Related papers: Le probl\`eme de Kadison-Singer (The Kadison-Singe…
We give an informal overview of the Kadison-Singer extension problem with emphasis on its initial connections to Dirac's formulation of quantum mechanics. Let H be an infinite dimensional separable Hilbert space, and B(H) the algebra of all…
Let $M_n$ denote the algebra of complex $n\times n $ matrices and write $M$ for the direct sum of the $M_n$. So a typical element of $M$ has the form \[x = x_1\oplus x_2 \... \oplus x_n \oplus \...,\] where $x_n \in M_n$ and $\|x\| =…
We construct a pure state on the C*-algebra $\mathcal B(\ell_2)$ of all bounded linear operators on $\ell_2$ which is not diagonalizable, i.e., it is not of the form $\lim_u\langle T(e_k), e_k\rangle$ for any orthonormal basis $(e_k)_{k\in…
The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open…
We give self-contained presentation of results related to the Kadison-Singer problem, which was recently solved by Marcus, Spielman, and Srivastava. This problem connects with unusually large number of areas including: operator algebras…
In these notes we develop a link between the Kadison-Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the…
Let H be a separable Hilbert space with a fixed orthonormal basis (e_n), n>=1, and B(H) be the full von Neumann algebra of the bounded linear operators T: H -> H. Identifying l^\infty = C(\beta N) with the diagonal operators, we consider…
In 2013, Marcus, Spielman, and Srivastava resolved the famous Kadison-Singer conjecture. It states that for $n$ independent random vectors $v_1,\cdots, v_n$ that have expected squared norm bounded by $\epsilon$ and are in the isotropic…
It is known that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. The recent…
This paper explores the intriguing connections between the invariant subspace problem, the Kadison-Singer problem, and the Borel conjecture. The Kadison-Singer problem, originally formulated in terms of pure states on C*-algebras, was later…
We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison-Singer problem to prove improved paving estimates. Working with Anderson's paving formulation of Kadison-Singer instead of Weaver's vector balancing…
In 1955 Kadison \cite{14} asked whether the analogue of the classical Burnside's theorem of the Linear Algebra holds in the infinite dimensional case. We use reproducing kernels method to solve the Kadison question. Namely, we prove that…
In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer…
We use the method of interlacing families of polynomials introduced to prove two theorems known to imply a positive solution to the Kadison--Singer problem. The first is Weaver's conjecture $KS_{2}$ \cite{weaver}, which is known to imply…
Through the lens of noncommutative function theory, we study restrictions of pure states to unital subspaces of $C^*$-algebras, in the spirit of the Kadison--Singer question. More precisely, given a unital subspace $M$ of a $C^*$-algebra…
Recently Marcus, Spielman and Srivastava proved Weaver's ${\rm{KS}}_r$ conjecture, which gives a positive solution to the Kadison-Singer problem. Cohen and Br\"and\'en independently extended this result to obtain the arbitrary-rank version…
Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem by showing that if u_1, ..., u_m are column vectors in C^d such that \sum u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so that \sum_{i…
The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph…
Marcus, Spielman, and Srivastava in their seminal work \cite{MSS13} resolved the Kadison-Singer conjecture by proving that for any set of finitely supported independently distributed random vectors $v_1,\dots, v_n$ which have "small"…
We show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of…