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arXiv · 1203.2854

Weak Paveability and the Kadison-Singer Problem

Charles A. Akemann·Joel Anderson·Betul Tanbay
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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.

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Charles A. Akemann, Joel Anderson, Betul Tanbay. 2012-03-13. Weak Paveability and the Kadison-Singer Problem. https://arxiv.org/abs/1203.2854

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