Relative positions of four subspaces in a Hilbert space and subfactors

Abstract

We study relative positions of four subspaces in a Hilbert space. Gelfand-Ponomarev gave a complete classification of indecomposable systems of four subspaces in a finite-dimensional space. In this note we show that there exist uncountably many indecomposable systems of four subspaces in an infinite-dimesional Hilbert space. We extend a numerical invariant, called defect, for a certain class of systems of four subspaces using Fredholm index. We show that the set of possible values of the defect is $\{\frac{n}{3};\ n \in \mathbf{Z}\}$.