# Quadrilaterals of Factors and Strong Singularity for Subfactors

### Pinhas Grossman (Cardiff)

Tuesday 20th October, 2009 16:00-17:00 214

#### Abstract

A quadrilateral of factors is a square of von Neumann algebras with trivial centers. For a noncommuting irreducible finite-index quadrilateral whose sides are "2-supertransitive'', there are severe restrictions on the possible configurations. In particular, there are exactly seven noncommuting, irreducible quadrilaterals with sides of index less than or equal to four, up to isomorphism of the standard invariant. This is joint work with Masaki Izumi. The notion of $\alpha$-strong singularity, $0 < \alpha \leq 1$, for a subalgebra of a II$_1$ factor, was introduced by Sinclair and Smith as an analytic property which implies singularity in the sense of Dixmier. A MASA turns out to be singular iff it is strongly singular with $\alpha = 1$ ( Sinclair, Smith, White, and Wiggins). For subfactors, however, this is not the case. A singular subfactor which is not $1$-strongly singular may be obtained by embedding the subfactor in a quadrilateral; the angle of the quadrilateral gives a bound on the maximal strong singularity constant. This is joint work with Alan Wiggins.