Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 363-374.

A note on the Kadison--Singer problem

Authors: Charles A. Akemann (1), Betul Tanbay (2), and Ali Ulger (3)
Author institution: (1) Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
(2) Department of Mathematics, Bogazici University , 34342 Istanbul, Turkey
(3) Department of Mathematics, Koc University, 34450 Sariyer-Istanbul,Turkey

Summary: Let

$H$ be a separable Hilbert space with a fixed orthonormal basis

$(e_{n})_{n\geqslant 1}$ and

$B(H)$ be the full von Neumann algebra of the bounded linear operators

$T:H\rightarrow H$ . Identifying

$\ell ^{\infty}=C(\beta N)$ with the diagonal operators, we consider

$C(\beta N)$ as a subalgebra of

$B(H)$ . For each

$t\in \beta N$ , let

$[\delta_{t}]$ be the \textit{set} of the states of

$B(H)$ that extend the Dirac measure

$\delta _{t}$ . Our main result shows that, for each

$t$ in

$\beta N$ , the set

$[\delta _{t}]$ either lies in a finite dimensional subspace of

$B(H)^*$ or else it must contain a homeomorphic copy of

$\beta N$ .

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