In recent joint work with V. Jones and D. Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans-Kawahigashi.
A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A_2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M_infinity-M_infinity bimodules are described by certain orbifolds (with gho…
Given a finite-index and finite-depth subfactor, we define the notion of \textit{quantum double inclusion} - a certain unital inclusion of von Neumann algebras constructed from the given subfactor - which is closely related to that of Ocneanu's asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor $R^H \subset R$ produces Drinfeld double of $H$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H \subset …
We define a canonical relative commutant planar algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.
We consider inclusions of type $(P\otimes A)^G\subset(P\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\subset B$. In the case $[A,B]=0$, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form $P(A\subset B)^G$, with $G$ acting in some natural sense on the bipartite graph algebra $P(A\subset B)$.
We show a close relationship between non-degenerate smooth commuting squares of $II_1$-factors with all inclusions of finite index and inclusions of subfactor planar algebras by showing that each leads to a construction of the other. One direction of this uses the Guionnet-Jones-Shlyakhtenko construction.
Let $\mathcal{N}\subset\mathcal{M}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-{$C^*$}-categorical/planar algebraic description of normal faithful conditional expectations $E:\mathcal{M}\to\mathcal{N}\subset\mathcal{M}$ with finite index and their duals $E':\mathcal{N}'\to\mathcal{M}'\subset\mathcal{N}'$ by means of the solutions of the conjugate equations for the inclusion morphism $ι:\mathcal{N}\to\mathcal{M}$ and its conjugate morphism $\overlineι:\mathcal{M}\to\mathcal{N}$. In particular, the theory of index for conditional expectations admits a 2-{$C^*$}-categori…
Given a subfactor planar algebra P and a Hilbert P-module of lowest weight 0 we build a bimodule over the symmetric enveloping inclusion associated to P. As an application we prove diagrammatically that the Temperley-Lieb-Jones standard invariants have the Haagerup property. This provides a new proof of a result due to Popa and Vaes.
Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.
An inclusion of II$_1$ factors $N \subset M$ with finite Jones index gives rise to a powerful set of invariants that can be approached successfully in a number of different ways. We describe Jones' pictorial description of the standard invariant of a subfactor as a so-called planar algebra and show how this point of view leads to new structure results for subfactors.
We define annular algebras for rigid $C^{*}$-tensor categories, providing a unified framework for both Ocneanu's tube algebra and Jones' affine annular category of a planar algebra. We study the representation theory of annular algebras, and show that all sufficiently large (full) annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set, hence have equivalent representation theories. Annular algebras admit a universal $C^{*}$-algebra closure analogous to the universal $C^{*}$-algebra for groups. These algebras have interesting cor…
We study 2-cabled analogs of Voiculescu's trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable definition, finiteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-$Γ$ non-$L^2$ rigid factors.
In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion} associated to a finite-index and finite-depth subfactor and studied the quantum double inclusion associated to the Kac algebra subfactor $R^H \subset R$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. In this article we analyse quantum double inclusions associated to the family of Kac algebra subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m$ times}} : m \geq 1 \}$. For each $…
To a weighted graph can be associated a bipartite graph planar algebra P. We construct and study the symmetric enveloping inclusion of P. We show that this construction is equivariant with respect to the automorphism group of P. The automorphism group of the weighted graph acts on P. We consider subgroups G of the automorphism group of the weighted graph such that the G-fixed point space P^G is a subfactor planar algebra. As an application we show that if G is amenable, then P^G is amenable as a subfactor planar algebra. We define the notions of a cocycle action of a Hecke pair on a tracial vo…
There is a natural construction which associates to a finitely generated, countable, discrete group $G$ and a 3-cocycle $ω$ of $G$ an inclusion of II$_1$ factors, the so-called diagonal subfactors (with cocycle). In the case when the cocycle is trivial these subfactors are well studied and their standard invariant (or planar algebra) is known. We give a description of the planar algebra of these subfactors when a cocycle is present. The action of Jones' planar operad involves the 3-cocycle $ω$ explicitly and some interesting identities for 3-cocycles appear when naturality of the action is ver…
We present more planar algebraic construction of subfactors than those of Guionet-Jones-Shlyakhtenko-Walker and Kodiyalam-Sunder which start from a subfactor planar algebra and give in a direct way a subfactor of the same standard invariant with the planar algebra. Our construction is based on using the ordinary concepts in planar algebras such as involution, inclusion and conditional expectation mappings as it is
Subfactors of the hyperfinite II$_1$ factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index $\frac{5+\sqrt{13}}{2}$, i.e. the index of the Haage…
Given a finite depth subfactor planar algebra $\mathcal{P}$ endowed with the graded $*$-algebra structures $\{Gr_k^+ \mathcal{P}\}_{k\in\mathbb{N}}$ of Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces $Tr_{k,+}$ on $Gr_k^+\mathcal{P}$ induced by the Temperley-Lieb diagrams and a sequence of trace-preserving embeddings into the bounded operators on a Hilbert space. Via these embeddings the $*$-algebras $\{Gr_k^+\mathcal{P}\}_{k\in \mathbb{N}}$ generate a tower of non-commutative probability spaces $\{M_{k,+}\}_{k\in\mathbb{N}}$ whose inclusions recover $\mathcal{P}$ as…
Using an analogue of the Guionnet-Jones-Shlaykhtenko construction for graphs we show that their construction applied to any subfactor planar algebra of finite depth yields an inclusion of interpolated free group factors with finite parameter, thereby giving another proof of their universality for finite depth planar algebras.
Let $(Γ,μ)$ be a bipartite graph together with a weight on its vertices. Assume that $μ$ is an eigenvector for the adjacency matrix of $Γ$. Let Aut$(Γ, μ)$ be the automorphism group of the bipartite graph $Γ$ that scales the weight $μ$. It is a locally compact totally disconnected group that acts on the bipartite graph planar algebra $P$ associated to $(Γ,μ)$. Consider a subgroup G < Aut$(Γ, μ)$ and the set of fixed points $P^G \subset P$ that we assume to be a subfactor planar algebra. If the closure of G inside Aut$(Γ, μ)$ satisfies an approximation property such as amenability, the Haagerup…
By a construction of Vaughan Jones, the bipartite graph $Γ(A)$ associated with the natural inclusion of $\mathbb C$ inside a finite-dimensional $C^*$-algebra $A$ gives rise to a planar algebra $\mathcal P^{Γ(A)}$. We prove that every subfactor planar subalgebra of $\mathcal P^{Γ(A)}$ is the fixed point planar algebra of a uniquely determined action of a compact quantum group $\mathbb G$ on $A$. We use this result to introduce a conceptual framework for the free wreath product operation on compact quantum groups in the language of planar algebras/standard invariants of subfactors. Our approach …
