In mathematics, the Kadison–Singer problem, posed in 1959, was a problem in functional analysis about whether certain extensions of certain linear functionals on certain C*-algebras were unique. The uniqueness was proven in 2013.
The statement arose from work on the foundations of quantum mechanics done by Paul Dirac in the 1940s and was formalized in 1959 by Richard Kadison and Isadore Singer. The problem was subsequently shown to be equivalent to numerous open problems in pure mathematics, applied mathematics, engineering and computer science. Kadison, Singer, and most later authors believed the statement to be false, but, in 2013, it was proven true by Adam Marcus, Daniel Spielman and Nikhil Srivastava, who received the 2014 Pólya Prize for the achievement.
The solution was made possible by a reformulation provided by Joel Anderson, who showed in 1979 that his "paving conjecture", which only involves operators on finite-dimensional Hilbert spaces, is equivalent to the Kadison–Singer problem. Nik Weaver provided another reformulation in a finite-dimensional setting, and this version was proved true using random polynomials.
Consider the separable Hilbert space ℓ2 and two related C*-algebras: the algebra of all continuous linear operators from ℓ2 to ℓ2, and the algebra of all diagonal continuous linear operators from ℓ2 to ℓ2.
A state on a C*-algebra is a continuous linear functional such that (where denotes the algebra's multiplicative identity) and for every . Such a state is called pure if it is an extremal point in the set of all states on (i.e. if it cannot be written as a convex combination of other states on ).
By the Hahn-Banach theorem, any functional on can be extended to . Kadison and Singer conjectured that, for the case of pure states, this extension is unique. That is, the Kadison–Singer problem consisted in proving or disproving the following statement:
- to every pure state on there exists a unique state on that extends .
This claim is in fact true.
Paving conjecture reformulation
The Kadison–Singer problem has a positive solution if and only if the following "paving conjecture" is true:
- For every there exists a natural number so that the following holds: for every and every linear operator on the -dimensional Hilbert space with zeros on the diagonal there exists a partition of into sets such that
Here denotes the orthogonal projection on the space spanned by the standard unit vectors corresponding to the elements of , so that the matrix of is obtained from the matrix of by replacing all rows and columns that don't correspond to the indices in by 0. The matrix norm is the spectral norm, i.e. the operator norm with respect to the Euclidean norm on .
Note that in this statement, may only depend on , not on .
Equivalent discrepancy statement
The following "discrepancy" statement, again equivalent to the Kadison–Singer problem because of previous work by Nik Weaver, was proven by Marcus/Spielman/Srivastava using a technique of random polynomials:
- Suppose vectors are given with (the identity matrix) and for all . Then there exists a partition of into two sets and such that
This statement implies the following:
- Suppose vectors are given with for all and
- Then there exists a partition of into two sets and such that, for :
Here the "discrepancy" becomes visible when α is small enough: the quadratic form on the unit sphere can be split into two roughly equal pieces, i.e. pieces whose values don't differ much from 1/2 on the unit sphere. In this form, the theorem can be used to derive statements about certain partition of graphs.
- Kadison, R.; Singer, I. (1959). "Extensions of pure states". American Journal of Mathematics. 81 (2): 383–400. doi:10.2307/2372748. JSTOR 2372748. MR 0123922.
- Casazza, P. G.; Fickus, M.; Tremain, J. C.; Weber, E. (2006). "The Kadison–Singer problem in mathematics and engineering: a detailed account". Operator theory, operator algebras, and applications. Contemporary Mathematics. 414. Providence, RI: American Mathematical Society. pp. 299–355. arXiv:math/0510024. doi:10.1090/conm/414/07820. ISBN 9780821839232. MR 2277219.
- Casazza, Peter G. (2015). "Consequences of the Marcus/Spielman/Stivastava solution to the Kadison–Singer Problem". arXiv:1407.4768 [math.FA].
- Marcus, Adam; Spielman, Daniel A.; Srivastava, Nikhil (2013). "Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem". arXiv:1306.3969 [math.CO].
- Srivastava, Nikhil (July 11, 2013). "Discrepancy, Graphs, and the Kadison–Singer Problem". Windows on Theory.
- Anderson, Joel (1979). "Restrictions and representations of states on C∗-algebras". Transactions of the American Mathematical Society. 249 (2): 303–329. doi:10.2307/1998793. JSTOR 1998793. MR 0525675.
- Weaver, Nik (2004). "The Kadison-Singer problem in discrepancy theory". Discrete Mathematics. 278 (1–3): 227–239. doi:10.1016/S0012-365X(03)00253-X.
- Nicholas J. A. Harvey (July 11, 2013). "An introduction to the Kadison–Singer Problem and the Paving Conjecture" (PDF).