WEAK PAVEABILITY AND THE KADISON-SINGER PROBLEM

The Kadison-Singer problem (hereinafter K-S) began with a problem in [5] and has since expanded to a very large number of equivalent problems in various fields (see [4] for an extensive discussion). 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 [3] if and only if K-S is affirmed (see definitions below). 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.