## Advanced Studies in Pure Mathematics

### Relative positions of four subspaces in a Hilbert space and subfactors

Yasuo Watatani

#### 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 $\{\frac{n}{3};\ n \in \mathbf{Z}\}$.

#### Article information

Dates
First available in Project Euclid: 1 January 2019

https://projecteuclid.org/ euclid.aspm/1546368841

Digital Object Identifier
doi:10.2969/aspm/03810319

Mathematical Reviews number (MathSciNet)
MR2059817

Zentralblatt MATH identifier
1065.46019

#### Citation

Watatani, Yasuo. Relative positions of four subspaces in a Hilbert space and subfactors. Operator Algebras and Applications, 319--328, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/03810319. https://projecteuclid.org/euclid.aspm/1546368841