arXiv subjects

Miek Messerschmidt

Publications and source records attributed to Miek Messerschmidt.

16 recordsLinked to original sources

arXiv2025

The raspberries in three dimensions with at most two sizes of berry

In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a catalog of all configurations of radii that permit the formation of raspberries that have at most two sizes of non-central spheres. Throughout, we discuss the construction of this catalog.

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arXiv2024

L-functional analysis

Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital $f$-algebra $\mathbb{L}$; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of $\mathbb{L}$-normed and $\mathbb{L}$-Banach spaces and bounded operators between them, we discuss the $\mathbb{L}$-valued analogues of the classical $\ell^p$-spaces, and we prove the analogue of the Hahn-Banach theorem. We also discuss the basics of the theory of $\mathbb{L}$-Hilbert spaces, including projections onto convex subsets, the Riesz Representation theorem, and representing $\mathbb{L}$-Hilbert spaces as a direct sum of $\ell^2$-spaces.

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arXiv2023

On compact packings of Euclidean space with spheres of finitely many sizes

For $d\in\mathbb{N}$, a compact sphere packing of Euclidean space $\mathbb{R}^{d}$ is a set of spheres in $\mathbb{R}^{d}$ with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial $d$-complex that covers all of $\mathbb{R}^{d}$. We are motivated by the question: For $d,n\in\mathbb{N}$ with $d,n\geq2$, how many configurations of numbers $0<r_{0}<r_{1}<\ldots<r_{n-1}=1$ can occur as the radii of spheres in a compact sphere packing of $\mathbb{R}^{d}$ wherein there occur exactly $n$ sizes of sphere? We introduce what we call `heteroperturbative sets' of labeled triangulations of unit spheres and we discuss the existence of non-trivial examples of heteroperturbative sets. For a fixed heteroperturbative set, we discuss how a compact sphere packing may be associated to the heteroperturbative set or not. We proceed to show, for $d,n\in\mathbb{N}$ with $d,n\geq2$ and for a fixed heteroperturbative set, that the collection of all configurations of $n$ distinct positive numbers that can occur as the radii of spheres in a compact packing is finite, when taken over all compact sphere packings of $\mathbb{R}^{d}$ which have exactly $n$ sizes of sphere and which are associated to the fixed heteroperturbative set.

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arXiv2022

The number of configurations of radii that can occur in compact packings of the plane with discs of $n$ sizes is finite

By a compact packing of the plane by discs, $P$, we mean a collection of closed discs in the plane with pairwise disjoint interior so that, for every disc $C\in P$, there exists a sequence of discs $D_{0},\ldots,D_{m-1}\in P$ so that each $D_{i}$ is tangent to both $C$ and $D_{i+1\mod m}.$ We prove, for every $n\in\mathbb N$, that there exist only finitely many tuples $(r_{0},r_{1},\ldots,r_{n-1})\in\mathbb{R}^{n}$ with $0<r_{0}<r_{1}\ldots<r_{n-1}=1$ that can occur as the radii of the discs in any compact packing of the plane with $n$ distinct sizes of disc.

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arXiv2019

A family of quotient maps of $\ell^\infty$ that do not admit uniformly continuous right inverses

Previously only two examples of Banach space quotient maps which do not admit uniformly continuous right inverses were known: one due to Aharoni and Lindenstrauss and one due to Kalton ($\ell^\infty\to\ell^\infty/c_{0}$). We show through an application of Kalton's Monotone Transfinite Sequence Theorem that a quotient map of a subspace of $\ell^\infty$ of sequences that converge to zero along an ideal in $\mathbb{N}$ toward another such subspace, provided one of the ideals is `much larger' than the other, cannot have a uniformly continuous right inverse. We show in general that pairs of ideals in $\mathbb{N}$, with one much larger than the other, occur in abundance. Some classical examples of ideals in $\mathbb{N}$ presented explicitly are: the finite subsets of $\mathbb{N}$, the subsets of $\mathbb{N}$ with convergent reciprocal series, and, the subsets of $\mathbb{N}$ with density zero, Banach density zero or Buck density zero.

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arXiv2019

On compact packings of the plane with circles of three radii

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$ We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ for which there exist a compact circle-packing of the plane consisting of circles with radii $s$, $r$ and $1$. We discuss computing the exact values of such $0<s<r<1$ as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing \emph{all} these values on contemporary consumer hardware with the methods employed in this paper.

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arXiv2019

Equivalence after extension and Schur coupling do not coincide, on essentially incomparable Banach spaces

In 1994 H. Bart and V.É. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC $\Rightarrow$ SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows from a complete description of EAE and SC for the case that the operators act on essentially incomparable Banach spaces, which also leads to a new characterization of the notion of essential incomparability. Concretely, the forward shift operators $U$ on $\ell^p$ and $V$ on $\ell^q$, for $1\leq p,q\leq \infty$, $p\neq q$, are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given $U$ and $V$ that are Atkinson or generalized invertible and EAE, we give a concrete operator $W$ that is SC to both $U$ and $V$, even if $U$ and $V$ are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.

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arXiv2018

Strong Klee-Andô Theorems through an Open Mapping Theorem for cone-valued multi-functions

A version of the classical Klee-Andô Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $α>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leqα\|x\|$. The conclusion of the Klee-Andô Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-Andô Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-Andô Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-Andô Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-Andô Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space $X$, ordered by a closed generating cone $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$.

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arXiv2017

A Pointwise Lipschitz Selection Theorem

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general.

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arXiv2017

The intrinsic metric on the unit sphere of a normed space

Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}π\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$.

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arXiv2015

Equivalence after extension for compact operators on Banach spaces

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators that are equivalent after extension, we prove the necessity of similar relationships between the $s$-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary $s$-functions. We investigate equivalence after extension for operators on $\ell^{p}$-spaces. We show that two operators that act on different $\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension, however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces $X$ and $Y$, we investigate consequences of an operator on $X$ being equivalent after extension to a compact operator on $Y$. We show that, in this case, a closed finite codimensional subspace of $Y$ must embed into $X$, and that certain general Banach space properties must transfer from $X$ to $Y$. We also show that no operator on $X$ can be equivalent after extension to an operator on $Y$, if $X$ and $Y$ are essentially incomparable Banach spaces.

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arXiv2015

Geometric duality theory of cones in dual pairs of vector spaces

This paper will generalize what may be termed the "geometric duality theory" of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual Banach space. We show that geometric duality theory is not restricted to real pre-ordered Banach spaces, as is done classically, but can be extended to real Banach spaces endowed with arbitrary collections of closed cones. We define geometric notions of normality, conormality, additivity and coadditivity for members of dual pairs of real vector spaces as certain possible interactions between two cones and two convex convex sets containing zero. We show that, thus defined, these notions are dual to each other under certain conditions, i.e., for a dual pair of real vector spaces $(Y,Z)$, the space $Y$ is normal (additive) if and only if its dual $Z$ is conormal (coadditive) and vice versa. These results are set up in a manner so as to provide a framework to prove results in the geometric duality theory of cones in real Banach spaces. As an example of using this framework, we generalize classical duality results for real Banach spaces pre-ordered by a single closed cone, to real Banach spaces endowed with an arbitrary collections of closed cones. As an application, we analyze some of the geometric properties of naturally occurring cones in C*-algebras and their duals.

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arXiv2015

Normality of spaces of operators and quasi-lattices

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that are not Banach lattices, but for which $B(X,Y)$ is normal. In particular, we show that a Hilbert space $\mathcal{H}$ endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and satisfies an identity analogous to the elementary Banach lattice identity $\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

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arXiv2014

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.

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arXiv2013

Crossed products of Banach algebras. III

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,α)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper we adapt the theory to the ordered context. We construct a pre-ordered crossed product of a Banach algebra from a pre-ordered Banach algebra dynamical system $(A,G,α)$ and a given uniformly bounded class $\mathcal{R}$ of continuous covariant representations of $(A,G,α)$. If $A$ has a positive bounded approximate left identity and $\mathcal{R}$ consists of non-degenerate continuous covariant representations, we establish a bijection between the positive non-degenerate bounded representations of the pre-ordered crossed product on pre-ordered Banach spaces with closed cones and the positive non-degenerate $\mathcal{R}$-continuous covariant representations of $(A,G,α)$ on such spaces. Under mild conditions, we show that this pre-ordered crossed product is the essentially unique pre-ordered Banach algebra for which such a bijection exists. Finally, we study pre-ordered generalized Beurling algebras. We show that they are bipositively topologically isomorphic to pre-ordered crossed products of Banach algebras associated with pre-ordered Banach algebra dynamical systems, and hence the general theory allows us to describe their positive representations on pre-ordered Banach spaces with closed cones.

math.FA↗

arXiv2013

Crossed products of Banach algebras. II

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,α)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper the theory is developed further. We consider the dependence of the crossed product on the class $\mathcal{R}$ and its essential uniqueness. Next we study generalized Beurling algebras: weighted Bochner spaces of $A$-valued functions on $G$ with a continuous multiplication. Though not Banach algebras in general, they are isomorphic to a crossed product of a Banach algebra, and the earlier work therefore predicts the structure of their representations. Classical results for the usual Beurling (Banach) algebras of scalar valued functions are then retrieved as special cases. We also show how, e.g., an anti-covariant pair of anti-representations of $A$ and $G$ can be viewed as a covariant pair for a related Banach algebra dynamical system, so that the earlier work becomes applicable to classes of such other pairs. After including material on the representations of the projective tensor product of Banach algebras, we combine this idea with the results already obtained and describe the two-sided modules over the generalized Beurling algebras, where again specializing to the scalars gives a classical result.

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