Journal of Operator Theory
Volume 63, Issue 2, Spring 2010 pp. 363-374.
A note on the Kadison--Singer problemAuthors: 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 (en)n⩾ 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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