A Note On The Kadison-Singer Problem

TL;DR

This paper investigates the structure of state extensions in the von Neumann algebra of bounded operators on a separable Hilbert space, revealing a dichotomy related to the Kadison-Singer problem and the topology of the Stone-Čech compactification.

Contribution

It establishes a new dichotomy for the sets of state extensions associated with points in eta N, connecting finite dimensionality with topological complexity in the context of the Kadison-Singer problem.

Findings

Each set of state extensions is either finite dimensional or contains a homeomorphic copy of eta N.

The result links the structure of state extensions to topological properties of eta N.

Provides insight into the nature of the Kadison-Singer problem through operator algebra topology.

Abstract

Let H be a separable Hilbert space with a fixed orthonormal basis (e_n), n>=1, and B(H) be the full von Neumann algebra of the bounded linear operators T: H -> H. Identifying l^\infty = C(\beta N) with the diagonal operators, we consider C(\beta N) as a subalgebra of B(H). For each t in \beta N, let [\delta_t] be the set of the states of B(H) that extend the Dirac measure \delta_t. Our main result shows that, for each t in \beta N, this set either lies in a finite dimensional subspace of B(H)* or else it must contain a homeomorphic copy of \beta N.