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 $\beta N$.

Keywords: Pure state extension, Kadison--Singer problem

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