Weak Paveability and the Kadison-Singer Problem

• Main Authors: Akemann, Charles A, Anderson, Joel, Tanbay, Betul
• Format: Journal Article
• Language: English
• Published: 13-03-2012
• Subjects: Mathematics - Operator Algebras
• DOI: 10.48550/arxiv.1203.2854

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.