Kadison–Singer problem

Builds on State on a C*-algebra Hilbert space

Problem

Let $H = \ell^2(\N)$ be the Hilbert space of square-summable sequences, and let $B(H)$ be the C*-algebra of bounded operators on $H$ . Let $D \subseteq B(H)$ be the C*-subalgebra of diagonal operators with respect to the standard basis, which is isomorphic to $\ell^\infty(\N)$ . A pure state on a C*-algebra is a state that is not a proper convex combination of two other states. Does every pure state on $D$ extend to a unique state on $B(H)$ ?

Remarks

Every state on $D$ extends to some state on $B(H)$ by the Hahn–Banach theorem, so the question is about uniqueness. Kadison and Singer posed the question in their 1959 paper "Extensions of pure states". Over the following decades it was shown to be equivalent to many problems in other fields, among them Anderson's paving conjecture, the Feichtinger conjecture on frames, and Weaver's discrepancy conjecture. Adam Marcus, Daniel Spielman and Nikhil Srivastava proved Weaver's conjecture, and so solved the problem with a positive answer; their proof was posted in 2013 and published in 2015. Their method uses interlacing families of polynomials.

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Tag	`C8TTRMVVS7`
URI	`m2://C8TTRMVVS7`
Kind	problem
Writing	W0 stub
MSC	46L30, 46L05, 42C15
Status	This copy of the site has no build data. Run `m2 status C8TTRMVVS7`.
About	State on a C*-algebra
Uses	State on a C*-algebra, Hilbert space
Same as	Wikidata Q25304456 Wikipedia Kadison–Singer problem
Changed	2 November 2026

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