Related papers: Le probl\`eme de Kadison-Singer (The Kadison-Singe…
In the summer of 2013 Marcus, Spielman, and Srivastava gave a surprising and beautiful solution to the Kadison--Singer problem. The current presentation is slightly more didactical than other versions that have appeared since; it hopes to…
Akemann and Anderson made a conjecture about ``paving'' projections in finite dimensional matrix algebras which, if true, would settle the well-known Kadison-Singer problem. We falsify their conjecture by an explicit seqence of…
We give a combinatorial form of the Kadison-Singer problem, a famous problem in C*-algebra. This combinatorial problem, which has several minor variations, is a discrepancy question about vectors in C^n. Some partial results can be easily…
The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable,…
This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in…
Marcus, Spielman and Srivastava (Annals of Mathematics 2014) solved the Kadison--Singer Problem by proving a strong form of Weaver's conjecture: they showed that for all $\alpha > 0$ and all lists of vectors of norm at most $\sqrt{\alpha}$…
It is shown that if $C_1$ and $C_2$ are maximal abelian self-adjoint subalgebras (masas) of C*-algebras $A_1$ and $A_2$, respectively, then the completion $C_1\otimes C_2$ of the algebraic tensor product $C_1\odot C_2$ of $C_1$ and $C_2$ in…
Anderson's paving conjectures are known to be equivalent to the Kadison-Singer problem. We prove some new equivalences of Anderson's conjectures that require the paving of smaller sets of matrices. We prove that if the strictly upper…
We investigate when a map on a selfadjoint operator space $E$ is an embedding, i.e., when its unitisation in the sense of Werner is completely isometric. Combining with results of Russell, of Ng, and of Dessi, the second and the last…
The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin…
Kadison's Pythagorean theorem (2002) provides a characterization of the diagonals of projections with a subtle integrality condition. Arveson (2007), Kaftal, Ng, Zhang (2009), and Argerami (2015) all provide different proofs of that…
Applying the solution to the Kadison-Singer problem, we show that every subset $\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\left\{ e^{i\lambda x}\right\} _{\lambda \in \Lambda}$ such that…
We show an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results,…
The theme of the paper is the question of existence and basic structure of transfer operators for endomorphisms of a unital C*-algebra. We establish a complete description of non-degenerate transfer operators, characterize complete transfer…
The famous Bernstein conjecture about optimal node systems in classical polynomial Lagrange interpolation, standing unresolved for about half a century, was solved by T. Kilgore in 1978. Immediately following him, also the additional…
Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then…
We establish the absolute continuity of the integrated density of states (IDS) for quasi-periodic Schr\"odinger operators with a large trigonometric potential and Diophantine frequency. This partially solves Eliasson's open problem in 2002.…
In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space $H^s(\mathbb{R}^n)$ that implies pointwise convergence for the solution of the linear Schr\"odinger equation. After progress by many…
We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume…
Kadison's transitivity theorem implies that, for irreducible representations of C*-algebras, every invariant linear manifold is closed. It is known that CSL algebras have this propery if, and only if, the lattice is hyperatomic (every…