DOCSLIB.ORG

Dimensions and C*-Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dimensions and C*-Algebras Von Neumann Algebras C*-Algebras Dimensions and C*-Algebras Nigel Higson Department of Mathematics Pennsylvania State University British Mathematics Colloquium, 2006 Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras C*-Algebras Outline 1 Von Neumann Algebras Classification into Types Group Measure Space Construction 2 C*-Algebras Dimensions and de Rham Theory The Baum-Connes Conjecture Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Rings of Operators Definition π :Γ → U(H) a unitary representation of a group. Commutant: M = { T ∈ B(H): T π(g) = π(g)T }. Commutant = algebra of operators—a von Neumann algebra: T ∈ M ⇒ T ∗ ∈ M. John von Neumann Tn ∈ M & Tnv → Tv, ∀v ⇒ T ∈ M. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Projections and Subrepresentations M = T ∈ B(H): T π(g) = π(g)T , ∀g ∈ Γ Subrepresentations of π correspond to projections in M. Equivalence of subrepresentations corresponds to Murray-von Neumann equivalence of projections: P ∼ Q ⇔ P = UV , Q = VU. Definition Denote by Dim(M) the set of equivalence classes of projections—it is a partially ordered set Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Factors and Isotypical Representations Definition A unitary representation π :Γ → U(H) is isotypical if any two subrepresentations have equivalent subsubrepresentations. Theorem Every representation decomposes canonically into isotypical representations. Theorem The commutants of isotypical representations are factors—von Neumann algebras with trivial centers. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Factors and Isotypical Representations Type I Isotypical Representations In simple situations, an isotypical representation is necessarily a multiple of a single irreducible representation. The corresponding factor is isomorphic to the algebra of all bounded operators on a Hilbert space K , where the dimension of K is equal to the multiplicity of the irreducible representation. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Factors and Isotypical Representations In general, isotypical representations can be far more complicated! Example Regular representation of a free group F on `2(F). It can be decomposed into irreducible representations in two (indeed many) totally disjoint ways. The complete structure of this representation is unknown. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Murray-von Neumann Classification Theorem If M is a factor, then the partially ordered set Dim(M) of equivalence classes of projections in M is isomorphic one of: {0, 1, 2,..., N} Type I [0, K ] Type II {0, ∞} Type III Remark Possibly N = ∞ and K = ∞. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Examples? (X, µ): measure space. G: discrete group of transformations of X (preserving the class of µ) Definition M(G, X) is the von Neumann algebra of operators on L2(G × X) generated by: ∞ Multiplication operators Mf , f ∈ L (X) Unitary translation operators Ug, g ∈ G such that ∗ UgMf Ug = Mg(f ). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Group Measure Space Construction Key Idea: The unitary operators Ug ∈ M(G, X) implement the action of G on the algebra of measurable functions on X. Theorem Assume the action of G on X is free. Then M(G, X) is a factor if and only if the action of G on X is ergodic. Theorem Assume that the action of G on X is ergodic. Then M(G, X) is of type I or II if and only if there is an invariant measure ν in the class of µ, in which case DimM(G, X) =∼ ν(E): E ⊆ X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Examples X = Z, G = Z, action by translations. Type I. X = S1, G = Z, action by irrational rotations. Type II. X = S1, G = SL(2, Z), action by projective transformations of 1 ∼ 1 S = RP . Type III. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture C*-Algebras Definition A C∗-algebra is a norm-closed ∗-algebra of operators on a Hilbert space. ∗ C -algebra: Tn ∈ A & kTn − T k → 0 ⇒ T ∈ A. Von Neumann algebra: Tn ∈ M & Tnv → Tv, ⇒ T ∈ M. Abelian C∗-algebra: C(X). Abelian von Neumann algebra: L∞(X). Abelian group C∗-algebra: C∗(G) = C(Gb). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Crossed Product Algebras X: smooth manifold. G: discrete group of diffeomorphisms of X. Definition ∗ ∗ The crossed product algebra Cλ(G, X) is the C -algebra of operators on L2(G × X) generated by: Multiplication operators Mf , where f is smooth and compactly supported. Unitary translation operators Ug, g ∈ G such that ∗ UgMf Ug = Mg(f ). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Traces and Dimensions Lemma Assume that there is a G-invariant measure ν on X. The formula X Z τ : Mfg Ug 7→ fe(x) dν(x) g∈G X ∗ defines a trace on Cλ(G, X): ∗ τ(xy) = τ(yx), ∀x, y ∈ Cλ(G, X). If p and q are equivalent projections, then τ(p) = τ(q). We can ask: what is the set of numerical dimensions ∗ τ(p): p ∈ Cλ(G, X) ? Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Traces and Dimensions Improved Question Trace τ : A → C What is abelian group generated by Dimτ A = ? τ(p), where p ∈ Mn(A) Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture The Irrational Rotation C*-Algebra Example Let X = S1, and let G = Z act on the circle by rotations through multiples of the angle 2πθ, where θ is irrational. The C∗-algebra ∗ Cλ(G, X) is the irrational rotation algebra Aθ. Theorem Dimτ Aθ = Z + θZ. Theorem (Rieffel) τ(p): p ∈ A = Z + θZ ∩ [0, 1]. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Twisted Product M: Connected manifold with fundamental group G. X: Manifold with smooth action of G. Definition M#X = (Me × X)/G. Me × X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Twisted Product Me × X M#X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Ruelle-Sullivan Current M: Connected manifold with fundamental group G. X: Manifold with smooth action of G. ν: Invariant measure on X Definition n Let n = dim(M). Define Cν :Ωc (M#X) → R as follows: If ω is supported in a small set W ⊆ M#X diffeomorphic to U × V ⊆ Me × X, Z Z ! Cν(ω) = ω dν(x). V U×{x} For general ω, use a partition of unity. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Ruelle-Sullivan Current Me × X U V Local Definition of Current Projection Cν(ω) = Z Z ! W ω dν(x). V U×{x} M#X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Conjecture on Dimensions Theorem n−1 Cν is closed: Cν(dη) = 0, for all η ∈ Ωc (M#X). n As a result, Cν defines a map Cν : Hc (M#X) → R. We are interested in values on integral cohomology classes . Conjecture (Approximate Form) ∗ Dimτ Cλ(G, X) = Cν(ω): ω closed and integral }. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Irrational Rotations Example — Irrational Rotation Cν(ω): ω closed, integral = Z + θZ Alternate Definition of Current L3 1 Z Cν(ω) = lim ω N→∞ N LN Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Some Consequences Corollary (of the Conjecture) ∗ ∗ Take X = pt, so that Cλ(G, X) = Cλ(G). Then ∗ ∗ Dimτ Cλ(G) = Z. As a result, Cλ(G) has no nontrivial projections. Corollary (of the Conjecture, due to Connes) If the group G = Z acts on X = S3 via a minimal ∗ diffeomorphism, then Dimτ Cλ(G, X) = Z. As a result, ∗ Cλ(G, X) has no nontrivial projections. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Von Neumann Algebras, Revisited Assume G acts on (X, µ). Define an action on (X × R, µ × Lebesgue) by dg µ g :(x, t) 7→ g · x, log ∗ (x) + t. dµ Definition The module of the given action of G on X is the pair Mod(G, X) = (A, α), where ∞ A = abelian von Neumann algebra = L (X × R)G α = action of R = translation action. Theorem The module is an invariant of M(G, X). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Type III Factors Assuming the action of G on X is ergodic, the module, Mod(G, X), is an ergodic flow.
Load more

Recommended publications
  • 2 Hilbert Spaces You Should Have Seen Some Examples Last Semester
    2 Hilbert spaces You should have seen some examples last semester. The simplest (finite-dimensional) ex- C • A complex Hilbert space H is a complete normed space over whose norm is derived from an ample is Cn with its standard inner product. It’s worth recalling from linear algebra that if V is inner product. That is, we assume that there is a sesquilinear form ( , ): H H C, linear in · · × → an n-dimensional (complex) vector space, then from any set of n linearly independent vectors we the first variable and conjugate linear in the second, such that can manufacture an orthonormal basis e1, e2,..., en using the Gram-Schmidt process. In terms of this basis we can write any v V in the form (f ,д) = (д, f ), ∈ v = a e , a = (v, e ) (f , f ) 0 f H, and (f , f ) = 0 = f = 0. i i i i ≥ ∀ ∈ ⇒ ∑ The norm and inner product are related by which can be derived by taking the inner product of the equation v = aiei with ei. We also have n ∑ (f , f ) = f 2. ∥ ∥ v 2 = a 2. ∥ ∥ | i | i=1 We will always assume that H is separable (has a countable dense subset). ∑ Standard infinite-dimensional examples are l2(N) or l2(Z), the space of square-summable As usual for a normed space, the distance on H is given byd(f ,д) = f д = (f д, f д). • • ∥ − ∥ − − sequences, and L2(Ω) where Ω is a measurable subset of Rn. The Cauchy-Schwarz and triangle inequalities, • √ (f ,д) f д , f + д f + д , | | ≤ ∥ ∥∥ ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∥ 2.1 Orthogonality can be derived fairly easily from the inner product.
    [Show full text]
  • Maxwell's Equations in Differential Form
    M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 71 CHAPTER Maxwell’s Equations in Differential Form 3 In Chapter 2 we introduced Maxwell’s equations in integral form. We learned that the quantities involved in the formulation of these equations are the scalar quantities, elec- tromotive force, magnetomotive force, magnetic flux, displacement flux, charge, and current, which are related to the field vectors and source densities through line, surface, and volume integrals. Thus, the integral forms of Maxwell’s equations,while containing all the information pertinent to the interdependence of the field and source quantities over a given region in space, do not permit us to study directly the interaction between the field vectors and their relationships with the source densities at individual points. It is our goal in this chapter to derive the differential forms of Maxwell’s equations that apply directly to the field vectors and source densities at a given point. We shall derive Maxwell’s equations in differential form by applying Maxwell’s equations in integral form to infinitesimal closed paths, surfaces, and volumes, in the limit that they shrink to points. We will find that the differential equations relate the spatial variations of the field vectors at a given point to their temporal variations and to the charge and current densities at that point. In this process we shall also learn two important operations in vector calculus, known as curl and divergence, and two related theorems, known as Stokes’ and divergence theorems. 3.1 FARADAY’S LAW We recall from the previous chapter that Faraday’s law is given in integral form by d E # dl =- B # dS (3.1) CC dt LS where S is any surface bounded by the closed path C.In the most general case,the elec- tric and magnetic fields have all three components (x, y,and z) and are dependent on all three coordinates (x, y,and z) in addition to time (t).
    [Show full text]
  • Some C*-Algebras with a Single Generator*1 )
    transactions of the american mathematical society Volume 215, 1976 SOMEC*-ALGEBRAS WITH A SINGLEGENERATOR*1 ) BY CATHERINE L. OLSEN AND WILLIAM R. ZAME ABSTRACT. This paper grew out of the following question: If X is a com- pact subset of Cn, is C(X) ® M„ (the C*-algebra of n x n matrices with entries from C(X)) singly generated? It is shown that the answer is affirmative; in fact, A ® M„ is singly generated whenever A is a C*-algebra with identity, generated by a set of n(n + l)/2 elements of which n(n - l)/2 are selfadjoint. If A is a separable C*-algebra with identity, then A ® K and A ® U are shown to be sing- ly generated, where K is the algebra of compact operators in a separable, infinite- dimensional Hubert space, and U is any UHF algebra. In all these cases, the gen- erator is explicitly constructed. 1. Introduction. This paper grew out of a question raised by Claude Scho- chet and communicated to us by J. A. Deddens: If X is a compact subset of C, is C(X) ® M„ (the C*-algebra ofnxn matrices with entries from C(X)) singly generated? We show that the answer is affirmative; in fact, A ® Mn is singly gen- erated whenever A is a C*-algebra with identity, generated by a set of n(n + l)/2 elements of which n(n - l)/2 are selfadjoint. Working towards a converse, we show that A ® M2 need not be singly generated if A is generated by a set con- sisting of four elements.
    [Show full text]
  • And Varifold-Based Registration of Lung Vessel and Airway Trees
    Current- and Varifold-Based Registration of Lung Vessel and Airway Trees Yue Pan, Gary E. Christensen, Oguz C. Durumeric, Sarah E. Gerard, Joseph M. Reinhardt The University of Iowa, IA 52242 {yue-pan-1, gary-christensen, oguz-durumeric, sarah-gerard, joe-reinhardt}@uiowa.edu Geoffrey D. Hugo Virginia Commonwealth University, Richmond, VA 23298 [email protected] Abstract 1. Introduction Registration of lung CT images is important for many radiation oncology applications including assessing and Registering lung CT images is an important problem adapting to anatomical changes, accumulating radiation for many applications including tracking lung motion over dose for planning or assessment, and managing respiratory the breathing cycle, tracking anatomical and function motion. For example, variation in the anatomy during radio- changes over time, and detecting abnormal mechanical therapy introduces uncertainty between the planned and de- properties of the lung. This paper compares and con- livered radiation dose and may impact the appropriateness trasts current- and varifold-based diffeomorphic image of the originally-designed treatment plan. Frequent imaging registration approaches for registering tree-like structures during radiotherapy accompanied by accurate longitudinal of the lung. In these approaches, curve-like structures image registration facilitates measurement of such variation in the lung—for example, the skeletons of vessels and and its effect on the treatment plan. The cumulative dose airways segmentation—are represented by currents or to the target and normal tissue can be assessed by mapping varifolds in the dual space of a Reproducing Kernel Hilbert delivered dose to a common reference anatomy and com- Space (RKHS). Current and varifold representations are paring to the prescribed dose.
    [Show full text]
  • Arxiv:1802.08667V5 [Stat.ML] 13 Apr 2021 Lwrta 1 Than Slower Asna Eoo N,Aogagvnsqec Fmodels)
    Manuscript submitted to The Econometrics Journal, pp. 1–49. De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers Victor Chernozhukov†, Whitney K. Newey†, and Rahul Singh† †MIT Economics, 50 Memorial Drive, Cambridge MA 02142, USA. E-mail: [email protected], [email protected], [email protected] Summary We provide adaptive inference methods, based on ℓ1 regularization, for regular (semi-parametric) and non-regular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects, and derivatives. Examples of non-regular functionals include average treatment effects, policy effects, and derivatives conditional on a co- variate subvector fixed at a point. We construct a Neyman orthogonal equation for the target parameter that is approximately invariant to small perturbations of the nui- sance parameters. To achieve this property, we include the Riesz representer for the functional as an additional nuisance parameter. Our analysis yields weak “double spar- sity robustness”: either the approximation to the regression or the approximation to the representer can be “completely dense” as long as the other is sufficiently “sparse”. Our main results are non-asymptotic and imply asymptotic uniform validity over large classes of models, translating into honest confidence bands for both global and local parameters. Keywords: Neyman orthogonality, Gaussian approximation, sparsity 1. INTRODUCTION Many statistical objects of interest can be expressed as a linear functional of a regression function (or projection, more generally). Examples include global parameters: average treatment effects, policy effects from changing the distribution of or transporting regres- sors, and average directional derivatives, as well as their local versions defined by taking averages over regions of shrinking volume.
    [Show full text]
  • Hilbert Space Theory and Applications in Basic Quantum Mechanics by Matthew R
    1 Hilbert Space Theory and Applications in Basic Quantum Mechanics by Matthew R. Gagne Mathematics Department California Polytechnic State University San Luis Obispo 2013 2 Abstract We explore the basic mathematical physics of quantum mechanics. Our primary focus will be on Hilbert space theory and applications as well as the theory of linear operators on Hilbert space. We show how Hermitian operators are used to represent quantum observables and investigate the spectrum of various linear operators. We discuss deviation and uncertainty and brieáy suggest how symmetry and representations are involved in quantum theory. APPROVAL PAGE TITLE: Hilbert Space Theory and Applications in Basic Quantum Mechanics AUTHOR: Matthew R. Gagne DATE SUBMITTED: June 2013 __________________ __________________ Senior Project Advisor Senior Project Advisor (Professor Jonathan Shapiro) (Professor Matthew Moelter) __________________ __________________ Mathematics Department Chair Physics Department Chair (Professor Joe Borzellino) (Professor Nilgun Sungar) 3 Contents 1 Introduction and History 4 1.1 Physics . 4 1.2 Mathematics . 7 2 Hilbert Space deÖnitions and examples 12 2.1 Linear functionals . 12 2.2 Metric, Norm and Inner product spaces . 14 2.3 Convergence and completeness . 17 2.4 Basis system and orthogonality . 19 3 Linear Operators 22 3.1 Basics . 22 3.2 The adjoint . 25 3.3 The spectrum . 32 4 Physics 36 4.1 Quantum mechanics . 37 4.1.1 Linear operators as observables . 37 4.1.2 Position, Momentum, and Energy . 42 4.2 Deviation and Uncertainty . 49 4.3 The Qubit . 51 4.4 Symmetry and Time Evolution . 54 1 Chapter 1 Introduction and History The development of Hilbert space, and its subsequent popularity, were a result of both mathematical and physical necessity.
    [Show full text]
  • Linear Algebra I: Vector Spaces A
    Linear Algebra I: Vector Spaces A 1 Vector spaces and subspaces 1.1 Let F be a field (in this book, it will always be either the field of reals R or the field of complex numbers C). A vector space V D .V; C; o;˛./.˛2 F// over F is a set V with a binary operation C, a constant o and a collection of unary operations (i.e. maps) ˛ W V ! V labelled by the elements of F, satisfying (V1) .x C y/ C z D x C .y C z/, (V2) x C y D y C x, (V3) 0 x D o, (V4) ˛ .ˇ x/ D .˛ˇ/ x, (V5) 1 x D x, (V6) .˛ C ˇ/ x D ˛ x C ˇ x,and (V7) ˛ .x C y/ D ˛ x C ˛ y. Here, we write ˛ x and we will write also ˛x for the result ˛.x/ of the unary operation ˛ in x. Often, one uses the expression “multiplication of x by ˛”; but it is useful to keep in mind that what we really have is a collection of unary operations (see also 5.1 below). The elements of a vector space are often referred to as vectors. In contrast, the elements of the field F are then often referred to as scalars. In view of this, it is useful to reflect for a moment on the true meaning of the axioms (equalities) above. For instance, (V4), often referred to as the “associative law” in fact states that the composition of the functions V ! V labelled by ˇ; ˛ is labelled by the product ˛ˇ in F, the “distributive law” (V6) states that the (pointwise) sum of the mappings labelled by ˛ and ˇ is labelled by the sum ˛ C ˇ in F, and (V7) states that each of the maps ˛ preserves the sum C.
    [Show full text]
  • Numerical Characterization of the Kähler
    Annals of Mathematics, 159 (2004), 1247–1274 Numerical characterization of the K¨ahler cone of a compact K¨ahler manifold By Jean-Pierre Demailly and Mihai Paun Abstract The goal of this work is to give a precise numerical description of the K¨ahler cone of a compact K¨ahler manifold. Our main result states that the K¨ahler cone depends only on the intersection form of the cohomology ring, the Hodge structure and the homology classes of analytic cycles: if X is a compact K¨ahler manifold, the K¨ahler cone K of X is one of the connected components of the set P of real (1, 1)-cohomology classes {α} which are numerically positive p on analytic cycles, i.e. Y α > 0 for every irreducible analytic set Y in X, p = dim Y . This result is new even in the case of projective manifolds, where it can be seen as a generalization of the well-known Nakai-Moishezon criterion, and it also extends previous results by Campana-Peternell and Eyssidieux. The principal technical step is to show that every nef class {α} which has positive n highest self-intersection number X α > 0 contains a K¨ahler current; this is done by using the Calabi-Yau theorem and a mass concentration technique for Monge-Amp`ere equations. The main result admits a number of variants and corollaries, including a description of the cone of numerically effective (1, 1)-classes and their dual cone. Another important consequence is the fact that for an arbitrary deformation X → S of compact K¨ahler manifolds, the K¨ahler cone of a very general fibre Xt is “independent” of t, i.e.
    [Show full text]
  • Lecture Notes and Exercises for PDE and Hilbert Spaces
    Math Tune-Up Louisiana State University August, 2008 Lectures on Partial Differential Equations and Hilbert Space 1. A linear partial differential equation of physics We begin by considering the simplest mathematical model of conduction of electricity in a material body. This is a part of electrostatics. In the body, there is an electric field E at equilibrium, which gives rise to a static electric current J (steady movement of charge). The electric field is conservative, meaning that its line integral about any closed curve in the body vanishes. Equivalently, it is generated by a scalar potential u: E = −∇u: (1.1) The electric field exerts a force on charged particles in the medium, causing a current den- sity J. The current and the electric field are related through a (real-valued) tensor σ; this is an example of a constitutive relation: J = σE: (1.2) In a static situation, the charge density ρ (charge per volume) is constant in time. This means that the charge flowing out of any region R of the material through its boundary @R must balance the charge entering into the body through an external source, which we denote by f (charge density per time): Z Z J · n dS = f: (1.3) @R R R R Since this holds for each region R, the Divergence Theorem @R J · n dS = R r·J dV , gives −∇·J = f: (1.4) In summary, the fields u, E, J, and f are related as follows: u = electric potential (energy/charge); (1.5) E = −∇u = electric field (force/charge); (1.6) J = σE = −σru = current density (charge/(time·area)); (1.7) f = −∇·σru = source (charge/(time·volume)): (1.8) The final equation is our inhomogeneous, linear, scalar, second-order, divergence-form, ellip- tic, partial differential equation: −∇·σru = f: (1.9) This equation can also be obtained through a variational principle: it is the \Euler- Lagrange" equation corresponding to a certain \Lagrangian density".
    [Show full text]
  • Integral Representation Theorems in Topological Vector Spaces
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 172, October 1972 INTEGRAL REPRESENTATION THEOREMS IN TOPOLOGICAL VECTOR SPACES BY ALAN H. SHUCHATi1) ABSTRACT. We present a theory of measure and integration in topological vector spaces and generalize the Fichtenholz-Kantorovich-Hildebrandt and Riesz representation theorems to this setting, using strong integrals. As an application, we find the containing Banach space of the space of continuous p-normed space-valued functions. It is known that Bochner integration in p-normed spaces, using Lebesgue measure, is not well behaved and several authors have developed integration theories for restricted classes of func- tions. We find conditions under which scalar measures do give well-behaved vector integrals and give a method for constructing examples. In recent years, increasing attention has been paid to topological vector spaces that are not locally convex and to vector-valued measures and integrals. In this article we present a theory of measure and integration in the setting of arbitrary topological vector spaces and corresponding generalizations of the Fichtenholz-Kantorovich-Hildebrandt and Riesz representation theorems. Our integral is derived from the Radon-type integral developed by Singer [20] and Dinculeanu [3] in Banach spaces. This often seems to lend itself to "softer" proofs than the Riemann-Stieltjes-type integral used by Swong [22] and Goodrich [8] in locally convex spaces. Also, although the tensor product notation is con- venient in certain places, we avoid the use of topological tensor products as in [22]. Since the spaces need not be locally convex, we must also avoid the use of the Hahn-Banach theorem as in [3], [20], for example.
    [Show full text]
  • The Holomorphic Functional Calculus Approach to Operator Semigroups
    Chapter 8 The Holomorphic Functional Calculus Approach to Operator Semigroups In Chapter 6 we have associated with a given C0-semigroup an operator, called its generator, and a functional calculus, the Hille{Phillips calculus. Here we address the converse problem: which operators are generators? The classical answer to this question and one of our goals in this chapter is the so- called Hille{Yosida theorem, which gives a characterization of the generator property in terms of resolvent estimates. However, instead of following the standard texts like [1], we shall approach the problem via a suitable functional calculus construction. Our method is generic, in that we learn here how to use Cauchy integrals over infinite con- tours in order to define functional calculi for unbounded operators satisfying resolvent estimates. (The same approach shall be taken later for defining holomorphic functional calculi for sectorial and strip-type operators.) This chapter is heavily based on the first part of [2], which in turn goes back to the unpublished note [3]. 8.1 Operators of Half-Plane Type Recall from Chapter 6 that if B generates a C0-semigroup of type (M; !), then s(B) is contained in the closed half-plane [ Re z ≤ ! ] and M kR(λ, B)k ≤ (Re λ > !): Re λ − ! In particular, if ! < α then R(·;A) is uniformly bounded for Re λ ≥ α. This \spectral data" is our model for what we shall call below an operator of \left half-plane type". Recall, however, that the associated Hille-Phillips calculus for the semi- group generated by B is actually a calculus for the negative generator 135 136 8 The Holomorphic Functional Calculus Approach to Operator Semigroups A := −B.
    [Show full text]
  • Low-Dimensional Electric Charges. Covariant Description 2 Which Is Explicitly Invariant Under the Diffeomorphism of the Manifold
    Low-dimensional electric charges. Covariant description Yakov Itin Institute of Mathematics, Hebrew University of Jerusalem and Jerusalem College of Technology E-mail: [email protected] Abstract. A compact and elegant description of the electromagnetic fields in media and in vacuum is attained in the differential forms formalism. This description is explicitly invariant under diffeomorphisms of the spacetime so it is suitable for arbitrary curvilinear coordinates. Moreover, it is independent of the geometry of the underline spacetime. The bulk electric charge and current densities are represented by twisted non-singular differential 3-forms. The charge and current densities with a support on the low dimensional submanifolds (surfaces, strings and points) naturally require singular differential forms. In this paper, we present a covariant metric-free description of the surface, string and point densities. It is shown that a covariant description requires Dirac’s delta- forms instead of delta-functions. Covariant metric-free conservation laws for the low-dimensional densities are derived. PACS numbers: 75.50.Ee, 03.50.De, 46.05.+b, 14.80.Mz 1. Introduction: Maxwell equations for differential forms The 3-dimensional description of the electromagnetic field in media is given in term of four vector fields [1]–[3]. Two of these fields, the electric field strength E and the magnetic field strength B, are of an intensity nature [1]. In the 4D-tensorial formalism, these fields are treated as the components of an antisymmetric second order tensor of the electromagnetic field strength Fij . Two other vector fields, the electric excitation D and the magnetic excitation H, are both of a quantity nature [1] and assembled in a tensorial density Hij .
    [Show full text]