Related papers: Positively Factorizable Maps
In this paper we give a first study of perfect copositive $n \times n$ matrices. They can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive…
This is a revised form of the previous paper in which we study cones of positive maps of B(H) into itself. We add the result that the dual cone of a symmetric mapping cone is itself a symmetric mapping cone. As applications we obtain…
Nonnegative matrix factorization (NMF) is a known unsupervised data-reduction method. The principle of the common cause (PCC) is a basic methodological approach in probabilistic causality, which seeks an independent mixture model for the…
This paper considers a restriction to non-negative matrix factorization in which at least one matrix factor is stochastic. That is, the elements of the matrix factors are non-negative and the columns of one matrix factor sum to 1. This…
We study equivariant linear maps between finite-dimensional matrix algebras, as introduced by Bhat. These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly…
We describe conditions that characterize amenability for groups in terms of positive definite functions valued in a von Neumann algebra.
We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras. Starting from a compact family of $\alpha$-admissible unit…
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
In this paper, we show that a $D$-type map $\Phi_D:M_n\rightarrow M_n$ with $D=(n-2)I_n+P_{\pi_1}+P_{\pi_2}$ induced by a pair $\{\pi_1,\pi_2\}$ of permutations of $(1,2,..., n)$ is positive if $\{\pi_1,\pi_2\}$ has property (C). The…
We investigate tensor products of matrix factorisations. This is most naturally done by formulating matrix factorisations in terms of bimodules instead of modules. If the underlying ring is C[x_1,...,x_N] we show that bimodule matrix…
A subalgebra $\mathcal{A}$ of a $C^*$-algebra $\mathcal{M}$ is logmodular (resp. has factorization) if the set $\{a^*a; a\text{ is invertible with }a,a^{-1}\in\mathcal{A}\}$ is dense in (resp. equal to) the set of all positive and…
We describe $\omega$-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other.…
We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability…
Gradient descent for matrix factorization exhibits an implicit bias toward approximately low-rank solutions. While existing theories often assume the boundedness of iterates, empirically the bias persists even with unbounded sequences. This…
We introduce twisted matrix factorizations for quantum complete intersections of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions…
Nuclear C*-algebras enjoy a number of approximation properties, most famously the completely positive approximation property. This was recently sharpened to arrange for the incoming maps to be sums of order-zero maps. We show that, in…
A rational map $\phi: \mathbb{P}^1 \to \mathbb{P}^1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat^{tm}_2$ can be parametrized by…
We construct a new class of positive indecomposable maps in the algebra of 'd x d' complex matrices. Each map is uniquely characterized by a cyclic bistochastic matrix. This class generalizes a Choi map for d=3. It provides a new reach…
We identify and characterize unital completely positive (UCP) maps on finite dimensional $C^*$-algebras for which the Choi-Effros product extended to the space generated by peripheral eigenvectors matches with the original product. We…
We give an overview of the question: which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive…