Kadison-Singer problem

The Kadison-Singer problem by Richard Kadison and Isadore Singer is a problem posed in 1959 in the theory of operator algebras . It asks whether certain extensions of linear functionals to certain operator algebras (certain C * algebras ) are unique. It was solved in 2013 by Adam W. Marcus , Daniel Spielman and Nikhil Srivastava .

Background and story

The problem has its roots in quantum mechanics, in which a quantum state is first characterized for a maximum number of mutually commuting self-adjoint operators (MASA, maximally abelian self adjoint algebra of operators). In quantum mechanics, the operators act on infinite-dimensional vector spaces of functions ( Hilbert spaces ). For this purpose, a probability distribution of the observables belonging to the MASA operators is established (so-called pure state ). The probability distribution is a linear functional of the MASA with certain positivity properties (because of the interpretation as probability). Then the pure state is extended to the other observables that correspond to the non-commuting operators. The question is whether this is clearly possible, as Paul Dirac suggested in his book The Principles of Quantum Mechanics .

The answer depends on what type of MASA is being considered. In the general case (diffuse MASA), Kadison and Singer showed that there are pure states that cannot be clearly continued. If the MASA is restricted to projection operators on one-dimensional subspaces (so-called atomic MASA) - in the case of operators on finite-dimensional vector spaces (matrices) the diagonal matrices would be - the question of the unambiguous continuation of pure states is known as the Kadison-Singer problem.

The Kadison-Singer problem is related to other open problems and to applications in, for example, signal theory, linear algebra, and graph theory. In particular, it was shown in 1979 by Joel Anderson as being equivalent to a paving conjecture in finite-dimensional vector spaces. In another finite-dimensional formulation, it was then solved in a positive sense by Marcus, Spielman and Srivastava in 2013.

formulation

The sequence space (a separable Hilbert space) and the C * -algebras of all continuous linear operators ( bounded linear operators) on and the diagonal continuous linear operators on are considered . ${\displaystyle \ell ^{2}}$ ${\displaystyle B}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle D}$ ${\displaystyle \ell ^{2}}$

A state of a C * -algebra is a continuous linear functional such that ( stands for the multiplicative identity of the algebra) and for each . The state is called pure if it is extremal (that is, no linear combination of other states). According to Hahn-Banach's theorem , a functional on can be extended to on. Kadison and Singer hypothesized that the corresponding extension of pure states is unambiguous. ${\displaystyle A}$ ${\displaystyle \varphi :A\to \mathbb {C} }$ ${\displaystyle \varphi (I)=1}$ ${\displaystyle I}$ ${\displaystyle \varphi (T)\geq 0}$ ${\displaystyle T\in A}$ ${\displaystyle D}$ ${\displaystyle B}$

Finally-dimensional formulation by Anderson

Joel Anderson proved the equivalence to a conjecture in finite-dimensional vector spaces, which he called paving conjecture :

For each there is a natural number , so that the following applies: For each and every linear operator (matrix) on the -dimensional Hilbert space , whose diagonal elements vanish, there is a division of into sets such that ${\displaystyle \epsilon >0}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle T}$ ${\displaystyle n}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \{1,\dots ,n\}}$ ${\displaystyle k}$ ${\displaystyle A_{1},\dots A_{k}}$

${\displaystyle \|P_{A_{j}}TP_{A_{j}}\|\leq \epsilon \|T\|\ \ {\text{für alle}}\,\,j=1,\ldots ,k.}$

The operator corresponds to the projection onto the sub-vector space that is spanned by the standard unit vectors according to the index set . is the matrix that emerges from the matrix by setting all elements of columns and rows that do not correspond to indices to zero. is the matrix norm . ${\displaystyle P_{A_{j}}}$ ${\displaystyle A_{j}}$ ${\displaystyle P_{A_{j}}TP_{A_{j}}}$ ${\displaystyle T}$ ${\displaystyle A_{j}}$ ${\displaystyle \|A\|}$

Finally dimensional formulation by Weaver and solution 2013

The problem was solved in 2013 by computer scientists Adam Marcus, Daniel Spielman and Nikhil Srivastava in a positive sense (an unambiguous continuation is possible). They solved the (according to Nik Weaver) equivalent problem from linear algebra:

Given and vectors ( ) with for each and . Then there is a breakdown of having ${\displaystyle \alpha >0}$ ${\displaystyle {\vec {v}}_{i}\in \mathbb {R} ^{n}}$ ${\displaystyle i=1,\cdots ,m}$ ${\displaystyle \sum _{i}{\langle {\vec {v}}_{i},{\vec {x}}\rangle }^{2}=1}$ ${\displaystyle \|{\vec {x}}\|=1}$ ${\displaystyle {\|{\vec {v}}_{i}\|}^{2}\leq \alpha }$ ${\displaystyle T_{1}\bigcup T_{2}}$ ${\displaystyle \{1,2,\cdots ,m\}}$

${\displaystyle |\sum _{i\in T_{j}}{\langle {\vec {v}}_{i},{\vec {x}}\rangle }^{2}-{\frac {1}{2}}|\leq 5\cdot {\sqrt {\alpha }}}$

for everyone and . ${\displaystyle \|{\vec {x}}\|=1}$ ${\displaystyle j=1,2}$

The formulation comes from the discrepancy theory, because the inequality can be interpreted as follows: A square shape (given by the ) can be broken down into two parts on the unit sphere so that they come as close as desired to the value on both parts . ${\displaystyle {\vec {v}}_{i}}$ ${\displaystyle {\tfrac {1}{2}}}$

For the proof, Spielman, Marcus and Srivastava only used elementary linear algebra, probability theory and elementary analysis and used random polynomials.

In this formulation, the theorem also applies to the decomposition of graphs (Marcus, Spielman, Srivastata simultaneously proved the existence of k-regular Ramanujan graphs for any k).

literature

• Kadison, Singer: Extensions of pure states, American Journal of Mathematics, Volume 81, 1959, pp. 383-400
• Marcus, Spielman, Srivastava: Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer Problem, Arxiv 2013
• Alain Valette, The Kadison-Singer Problem , Snapshots of modern mathematics, Oberwolfach 2014

Individual evidence

1. ↑ For the corresponding terms in quantum mechanics see pure and mixed state
2. ↑ PG Casazza, M. Fickus, JC Tremain, E. Weber: 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, 2006, pp. 299-355, Arxiv
3. ^ PG Casazza, Consequences of the Marcus / Spielman / Stivastava solution to the Kadison-Singer Problem , Arxiv
4. ↑ Joel Anderson Restrictions and representations of states on C * - algebras , Transactions of the American Mathematical Society, Volume 249, 1979, pp. 303-329
5. ↑ Marcus, Spielman, Srivastava, Ramanujan Graphs and the Solution of the Kadison-Singer Problem (Arxiv), Proc. ICM 2014
6. ^ Weaver, The Kadison-Singer problem in discrepancy theory, Discrete Mathematics, Volume 278, 2004, pp. 227-239