arXiv subjects

Christopher Ramsey

Publications and source records attributed to Christopher Ramsey.

At least 19 recordsLinked to original sources

arXiv2026

The 1-2-3 conjecture for polygonal tilings

The 1-2-3 conjecture has been solved positively in 2024 for finite graphs and by extension for infinite graphs which are locally finite. The solution is non-constructive, and finding explicit solutions for large (or infinite) graphs is very hard. By exploiting the extra structure present in many non-periodic tilings, we find explicit solutions for the Chair (all three vertex placements), Non-Pinwheel, Pinwheel, Half-hex, Ammann-Beenker (two versions), Penrose Rhomb, and the Domino tilings. We prove that for any fully periodic tiling of the plane there exists a fully periodic solution, and provide an algorithm for finding such a solution. We give solutions for the fully periodic square, triangle and hexagonal lattices.

math.CO↗

arXiv2026

A C*-cover lattice dichotomy

In this paper, we show that the lattice of C*-covers of a non-selfadjoint operator algebra is either one point or uncountable. We prove that there are non-selfadjoint operator algebras with a one-point lattice in two ways: as an explicit subalgebra of the C*-algebra of a universal contraction, and via a direct limit construction inspired by the work of Kirchberg and Wassermann for operator systems. We also establish that the C*-envelope need not have an immediate successor C*-cover in the lattice, and that a semi-Dirichlet non-selfadjoint operator algebra never has a one-point lattice.

math.OA↗

arXiv2026

On the spectrum of non-ergodic measures

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure.

math.DS↗

arXiv2025

Nondegenerate actions on Hilbert C*-modules

Let X be a (right) Hilbert C*-module and let B be a C*-algebra acting on X from the left via adjointable operators. In this note we establish the equivalence of two notions of nondegeneracy for such an action of B on X. Furthermore, we explain how techniques from an earlier paper of Kim can be used to give a short proof of a recent result of Dessi, Kakariadis and Paraskevas.

math.OA↗

arXiv2025

Couniversality for C*-algebras of residually finite-dimensional operator algebras

The C*-envelope of a non self-adjoint operator algebra is known to encode many properties of the underlying subalgebra. However, the C*-envelope does not always encode the residual finite-dimensionality of an operator algebra. To elucidate this failure, we study couniversal existence in the space of residually finite-dimensional (RFD) C*-algebras attached to a fixed operator algebra. We construct several examples of residually finite-dimensional operator algebras for which there does not exist a minimal RFD C*-algebra, answering a question of the first two authors. For large swathes of tensor algebras of C*-correspondences, we also prove that the space of RFD C*-algebras fails to be closed under infima of C*-covers. In the case of the disc algebra, we are able to achieve this failure for a single pair of RFD C*-algebras.

math.OA↗

arXiv2024

Relative position in binary substitutions

Given an infinite word ${\bf w}$ on a finite alphabet, an immediate question arises:~can we understand the frequency of letters in ${\bf w}$\,? For words that are the fixed points of substitutions, the answer to this question is often `yes' -- the details and methods of these answers have been well-documented. In this paper, toward a better-understanding of the fixed points of binary substitutions, we delve deeper by investigating, in fine detail, the position of letters by defining various position functions and proving results about their behavior. Our analysis reveals new information about the Fibonacci substitution and the extended Pisa family of substitutions, as well as a new characterization of the Thue--Morse sequence.

math.CO↗

arXiv2024

Diffraction of the primes and other sets of zero density

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

math.FA↗

arXiv2024

On numerical diameters and linear maps

This paper studies the diameter of the numerical range of bounded operators on Hilbert space and the induced seminorm, called the numerical diameter, on bounded linear maps between operator systems which is sensible in the case of unital maps and their scalar multiples. It is shown that the completely bounded numerical diameter is a norm that is comparable but not equal to the completely bounded norm. This norm is particularly interesting in the case of unital completely positive maps and their sections.

math.FA↗

arXiv2023

Generating functions of substitutions

In this paper we study substitutions and some of their associated generating functions. This association takes aperiodicity to transcendence, and vice-versa. These generating functions have a recursive structure arising from the substitution which we use to study the roots of the characteristic function of the letter a in the Fibonacci substitution.

math.CO↗

arXiv2023

Crossed products and C*-covers of semi-Dirichlet operator algebras

In this paper, we show that the semi-Dirichlet C*-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet C*-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.

math.OA↗

arXiv2023

The Lattice of C*-covers of an operator algebra

In this paper it is shown that the lattice of C*-covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C*-covers are developed and proven to be distinct. The lattice of C*-covers of direct sums and tensor products are studied. Along the way key examples are found of an operator algebra that generates exactly n C*-algebras up to *-isomorphism and a simple operator algebra that is not similar to a C*-algebra.

math.OA↗

arXiv2022

Colourings of aperiodic tilings

We find explicit optimal vertex, edge and face coulourings for the chair tiling, the Ammann--Beenker tiling, the rational pinwheel tiling and the pinwheel tiling.

math.CO↗

arXiv2022

The isomorphism problem for tensor algebras of multivariable dynamical systems

We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.

math.OA↗

arXiv2022

Symbolic Substitutions, Bratteli Diagrams and Operator Algebras

In this paper we look at symbolic substitutions and their relationship to Bratteli diagrams and their associated operator algebras. In particular, we consider the equivalence relation on substitutions induced by telescope equivalence of Bratteli diagrams. Such an equivalence preserves pure aperiodicity and primitivity but fails to preserve rank, order, and number of letters. In a similar manner, we consider the equivalence relation on substitutions induced by telescope equivalence of ordered Bratteli diagrams. This results in a finer equivalence but fails to provide a complete invariant. An application to Fibonacci-like substitutions is developed.

math.OA↗

arXiv2022

Bistochastic operators and quantum random variables

Given a positive operator-valued measure $ν$ acting on the Borel sets of a locally compact Hausdorff space $X$, with outcomes in the algebra $\mathcal B(\mathcal H)$ of all bounded operators on a (possibly infinite-dimensional) Hilbert space $\mathcal H$, one can consider $ν$-integrable functions $X\rightarrow \mathcal B(\mathcal H)$ that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.

math.FA↗

arXiv2021

L$^p$ spaces of operator-valued functions

We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L^p space that is shown to be complete. Various other norm candidates are considered as well as generalizations of Hölder's inequality to this new context.

math.FA↗

arXiv2019

The non-selfadjoint approach to the Hao-Ng isomorphism

In an earlier work, the authors proposed a non-selfadjoint approach to the Hao-Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of C*-correspondences, each one of these conjectures is equivalent to the Hao-Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C*-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of C*-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana's injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao-Ng isomorphism for the reduced crossed product and all hyperrigid C*-correspondences. A culmination of these results is the resolution of the Hao-Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg and Spielberg.

math.OA↗

arXiv2019

On Operator Valued Measures

We consider positive operator valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator valued measure seen in the quantum information theory literature. We define the Radon-Nikodym derivative of a positive operator valued measure with respect to a complex measure induced by a given quantum state; this derivative does not always exist when the Hilbert space is infinite dimensional in so much as its range may include unbounded operators. We define integrability of a positive quantum random variable with respect to a positive operator valued measure. Emphasis is put on the structure of operator valued measures, and we develop positive operator valued versions of the Lebesque decomposition theorem and Johnson's atomic and nonatomic decomposition theorem. Beyond these generalizations, we make connections between absolute continuity and the "cleanness" relation defined on positive operator valued measures as well as to the notion of atomic and nonatomic measures.

math.FA↗