154 research outputs found
Semi-regular masas of transfinite length
In 1965 Tauer produced a countably infinite family of semi-regular masas in
the hyperfinite 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 , finding
masas for which this procedure terminates at the -th stage. Such masas are
said to have length . 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
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
that are -nearly satisfying the
Parseval's condition and the equal norm condition, is it close to a set of
vectors that exactly satisfy the Parseval's
condition and the equal norm condition? Given , the squared
distance (to the set of exact solutions) is defined as where the infimum is over the set of exact solutions.
Previous results show that the squared distance of any -nearly
solution is at most and there are
-nearly solutions with squared distance at least .
The fundamental open question is whether the squared distance can be
independent of the number of vectors .
We answer this question affirmatively by proving that the squared distance of
any -nearly solution is . 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 -nearly solution is . Then, we show that by randomly perturbing the input vectors, the
dynamical system will converge faster and the squared distance of an
-nearly solution is when is large enough
and 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
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