Related papers: A planar calculus for infinite index subfactors

Generalizing Jones's notion of a planar algebra, we have previously introduced an A_2-planar algebra capturing the structure contained in the double complex pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system.…

We present a purely planar algebraic proof of the main result of a paper of Guionnet-Jones-Shlaykhtenko which constructs an extremal subfactor from a subfactor planar algebra whose standard invariant is given by that planar algebra.

For any finite dimensional C^*-algebra A, we give an endomorphism \Phi of the hyperfinite II_1 factor R of finite Jones index such that: for all k \in \mathbb {N}, \Phi^k (R)' \cap R= \otimes^k A. The Jones index [R: \Phi (R)]= (rank…

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the…

We prove that every rigid C*-bicategory with finite-dimensional centers (finitely decomposable horizontal units) can be realized as Connes' bimodules over finite direct sums of II$_1$ factors. In particular, we realize every multitensor…

We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a…

In this paper we consider a semigroup of completely positive maps $\tau=(\tau_t,t \ge 0)$ with a faithful normal invariant state $\phi$ on a type-$II_1$ factor $\cla_0$ and propose an index theory. We :achieve this via a more general…

We construct inclusions of the form $(B_0\otimes P)^G\subset (B_1\otimes P)^G$, where $G$ is a compact quantum group of Kac type acting on an inclusion of finite dimensional $\c^*$-algebras $B_0\subset B_1$ and on a $II_1$ factor $P$. Under…

We examine the notion of $\alpha$-strong singularity for subfactors of a \IIi factor, which is a metric quantity that relates the distance between a unitary in the factor and a subalgebra with the distance between that subalgebra and its…

We study II_1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result: every finite index M-N-bimodule (in…

The notion of index for arbitrary real factors is introduced and investigated. The main tool in our approach is the reduction of real factors to involutive *-anti-automorphisms of their complex enveloping von Neumann algebras. Similar to…

We give the classification of subfactor planar algebras at index exactly 5. All the examples arise as standard invariants of subgroup subfactors. Some of the requisite uniqueness results come from work of Izumi in preparation. The…

In this paper, we explicitly work out the subfactor planar algebra $P^{(N \subset Q)}$ for an intermediate subfactor $N \subset Q \subset M$ of an irreducible subfactor $N \subset M$ of finite index. We do this in terms of the subfactor…

We prove that a regular subfator of type $II_1$ with finite Jones index always admits a two-sided Pimsner-Popa basis. This is preceeded by a pragmatic revisit of Popa's notion of orthogonal systems.

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…

We call a subfactor trivial if it is isomorphic with the obvious inclusion of N into matrices over N. We prove the existence of type II_1 factors M without non-trivial finite index subfactors. Equivalently, every M-M-bimodule with finite…

Guionnet et al. gave a construction of a II_1 factor associated to a subfactor planar algebra. In this paper we define an unshaded planar algebra. To any unshaded planar algebra P we associate a finite von Neumann algebra M_P. We prove that…

We prove that any separable II$_1$ factor $M$ admits a {\it coarse decomposition} over the hyperfinite II$_1$ factor $R$, i.e., there exists an embedding $R\hookrightarrow M$ such that $L^2M\ominus L^2R$ is a multiple of the coarse Hilbert…

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…

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…