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Journal of Operator Theory

Volume 71, Issue 1, Winter 2014  pp. 295-300.

Weak paveability and the Kadison-Singer problem

Authors:  Charles A. Akemann (1), Joel Anderson (2), and Betul Tanbay (3)
Author institution: (1) Department of Mathematics, University of\break California, Santa Barbara, CA 93106, U.S.A.
(2) Department of Mathematics, Penn State University, PA 16802, U.S.A.
(3) Department of Mathematics, Bogazici University , 34342 Istanbul, Turkey

Summary:  The Kadison-Singer problem (hereinafter K-S) began with a problem in R.V. Kadison and I.M. Singer, \textit{Amer. J. Math.} \textbf{81}(1959), 383--400, and has since expanded to a very large number of equivalent problems in various fields (see P.G. Casazza and D. Edidin, \textit{Equivalents of the Kadison-Singer Problem. Function Spaces}, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI 2007, pp. 123--142, for an extensive discussion). In the present paper we will introduce the notion of \textit{weak paveability} for positive elements of a von Neumann algebra $M$. This new formulation implies the traditional version of paveability as in J. Anderson, Extensions, restrictions, and representations of states on $C^*$-algebras, \textit{Trans. Amer. Math. Soc.} \textbf{249}(1979), 303--329, 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.

Keywords:  pure state extension, Kadison-Singer problem

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