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Kenneth R. Davidson

Publications and source records attributed to Kenneth R. Davidson.

At least 19 recordsLinked to original sources
arXiv2025

Exactness and LLP Results via Operator System Methods

In this paper, we employ operator system techniques to investigate structural properties of C*-algebras. In particular, we provide more direct proofs of results concerning exactness and the local lifting property (LLP) of group C$^*$-algebras that avoid relying on the traditional heavy machinery of C$^*$-algebra theory. Briefly, these methods allow us to deduce that any C*-algebra containing $n$ unitaries whose $\textit{ joint numerical radius}$, in the sense defined by \cite{FKP}, is strictly less than $n$, must fail certain of these properties.

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arXiv2024

Large Perturbations of Nest Algebras

Let $\mathcal{M}$ and $\mathcal{N}$ be nests on separable Hilbert space. If the two nest algebras are distance less than 1 ($d(\mathcal{T}(\mathcal{M}),\mathcal{T}(\mathcal{N})) < 1$), then the nests are distance less than 1 ($d(\mathcal{M},\mathcal{N})<1$). If the nests are distance less than 1 apart, then the nest algebras are similar, i.e. there is an invertible $S$ such that $S\mathcal{M} = \mathcal{N}$, so that $S \mathcal{T}(\mathcal{M})S^{-1} = \mathcal{T}(\mathcal{N})$. However there are examples of nests closer than 1 for which the nest algebras are distance 1 apart.

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arXiv2024

Completely Bounded Norms of $k$-positive Maps

Given an operator system $\mathcal{S}$, we define the parameters $r_k(\mathcal{S})$ (resp. $d_k(\mathcal{S})$) defined as the maximal value of the completely bounded norm of a unital $k$-positive map from an arbitrary operator system into $\mathcal{S}$ (resp. from $\mathcal{S}$ into an arbitrary operator system). In the case of the matrix algebras $M_n$, for $1 \leq k \leq n$, we compute the exact value $r_k(M_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters $d_k(M_n)$. Moreover, when $\mathcal{S}$ is a finite-dimensional operator system, adapting recent results of Passer and the 4th author, we show that the sequence $(r_k( \mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ is exact and that the sequence $(d_k(\mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ has the lifting property.

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arXiv2023

On boundary representations

Let $S$ be an operator system sitting in its C*-envelope $C^*_{\mathrm{min}}(S)$. Starting with a pure state on $S$, let $F$ be the face of state extensions to $C^*_{\mathrm{min}}(S)$. The dilation theorem of Davidson-Kennedy shows that the GNS representations corresponding to some of the extreme states of $F$ are boundary representations. We construct an explicit example in which $F$ is an interval and only one of the two extreme points yields a boundary representation.

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arXiv2023

Positive maps and entanglement in real Hilbert spaces

The theory of positive maps plays a central role in operator algebras and functional analysis, and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert spaces, and little is known about its variant on real Hilbert spaces. In this article we study positive maps acting on a full matrix algebra over the reals, pointing out a number of fundamental differences with the complex case and discussing their implications in quantum information. We provide a necessary and sufficient condition for a real map to admit a positive complexification, and connect the existence of positive maps with non-positive complexification with the existence of mixed states that are entangled in real Hilbert space quantum mechanics, but separable in the complex version, providing explicit examples both for the maps and for the states. Finally, we discuss entanglement breaking and PPT maps, and we show that a straightforward real version of the PPT-squared conjecture is false even in dimension 2. Nevertheless, we show that the original PPT-squared conjecture implies a different conjecture for real maps, in which the PPT property is replaced by a stronger property of invariance under partial transposition (IPT). When the IPT property is assumed, we prove an asymptotic version of the conjecture.

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arXiv2022

Interpolation and duality in algebras of multipliers on the ball

We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact $\operatorname{Mult}(\mathcal{H})$-totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is $\operatorname{Mult}(\mathcal{H})$-totally null.

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arXiv2022

Counterexamples to the extendibility of positive unital norm-one maps

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.

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arXiv2021

Choquet order and hyperrigidity for function systems

We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the non-separable setting, as well as a non-separable version of Šaškin's theorem from approximation theory. We show that a slight variant of this order characterizes the representations of a commutative C*-algebra that have the unique extension property relative to a set of generators. This reduces the commutative case of Arveson's hyperrigidity conjecture to the question of whether measures that are maximal with respect to the classical Choquet order are also maximal with respect to this new order. An example shows that these orders are not the same in general.

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arXiv2020

Strongly peaking representations and compressions of operator systems

We use Arveson's notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets which admit minimal presentations. A fully compressed separable operator system necessarily generates the C*-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.

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arXiv2019

Structure of free semigroupoid algebras

A free semigroupoid algebra is the closure of the algebra generated by a TCK family of a graph in the weak operator topology. We obtain a structure theory for these algebras analogous to that of free semigroup algebra. We clarify the role of absolute continuity and wandering vectors. These results are applied to obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with reflexivity, a Kaplansky density theorem and classification for free semigroupoid algebras. Several classes of examples are discussed and developed, including self-adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence.

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arXiv2019

Noncommutative Choquet theory

We introduce a new and extensive theory of noncommutative convexity along with a corresponding theory of noncommutative functions. We establish noncommutative analogues of the fundamental results from classical convexity theory, and apply these ideas to develop a noncommutative Choquet theory that generalizes much of classical Choquet theory. The central objects of interest in noncommutative convexity are noncommutative convex sets. The category of compact noncommutative sets is dual to the category of operator systems, and there is a robust notion of extreme point for a noncommutative convex set that is dual to Arveson's notion of boundary representation for an operator system. We identify the C*-algebra of continuous noncommutative functions on a compact noncommutative convex set as the maximal C*-algebra of the operator system of continuous noncommutative affine functions on the set. In the noncommutative setting, unital completely positive maps on this C*-algebra play the role of representing measures in the classical setting. The continuous convex noncommutative functions determine an order on the set of unital completely positive maps that is analogous to the classical Choquet order on probability measures. We characterize this order in terms of the extensions and dilations of the maps, providing a powerful new perspective on the structure of completely positive maps on operator systems. Finally, we establish a noncommutative generalization of the Choquet-Bishop-de Leeuw theorem asserting that every point in a compact noncommutative convex set has a representing map that is supported on the extreme boundary. In the separable case, we obtain a corresponding integral representation theorem.

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arXiv2018

Nevanlinna-Pick Families and Singular Rational Varieties

The goal of this note is to apply ideas from commutative algebra (a.k.a. affine algebraic geometry) to the question of constrained Nevanlinna-Pick interpolation. More precisely, we consider subalgebras $A \subset \mathbb{C}[z_1,\ldots,z_d]$, such that the map from the affine space to the spectrum of $A$ is an isomorphism except for finitely many points. Letting $\mathfrak{A}$ be the weak-$*$ closure of $A$ in $\mathcal{M}_d$ -- the multiplier algebra of the Drury-Arveson space. We provide a parametrization for the Nevanlinna-Pick family of $M_k(\mathfrak{A})$ for $k \geq 1$. In particular, when $k=1$ the parameter space for the Nevanlinna-Pick family is the Picard group of $A$.

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arXiv2017

A proof of Boca's Theorem

We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application we give a dilation theoretic proof of Boca's Theorem.

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arXiv2016

Complete spectral sets and numerical range

We define the complete numerical radius norm for homomorphisms from any operator algebra into ${\mathcal B}({\mathcal H})$, and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if $K$ is a complete $C$-spectral set for an operator $T$, then it is a complete $M$-numerical radius set, where $M=\frac12(C+C^{-1})$. In particular, in view of Crouzeix's theorem, there is a universal constant $M$ (less than 5.6) so that if $P$ is a matrix polynomial and $T \in {\mathcal B}({\mathcal H})$, then $w(P(T)) \le M \|P\|_{W(T)}$. When $W(T) = \overline{\mathbb D}$, we have $M = \frac54$.

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arXiv2016

Ideals in a multiplier algebra on the ball

We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.

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arXiv2016

Absolute continuity for commuting row contractions

Absolutely continuous commuting row contractions admit a weak-$*$ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury-Arveson space, we give a complete characterization of these commuting row contractions in measure theoretic terms. We also establish that completely non-unitary row contractions are necessarily absolutely continuous, in direct parallel with the case of a single contraction. Finally, we consider refinements of this question for row contractions that are annihilated by a given ideal.

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arXiv2016

Duality, convexity and peak interpolation in the Drury-Arveson space

We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of $\mathcal{A}^*_d$.

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