Weak Paveability and the Kadison-Singer Problem

TL;DR

This paper introduces the concept of weak paveability in von Neumann algebras, linking it to the Kadison-Singer problem, and shows its potential as an easier approach to resolving the problem.

Contribution

It defines weak paveability for positive elements, proves its density and openness, and relates it to the Kadison-Singer problem, suggesting a new pathway for its resolution.

Findings

Weakly paveable positive elements form an open, dense set in the algebra.

Affirming K-S reduces to showing projections with compact diagonal are weakly paveable.

Weak paveability may either contain a counterexample or serve as an easier route to prove K-S.

Abstract

The Kadison-Singer Problem (K-S) has expanded since 1959 to a very large number of equivalent problems in various fields. In the present paper we will introduce the notion of weak paveability for positive elements of a von Neumann algebra M. This new formulation implies the traditional version of paveability iff K-S is affirmed. We show that the set of weakly paveable positive elements of $M^+$ is open and norm dense in $M^+$. Finally, we show that to affirm K-S it suffices to show that projections with compact diagonal are weakly paveable. Therefore weakly paveable matrices will either contain a counterexample, or else weak paveability must be an easier route to affirming K-S.