Banach Bundles Research Articles

On the reflexivity properties of Banach bundles and Banach modules

In this paper, we investigate some reflexivity-type properties of separable measurable Banach bundles over a σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-finite measure space. Our two main results are the following:The fibers of a bundle are uniformly convex (with a common modulus of convexity) if and only if the space of its Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-sections is uniformly convex for every p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (1,\\infty )$$\\end{document}.The fibers of a bundle are reflexive if and only if the space of its Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-sections is reflexive for every p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (1,\\infty )$$\\end{document}. They generalise well-known results for Lebesgue–Bochner spaces.

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue differentiation theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon–Nikodým property.

Uniform enveloping semigroupoids for groupoid actions

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems.We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter—Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.

On the Topological Structure of Nonlocal Continuum Field Theories

An alternative to conventional spacetime is proposed and rigorously formulated for nonlocal continuum field theories through the deployment of a fiber bundle-based superspace extension method. We develop, in increasing complexity, the concept of nonlocality starting from general considerations, going through spatial dispersion, and ending up with a broad formulation that unveils the link between general topology and nonlocality in generic material media. It is shown that nonlocality naturally leads to a Banach (vector) bundle structure serving as an enlarged space (superspace) inside which physical processes, such as the electromagnetic ones, take place. The added structures, essentially fibered spaces, model the topological microdomains of physics-based nonlocality and provide a fine-grained geometrical picture of field–matter interactions in nonlocal metamaterials. We utilize standard techniques in the theory of smooth manifolds to construct the Banach bundle structure by paying careful attention to the relevant physics. The electromagnetic response tensor is then reformulated as a superspace bundle homomorphism and the various tools needed to proceed from the local topology of microdomains to global domains are developed. For concreteness and simplicity, our presentations of both the fundamental theory and the examples given to illustrate the mathematics all emphasize the case of electromagnetic field theory, but the superspace formalism developed here is quite general and can be easily extended to other types of nonlocal continuum field theories. An application to fundamental theory is given, which consists of utilizing the proposed superspace theory of nonlocal metamaterials in order to explain why nonlocal electromagnetic materials often require additional boundary conditions or extra input from microscopic theory relative to local electromagnetism, where in the latter case such extra input is not needed. Real-life case studies quantitatively illustrating the microdomain structure in nonlocal semiconductors are provided. Moreover, in a series of connected appendices, we outline a new broad view of the emerging field of nonlocal electromagnetism in material domains, which, together with the main superspace formalism introduced in the main text, may be considered a new unified general introduction to the physics and methods of nonlocal metamaterials.

The dynamics of non-perturbative phases via Banach bundles

Strongly coupled Dyson–Schwinger equations generate infinite power series of running coupling constants together with Feynman diagrams with increasing loop orders as coefficients. Theory of graphons for sparse graphs can address a new useful approach for the study of graph limits of sequences of partial sums corresponding to these infinite power series in the context of Feynman graphons and the cut-distance topology. Graphon models enable us to associate some new analytic graphs to non-perturbative solutions of Dyson–Schwinger equations. Homomorphism densities of Feynman graphons provide a new way of analyzing non-perturbative phase transitions. We explain the structures of topological renormalization quotient Hopf algebras of Feynman graphons which encode gauge symmetries Hopf ideals in the context of the weakly isomorphic equivalence classes corresponding to the Slavnov–Taylor / Ward–Takahashi elements. We apply Feynman graphon representations of combinatorial Dyson–Schwinger equations underlying the Connes–Kreimer renormalization Hopf algebra to construct a new class of Banach bundles for the study of the dynamics of non-perturbative phases in strongly coupled gauge field theories.

Analytic bundle structure on the idempotent manifold

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and $$\mathcal {I}(X)$$ be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from $$\mathcal {L}(X)$$ , the subset $$\mathcal {I}(X)$$ is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $$\mathscr {G}(X)$$ , via the map $$\kappa $$ that sends $$Q\in \mathcal {I}(X)$$ to Q(X), such that the affine-Banach space structure on each fiber is the one induced from $$\mathcal {L}(X)$$ . Using this, we show that if H is a real or complex Hilbert space, then the assignment $$\begin{aligned} (E,T)\mapsto T^*\circ P_{E^\bot } + P_{E}, \quad \text {where}\quad E\in \mathscr {G}(H)\quad \text {and}\quad T\in \mathcal {L}(E,E^\bot ), \end{aligned}$$ induces a real bi-analytic bijection from the total space of the tangent bundle, $$\mathbf {T}(\mathscr {G}(H))$$ , of $$\mathscr {G}(H)$$ onto $$\mathcal {I}(H)$$ (here, $$E^\bot $$ is the orthogonal complement of E, $$P_E\in \mathcal {L}(H)$$ is the orthogonal projection onto E, and $$T^*$$ is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every $$E\in \mathscr {G}(H)$$ , we identify $$\mathcal {L}(E,E^\bot )$$ with a subspace of $$\mathcal {L}(H)$$ via the embedding $$S\mapsto S\circ P_E$$ , then the inclusion map from $$\mathbf {T}(\mathscr {G}(H))$$ to the trivial Banach bundle $$\mathscr {G}(H)\times \mathcal {L}(H)$$ is a real analytic immersion. Through this, we give a concrete idempotent in $$M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )$$ that represents the K-theory class of the tangent bundle $$\mathbf {T}(\mathscr {G}(\mathbb {K}^n))$$ , when $$\mathbb {K}$$ is either the real field or the complex field.

Projective limit of a sequence of compatible weak symplectic forms on a sequence of Banach bundles and Darboux Theorem

Given a projective sequence of Banach bundles, each one provided with of a weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on projective limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar's results ([15]). In particular we give an analog result of Kumar' s one with weaker assumptions and we give an example for which such weaker conditions are satisfied. More generally, we produce examples of projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true and an example for which the Darboux Theorem is true on each manifold, but is not true on the projective limit of these manifolds.

Gelfand-type theorems for dynamical Banach modules

The representation theorems of Gelfand and Kakutani for commutative C*-algebras and AM- and AL-spaces are the basis for the Koopman linearization of topological and measure-preserving dynamical systems. In this article we prove versions of these results for dynamics on topological and measurable Banach bundles and the corresponding weighted Koopman representations on Banach modules.

Local Lorentz invariance and a new theory of gravitation equivalent to general relativity

A gauge theory of the Lorentz group with a mass-dimension one gauge field coupling to matter of any spin is developed. As a completely new feature the ‘Vierbein’ assuring local gauge invariance enters not as an independent dynamical field, but emerges as a functional of the Lorentz gauge field. The underlying geometry of the theory turns out to be a SO(1,3) Banach bundle. The most general action which is renormalizable by power-counting is constructed in terms of the gauge field and its first derivatives. It contains no higher derivative terms in the gauge field which destroy unitarity in the usual renormalizable R2-theories of gravitation. Finally equivalence of the Lorentz gauge field theory coupled to spin zero matter with general relativity is established.

The Serre–Swan theorem for normed modules

The aim of this note is to analyse the structure of the $$L^0$$ -normed $$L^0$$ -modules over a metric measure space. These are a tool that has been introduced by Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an $$L^0$$ -normed $$L^0$$ -module can be viewed as the space of sections of a suitable measurable Banach bundle and in which sense such correspondence can be actually made into an equivalence of categories.

The Selgrade Decomposition for Linear Semiflows on Banach Spaces

We extend Selgrade's Theorem, Morse spectrum, and related concepts to the setting of linear skew product semiflows on a separable Banach bundle. We recover a characterization, well-known in the finite-dimensional setting, of exponentially separated subbundles as attractor-repeller pairs for the associated semiflow on the projective bundle.

A selection theorem for Banach bundles and applications

It is shown that certain lower semi-continuous maps from a paracompact space to the family of closed subsets of the bundle space of a Banach bundle admit continuous selections. This generalization of the theorem of Douady, dal Soglio-Herault, and Hofmann on the fullness of Banach bundles has applications to establishing conditions under which the induced maps between the spaces of sections of Banach bundles are onto. Another application is to a generalization of the theorem of Bartle and Graves [3] for Banach bundle maps that are onto their images. Other applications of the selection theorem are to the study of the M-ideals of the space of bounded sections begun in [4] and continued in [9]. A class of Banach bundles that generalizes the class of trivial Banach bundles is introduced and some properties of these Banach bundles are discussed.

Open quotients of trivial vector bundles

Given an arbitrary topological complex vector space A, a quotient vector bundle for A is a quotient of a trivial vector bundle π2:A×X→X by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space X. Hyperspaces consisting of linear subspaces of A, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grassmannians. If A is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bundles, and such that bundles of constant finite rank must be locally trivial.

Deformations of Fell bundles and twisted graph algebras

AbstractWe consider Fell bundles over discrete groups, and theC*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles with the same underlying Banach bundle but with the multiplication deformed by a two-cocycle on the group. Every graph algebra can be viewed as theC*-algebra of a Fell bundle, and there are many cocycles of interest with which to deform them. We thus obtain many of the twisted graph algebras of Kumjian, Pask and Sims. We demonstate the utility of our approach to these twisted graph algebras by proving that the deformations associated to different cocycles can be assembled as the fibres of aC*-bundle.

Distribution of finite-dimensional and separable stalks of an ample Banach bundle

The topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle are finite-dimensional or separable. A connection is established between the property of the stalks of a bundle to be finite-dimensional or separable with the analogous property of the stalks of the ample hull of the bundle. A new criterion is obtained for existence of the dual bundle.

Finite dimensionality and separability of the stalks of Banach bundles

Topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable. Some connection is established between finite dimensionality or separability of the stalks of a bundle and the analogous properties of the stalks of the ample hull of the bundle. We obtain a new criterion for existence of a dual bundle.

GEOMETRIC FORMULATION OF NON-AUTONOMOUS MECHANICS

We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fiber bundles over a time axis ℝ. Connections on these bundles describe reference frames. Quantum non-autonomous mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization.

Banach lattices of continuous sections

The aim of this note is to outline some application of ample continuous Banach bundles to the theory of Banach lattices.

Almost Lie structures on an anchored Banach bundle

Under appropriate assumptions, we generalize the concept of linear almost Poisson structures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the “Hilbert snake” as exposed in Pelletier and Saffidine (2011) [9].

ISOMETRIC EMBEDDINGS OF BANACH BUNDLES

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.