The Kadison-Singer Problem in Mathematics and Engineering

Abstract

We will show that the famous, intractible 1959 Kadison-Singer problem in $C^{\ast }$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.

References in corpus (1)

• Decoding by Linear Programming

Cited by in corpus (35)

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• A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators
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• Fusion Frames: Existence and Construction
• Traces, Ultrapowers and the Pedersen-Petersen C*-Algebras
• Twice-Ramanujan Sparsifiers
• Mixed Determinants and the Kadison-Singer problem
• A Survey of Fusion Frames in Hilbert Spaces
• Sparse Signal Processing with Frame Theory
• Algorithmic constructions of unitary matrices and tight frames
• A short note on the Feichtinger Conjecture
• The paving conjecture is equivalent to the paving conjecture for triangular matrices
• Minimal sequences and the Kadison-Singer problem
• The Feichtinger conjecture for reproducing kernels in model subspaces
• The Kadison-Singer problem in discrepancy theory, II
• The Kadison-Singer Problem
• The random paving property for uniformly bounded matrices
• Parseval frames for ICC groups
• A Note On The Kadison-Singer Problem
• The Feichtinger Conjecture for Exponentials
• Concrete constructions of non-pavable projections
• Syndetic Sets, Paving, and the Feichtinger Conjecture
• Iterative actions of normal operators
• Random Reordering in SOR-Type Methods
• Uniform partitions of frames of exponentials into Riesz sequences
• Spectral Envelopes - A Preliminary Report
• The Bourgain-Tzafriri conjecture and concrete constructions of non-pavable projections
• Invertibility threshold for $H^{\infty }$ trace algebras, and effective matrix inversions
• Set theory and C*-algebras
• Kadison-Singer from mathematical physics: An introduction
• A Decomposition Theorem for frames and the Feichtinger Conjecture
• Sampling and interpolation in radial weighted spaces of analytic functions
• Tutorial on Rational Rotation $C^{\ast }$--Algebras