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Partly inspired by quantum mechanics, in 1959 Richard Kadison and Isadore Singer studied the possible uniqueness of extensions of certain functionals (i.e. pure states) on commutative operator algebras on Hilbert space. This hinges on two key examples: for the first they proved lack of uniqueness, but for the second they left the question open (“We incline to the view that such extension is non-unique”). This problem was subsequently related to various other areas of mathematics, such as linear algebra and probability theory. In 2013 Adam Marcus, Daniel Spielman and Nikhil Srivastava finally proved that the answer for the open case was actually positive, for which they received the 2014 Pólya Prize. In this article, Klaas Landsman and Marco Stevens discuss the conjecture (and its proof ) in the light of a more general question that Kadison and Singer had in mind.