Semi-regular masas of transfinite length

In 1965 Tauer produced a countably infinite family of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each $n\in\mathbb N$, finding masas for which this procedure terminates at the $n$-th stage. Such masas are said to have length $n$. In this paper we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalising tower.Comment: 14 page

The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \ldots, u_n \in \mathbb R^d$ that are $\epsilon$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, \ldots, v_n \in \mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1, \ldots, u_n$, the squared distance (to the set of exact solutions) is defined as $\inf_{v} \sum_{i=1}^n \| u_i - v_i \|_2^2$ where the infimum is over the set of exact solutions. Previous results show that the squared distance of any $\epsilon$-nearly solution is at most $O({\rm{poly}}(d,n,\epsilon))$ and there are $\epsilon$-nearly solutions with squared distance at least $\Omega(d\epsilon)$. The fundamental open question is whether the squared distance can be independent of the number of vectors $n$. We answer this question affirmatively by proving that the squared distance of any $\epsilon$-nearly solution is $O(d^{13/2} \epsilon)$. Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any $\epsilon$-nearly solution is $O(d^2 n \epsilon)$. Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an $\epsilon$-nearly solution is $O(d^{5/2} \epsilon)$ when $n$ is large enough and $\epsilon$ is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor changes in various place

On stable finiteness of group rings

For an arbitrary field or division ring K and an arbitrary group G, stable finiteness of K[G] is equivalent to direct finiteness of K[G×H] for all finite groups H

Exactness of the Fock space representation of the q-commutation relations

We show that for all q in the interval (-1,1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (-1,1).Comment: 20 page

Isomorphisms of Cayley graphs of surface groups

A combinatorial proof is given for the fact that the Cayley graph of the fundamental group Γg of the closed, orientable surface of genus g ≥ 2 with respect to the usual generating set is isomorphic to the Cayley graph of a certain Coxeter group generated by 4g elements