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Kadison-Singer problem

Context

Functional analysis

Overview diagrams

Basic concepts

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Theorems

Topics in Functional Analysis

Operator algebra

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Concepts

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Theorems

States and observables

Operator algebra

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Local QFT

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Perturbative QFT

Contents

Idea

Given a separable Hilbert space β„‹\mathcal{H}, such as the sequence space 𝓁 2\mathcal{l}_2, write B(β„‹)B(\mathcal{H}) for its C*-algebra of bounded linear operators and D(β„‹)D(\mathcal{H}) for the subalgebra of diagonal operators.

The Kadison-Singer problem is the question:

Does every pure state ψ\psi on D(β„‹)D(\mathcal{H}) extend to a state ρ\rho on B(β„‹)B(\mathcal{H})?

D(β„‹) ⟢ψ β„‚ ↓ β†— βˆƒΟ B(β„‹) \array{ D(\mathcal{H}) &\overset{\psi}{\longrightarrow}& \mathbb{C} \\ \downarrow & \nearrow_{\mathrlap{\exists \rho}} \\ B(\mathcal{H}) }

This was proven to be the case in Marcus-Spielman-Srivastava 13.

Other theorems about the foundations and interpretation of quantum mechanics include:

References

The problem was stated in

  • Richard V. Kadison and Isadore Singer, Extensions of pure states Americal Journal of Mathematics, 81(2):383–400, 1959

and its solution was proven in

  • Adam Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem, 2013. (arXiv:1306.3969)

Review includes

  • Nicholas Harvey, An introduction to the Kadison-Singer conjecture and the paving conjecture (pdf)

Last revised on December 11, 2017 at 08:47:05. See the history of this page for a list of all contributions to it.