

For infinite index the picture changes structurally. The Jones tower still exists, but $A_n$ is a type $II_\infty$ factor (a factor whose semifinite trace takes the value $+\infty$ on the identity) for all $n\ge 2$, and Burns observed that the centralizer algebras $A_0'\cap A_n$ and the central $L^2$-vectors $A_0'\cap L^2(A_n)$ no longer coincide. Penneys built a planar calculus for $II_1$-factor bimodules of arbitrary index, where a Hilbert bimodule ${}_AH_A$ is a Hilbert space with commuting normal left and right actions of $A$. For a subfactor $A\subset B$ the relevant bimodule is $H=L^2(B)$ with its distinguished $A$-central vector $\hat{1}$, and one defines $Q_n=A'\cap(A^{op})'\cap\mathcal{B}(H^{\otimes_A n})=\mathrm{End}_{A-A}(H^{\otimes_A n}),\qquad P_n=A'\cap H^{\otimes_A n},$ where $H^{\otimes_A n}$ is the Connes relative tensor product over $A$ and $A^{op}$ is $A$ with reversed multiplication acting on the right; $Q_n$ consists of the $A$-$A$ bilinear endomorphisms of $H^{\otimes_A n}$ and $P_n$ of its $A$-central vectors. Jones-Penneys propose the pair of towers $(Q_\bullet,P_\bullet)$ with their compatible planar actions as the standard invariant of an infinite-index subfactor or $II_1$-factor bimodule; in the subfactor case $Q_n$ identifies with the centralizer algebra $A_0'\cap A_{2n}$.

The GICAR algebra — for gauge-invariant canonical anticommutation relations, also called the fermion algebra — is the $C^*$-algebra generated by the creation and annihilation operators on fermionic Fock space $\mathcal{F}(\mathcal{H})=\bigoplus_{n\ge0}\Lambda^n \mathcal{H}$ over a one-particle Hilbert space $\mathcal{H}$ (the antisymmetric tensor powers $\Lambda^n \mathcal{H}$ describe $n$ identical fermions). For a one-particle space $\mathcal{H}_n$ of dimension $n$, $\mathrm{GICAR}(\mathcal{H}_n)\cong\bigoplus_{k=0}^n M_{\binom{n}{k}}(\mathbb{C})$ has dimension $\sum_{k=0}^n\binom{n}{k}^2$ and its Bratteli diagram (the graph recording how the simple matrix-algebra summands include into one another along a tower) is Pascal's triangle; diagrammatically this tower is the planar rook algebra, spanned by diagrams of partial permutations (equivalently, placements of rooks on a square board). Connes and Evans found these algebras embedded in the Temperley-Lieb algebras of finite-index subfactors. Jones-Penneys prove the arbitrary-index analogue: because the odd trace-preserving operator-valued weights (the analogues of conditional expectations from a larger algebra onto a smaller one, which at infinite index need not be normalized) in the infinite-index Jones tower are conditional expectations (a theorem of Burns), odd Jones projections exist in every $Q_n$, and together with the partial isometries witnessing their Murray-von Neumann equivalence (equivalence of projections mediated by a partial isometry) they generate a faithful copy of $\mathrm{GICAR}(\mathcal{H}_n)$ inside $Q_n$. Consequently $Q_n$ is nonabelian for $n\ge 2$ and $\dim Q_n\ge\sum_k\binom{n}{k}^2$, and the rectangular GICAR category $RG$ (the $*$-tensor category generated by the GICAR tower) acts faithfully on the bimodules $H^{\otimes_A n}$. The GICAR/planar-rook tower is therefore a universal lower bound for every infinite-index standard invariant — the infinite-index counterpart of the Temperley-Lieb tower.

This lower bound is sharp for bimodules: for an outer action $\sigma:\mathbb{Z}\to\mathrm{Out}(R)$ of the integers on the hyperfinite $II_1$ factor $R$ (the hyperfinite $II_1$ factor is the one generated by an increasing union of finite-dimensional matrix algebras; outer meaning not implemented by unitary conjugation), the bimodule $K=L^2(R)_\sigma$ has irreducible, pairwise non-isomorphic tensor powers, and for $H=L^2(R)\oplus K$ one gets $\mathrm{End}_{R-R}(H^{\otimes n})=G_n$ and $\dim P_n=1$ for all $n$ — the smallest standard invariant possible. However, this example does not come from a subfactor: at the time of the paper no infinite-index subfactor was known to attain the bound, and the question whether one exists, or whether an obstruction forbids it, is the question posed by the authors.

Problem Statement

Does there exist an inclusion $A\subset B$ of $II_1$ factors with infinite Jones index whose standard invariant, in the sense of Jones-Penneys, is exactly the GICAR (planar rook) tower? Concretely, with $H=L^2(B)$ and the towers $Q_n=\mathrm{End}_{A-A}(H^{\otimes_A n})$ (which identifies with the centralizer algebra $A_0'\cap A_{2n}$ in the subfactor case) and $P_n=A'\cap H^{\otimes_A n}$: is there such an $A\subset B$ for which, for every $n\ge 0$, $Q_n$ equals exactly the image of the GICAR algebra $\mathrm{GICAR}(\mathcal{H}_n)$ (the planar rook algebra $G_n$), and $\dim P_n=1$?

In the bimodule form of Questions 6.6 of the source paper: does there exist a Hilbert bimodule $K$ over a $II_1$ factor $A$ such that, writing $H=L^2(A)\oplus K$, one has $\mathrm{End}_{A-A}(H^{\otimes_A n})=G_n$ and $\dim(A'\cap H^{\otimes_A n})=1$ for all $n\ge 0$, and such that $K$ is either symmetric — admitting a conjugate-linear isomorphism $J$ with $J(a\xi b)=b^*(J\xi)a^*$ and $J^2=\mathrm{id}$ — or of the form $K=L^2(B)\ominus L^2(A)$ for a (necessarily infinite-index) $II_1$-subfactor $A\subset B$? The subfactor-complement branch is exactly the question of the previous paragraph, since $L^2(A)\oplus(L^2(B)\ominus L^2(A))\cong L^2(B)$ as $A$-$A$ Hilbert bimodules; the symmetry branch is a separate sufficient route posed by the authors, which would not by itself produce a subfactor.

Either resolution of the subfactor question is admitted: an explicit construction of such an inclusion $A\subset B$ — equivalently, of such a bimodule of the form $L^2(B)\ominus L^2(A)$ — or a rigorous nonexistence theorem showing that every infinite-index $II_1$-subfactor's standard invariant strictly exceeds the GICAR/planar-rook lower bound at some finite level; a bimodule that is symmetric but not of the form $L^2(B)\ominus L^2(A)$ resolves only the bimodule branch of Questions 6.6, leaving the subfactor question open. Both directions were explicitly contemplated by the authors: "We will continue to search for an example of an infinite index subfactor with the simplest possible standard invariant, or to attempt to show no such example exists."

References

1. Vaughan F. R. Jones and David Penneys, Infinite index subfactors and the GICAR categories, Communications in Mathematical Physics 339 (2015), no. 2, 729-768. DOI: 10.1007/s00220-015-2407-8; arXiv:1410.0856, https://arxiv.org/abs/1410.0856
2. V. F. R. Jones, Index for subfactors, Inventiones Mathematicae 72 (1983), no. 1, 1-25. DOI: 10.1007/BF01389127, https://doi.org/10.1007/BF01389127
3. Sorin Popa, Markov traces on universal Jones algebras and subfactors of finite index, Inventiones Mathematicae 111 (1993), 375-405. DOI: 10.1007/BF01231293, https://doi.org/10.1007/BF01231293
4. Alain Connes and David E. Evans, Embeddings of U(1)-current algebras in non-commutative algebras of classical statistical mechanics, Communications in Mathematical Physics 121 (1989), no. 3, 507-525. DOI: 10.1007/BF01217736, https://doi.org/10.1007/BF01217736
5. Michael Burns, Subfactors, Planar Algebras and Rotations (2011), arXiv:1111.1362, https://arxiv.org/abs/1111.1362
6. David Penneys, A planar calculus for infinite index subfactors (2011), arXiv:1110.3504, https://arxiv.org/abs/1110.3504
7. Arthur Bartels, Christopher L. Douglas, and André Henriques, Dualizability and index of subfactors, Quantum Topology 5 (2014), 289-345 (preprint 2011, revised 2014). arXiv:1110.5671, https://arxiv.org/abs/1110.5671
8. Ryosuke Sato, GICAR algebras and dynamics on determinantal point processes: discrete orthogonal polynomial ensemble case (2023), arXiv:2309.01117, https://arxiv.org/abs/2309.01117
9. Dietmar Bisch and Julio Cáceres, New hyperfinite subfactors with infinite depth (2025), arXiv:2509.00227, https://arxiv.org/abs/2509.00227