Mathematical Description of Physical Systems Federico Zalamea
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Abelian, Amenable Operator Algebras Are Similar to C∗-Algebras
ABELIAN, AMENABLE OPERATOR ALGEBRAS ARE SIMILAR TO C∗-ALGEBRAS LAURENT W. MARCOUX1 AND ALEXEY I. POPOV Abstract. Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C∗-algebra. We do this by showing that if A ⊆ B(H) is an abelian algebra with the property that given any bounded representation % : A!B(H%) of A on a Hilbert space H%, every invariant subspace of %(A) is topologically complemented by another invariant subspace of %(A), then A is similar to an abelian C∗-algebra. 1. Introduction. 1.1. Let A be a Banach algebra and X be a Banach space which is also a bimodule over A. We say that X is a Banach bimodule over A if the module operations are continuous; that is, if there exists κ > 0 so that kaxk 6 κkak kxk, and kxbk 6 κkxk kbk for all a; b 2 A and x 2 X. Given a Banach bimodule X over A, we introduce an action of A upon the dual space X∗ of X under which X∗ becomes a dual Banach A-bimodule. This is the so-called dual action: (ax∗)(x) = x∗(xa) and (x∗a)(x) = x∗(ax) for all a 2 A, x 2 X, x∗ 2 X∗. A (continuous) derivation from a Banach algebra A into a Banach A-bimodule X is a continuous linear map δ : A! X satisfying δ(ab) = aδ(b) + δ(a)b for all a; b 2 A. -
On the Space of Generalized Fluxes for Loop Quantum Gravity
Home Search Collections Journals About Contact us My IOPscience On the space of generalized fluxes for loop quantum gravity This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2013 Class. Quantum Grav. 30 055008 (http://iopscience.iop.org/0264-9381/30/5/055008) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 194.94.224.254 The article was downloaded on 07/03/2013 at 11:02 Please note that terms and conditions apply. IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 30 (2013) 055008 (24pp) doi:10.1088/0264-9381/30/5/055008 On the space of generalized fluxes for loop quantum gravity B Dittrich1,2, C Guedes1 and D Oriti1 1 Max Planck Institute for Gravitational Physics, Am Muhlenberg¨ 1, D-14476 Golm, Germany 2 Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo ON N2 L 2Y5, Canada E-mail: [email protected], [email protected] and [email protected] Received 5 September 2012, in final form 13 January 2013 Published 5 February 2013 Online at stacks.iop.org/CQG/30/055008 Abstract We show that the space of generalized fluxes—momentum space—for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections—position space— due to the non-Abelianess of the gauge group SU(2). From the Abelianization of SU(2),U(1)3, we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse graining of the underlying graphs. -
Chapter 2 C -Algebras
Chapter 2 C∗-algebras This chapter is mainly based on the first chapters of the book [Mur90]. Material bor- rowed from other references will be specified. 2.1 Banach algebras Definition 2.1.1. A Banach algebra C is a complex vector space endowed with an associative multiplication and with a norm k · k which satisfy for any A; B; C 2 C and α 2 C (i) (αA)B = α(AB) = A(αB), (ii) A(B + C) = AB + AC and (A + B)C = AC + BC, (iii) kABk ≤ kAkkBk (submultiplicativity) (iv) C is complete with the norm k · k. One says that C is abelian or commutative if AB = BA for all A; B 2 C . One also says that C is unital if 1 2 C , i.e. if there exists an element 1 2 C with k1k = 1 such that 1B = B = B1 for all B 2 C . A subalgebra J of C is a vector subspace which is stable for the multiplication. If J is norm closed, it is a Banach algebra in itself. Examples 2.1.2. (i) C, Mn(C), B(H), K (H) are Banach algebras, where Mn(C) denotes the set of n × n-matrices over C. All except K (H) are unital, and K (H) is unital if H is finite dimensional. (ii) If Ω is a locally compact topological space, C0(Ω) and Cb(Ω) are abelian Banach algebras, where Cb(Ω) denotes the set of all bounded and continuous complex func- tions from Ω to C, and C0(Ω) denotes the subset of Cb(Ω) of functions f which vanish at infinity, i.e. -
Banach Algebras
Banach Algebras Yurii Khomskii Bachelor Thesis Department of Mathematics, Leiden University Supervisor: Dr. Marcel de Jeu April 18, 2005 i Contents Foreword iv 1. Algebraic Concepts 1 1.1. Preliminaries . 1 1.2. Regular Ideals . 3 1.3. Adjoining an Identity . 4 1.4. Quasi-inverses . 8 2. Banach Algebras 10 2.1. Preliminaries of Normed and Banach Algebras . 10 2.2. Inversion and Quasi-inversion in Banach Algebras . 14 3. Spectra 18 3.1. Preliminaries . 18 3.2. Polynomial Spectral Mapping Theorem and the Spectral Radius Formula . 22 4. Gelfand Representation Theory 25 4.1. Multiplicative Linear Functionals and the Maximal Ideal Space . 25 4.2. The Gelfand Topology . 30 4.3. The Gelfand Representation . 31 4.4. The Radical and Semi-simplicity . 33 4.5. Generators of Banach algebras . 34 5. Examples of Gelfand Representations 36 5.1. C (X ) for X compact and Hausdorff . 36 5.2. C 0(X ) for X locally compact and Hausdorff. 41 5.3. Stone-Cecˇ h compactification . 42 5.4. A(D) . 44 5.5. AC (Γ) . 46 5.6. H 1 . 47 ii iii Foreword The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard exam- ple was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infin- ity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. -
Mathematical Tripos Part III Lecture Courses in 2013-2014
Mathematical Tripos Part III Lecture Courses in 2013-2014 Department of Pure Mathematics & Mathematical Statistics Department of Applied Mathematics & Theoretical Physics Notes and Disclaimers. • Students may take any combination of lectures that is allowed by the timetable. The examination timetable corresponds to the lecture timetable and it is therefore not possible to take two courses for examination that are lectured in the same timetable slot. There is no requirement that students study only courses offered by one Department. • The code in parentheses after each course name indicates the term of the course (M: Michaelmas; L: Lent; E: Easter), and the number of lectures in the course. Unless indicated otherwise, a 16 lecture course is equivalent to 2 credit units, while a 24 lecture course is equivalent to 3 credit units. Please note that certain courses are non-examinable, and are indicated as such after the title. Some of these courses may be the basis for Part III essays. • At the start of some sections there is a paragraph indicating the desirable previous knowledge for courses in that section. On one hand, such paragraphs are not exhaustive, whilst on the other, not all courses require all the pre-requisite material indicated. However you are strongly recommended to read up on the material with which you are unfamiliar if you intend to take a significant number of courses from a particular section. • The courses described in this document apply only for the academic year 2013-14. Details for subsequent years are often broadly similar, but not necessarily identical. The courses evolve from year to year. -
Spectral Theory
SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensional case. Namely, we first show that an eigenvector exists, and then we show that there is an orthonormal basis of eigenvectors by an inductive argument (in the form of an application of Zorn's lemma). We first recall a few facts about self-adjoint operators. Proposition 1.1. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. (1) If W ⊂ V is a T -invariant subspace, then W ? is T -invariant. (2) For all v 2 V , hT v; vi 2 R. (3) Let Vλ = fv 2 V j T v = λvg. Then Vλ ? Vµ whenever λ 6= µ. We will need one further technical fact that characterizes the norm of a self-adjoint operator. Lemma 1.2. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. Then kT k = supfjhT v; vij j kvk = 1g. Proof. Let α = supfjhT v; vij j kvk = 1g. Evidently, α ≤ kT k. We need to show the other direction. Given v 2 V with T v 6= 0, setting w0 = T v=kT vk gives jhT v; w0ij = kT vk. -
MAT 449 : Representation Theory
MAT 449 : Representation theory Sophie Morel December 17, 2018 Contents I Representations of topological groups5 I.1 Topological groups . .5 I.2 Haar measures . .9 I.3 Representations . 17 I.3.1 Continuous representations . 17 I.3.2 Unitary representations . 20 I.3.3 Cyclic representations . 24 I.3.4 Schur’s lemma . 25 I.3.5 Finite-dimensional representations . 27 I.4 The convolution product and the group algebra . 28 I.4.1 Convolution on L1(G) and the group algebra of G ............ 28 I.4.2 Representations of G vs representations of L1(G) ............. 33 I.4.3 Convolution on other Lp spaces . 39 II Some Gelfand theory 43 II.1 Banach algebras . 43 II.1.1 Spectrum of an element . 43 II.1.2 The Gelfand-Mazur theorem . 47 II.2 Spectrum of a Banach algebra . 48 II.3 C∗-algebras and the Gelfand-Naimark theorem . 52 II.4 The spectral theorem . 55 III The Gelfand-Raikov theorem 59 III.1 L1(G) ....................................... 59 III.2 Functions of positive type . 59 III.3 Functions of positive type and irreducible representations . 65 III.4 The convex set P1 ................................. 68 III.5 The Gelfand-Raikov theorem . 73 IV The Peter-Weyl theorem 75 IV.1 Compact operators . 75 IV.2 Semisimplicity of unitary representations of compact groups . 77 IV.3 Matrix coefficients . 80 IV.4 The Peter-Weyl theorem . 86 IV.5 Characters . 87 3 Contents IV.6 The Fourier transform . 90 IV.7 Characters and Fourier transforms . 93 V Gelfand pairs 97 V.1 Invariant and bi-invariant functions . -
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Arxiv:Math/0203199V1 [Math.FA] 19 Mar 2002
Amenability for dual Banach algebras Volker Runde Abstract ∗ We define a Banach algebra A to be dual if A = (A∗) for a closed submodule A∗ of A∗. The class of dual Banach algebras includes all W ∗-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Ba- nach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability — Connes-amenability and strong Connes-amenability — which take the w∗-topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes-amenability of A∗∗; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connes-amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for C∗-algebras. 2000 Mathematics Subject Classification: 43A10, 46B28, 46H25, 46H99 (primary), 46L10, 46M20. 1 Introduction arXiv:math/0203199v1 [math.FA] 19 Mar 2002 Amenable Banach algebras were introduced by B. E. Johnson in [Joh 2], and have since then turned out to be extremely interesting objects of research. -
Lecture 2: Operator Amenability of the Fourier Algebra
Lecture 2: Operator Amenability of the Fourier Algebra Zhong-Jin Ruan at Leeds, Tuesday 18 May , 2010 1 Amenability of Groups Let us first recall that a locally compact group G is amenable if there exists a left invariant mean on L∞(G), i.e. there exists a positive linear functional m : L∞(G) → C such that m(1) = 1 and m(sh) = m(h) for all s ∈ G and h ∈ L∞(G), where we define sh(t) = h(st). Theorem: The following are equivalent: 1. G is amenable; 2. G satisfies the Følner condition: for every ε > 0 and compact subset C ⊆ G, there exists a compact subset K ⊆ G such that |K∆sK| < ε for all s ∈ C; µ(K) 3. A(G) has a bounded (or contractive) approximate identity. 2 Amenability of Banach Algebras In the early 1970’s, B. Johnson introduced the bounded Hochschild cohomology and the amenability for Banach algebras. A Banach algebra A is called amenable if for every bounded A-bimodule V , every bounded derivation D : A → V ∗ is inner, i.e. there exists f ∈ V ∗ such that D(a) = a · f − f · a. A linear map D : A → V ∗ is a derivation if it satisfies D(ab) = D(a) · b + a · D(b) for all a, b ∈ A. Theorem [B. Johnson 1972] : A locally compact group G is amenable if and only if L1(G) is an amenable Banach algebra. 3 Amenability of A(G) Question: Since A(G) is the natural dual object of L1(G), it is natural to ask whether G is amenable if and only if A(G) is amenable ?? B. -
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Arxiv:1811.10987V1 [Math.FA]
FRECHET´ ALGEBRAS IN ABSTRACT HARMONIC ANALYSIS Z. ALIMOHAMMADI AND A. REJALI Abstract. We provide a survey of the similarities and differences between Banach and Fr´echet algebras including some known results and examples. We also collect some important generalizations in abstract harmonic anal- ysis; for example, the generalization of the concepts of vector-valued Lips- chitz algebras, abstract Segal algebras, Arens regularity, amenability, weak amenability, ideal amenability, etc. 0. Introduction The class of Fr´echet algebras which is an important class of locally convex algebras has been widely studied by many authors. For a full understanding of Fr´echet algebras, one may refer to [27, 36]. In the class of Banach algebras, there are many concepts which were generalized to the Fr´echet case. Examples of these concepts are as follows. Biprojective Banach algebras were studied by several authors, notably A. Ya. Helemskii [35] and Yu. V. Selivanov [62]. A number of papers appeared in the literature concerned with extending the notion of biprojectivity and its results to the Fr´echet algebras; see for example [48, 63]. Flat cyclic Fr´echet modules were investigated by A. Yu. Pirkovskii [49]. Let A be a Fr´echet algebra and I be a closed left ideal in A. Pirkovskii showed that there exists a necessary and sufficient condition for a cyclic Fr´echet A-module A♯/I to be strictly flat. He also generalized a number of characterizations of amenable Ba- nach algebras to the Fr´echet algebras (ibid.). In 2008, P. Lawson and C. J. Read [43] introduced and studied the notions of approximate amenability and approxi- mate contractibility of Fr´echet algebras. -
Israel Moiseevich Gelfand, (1913 - 2009)
1 Israel Moiseevich Gelfand, (1913 - 2009) V.S.Sunder The Institute of Mathematical Sciences, Taramani, Chennai 600113, India e-mail: [email protected] January 17, 2011 When Gelfand passed away on October 5th, 2009, the world might have seen the last of the classical scholars (in the mould of Henri Poincare or John von Neumann) whose accomplishments/scholarship were not confined by artificial borders. The wikipedia ‘paraphrases’ his work thus: ‘Israel Gelfand is known for many developments including: • the Gelfand representation in Banach algebra theory; • the Gelfand Mazur theorem in Banach algebra theory; • the Gelfand Naimark theorem; • the Gelfand Naimark Segal construction; • Gelfand Shilov spaces • the Gelfand Pettis integral; • the representation theory of the complex classical Lie groups; • contributions to the theory of Verma modules in the represen- tation theory of semisimple Lie algebras (with I.N. Bernstein and S.I. Gelfand); • contributions to distribution theory and measures on infinite- dimensional spaces; • the first observation of the connection of automorphic forms with representations (with Sergei Fomin); • conjectures about the index theorem; 2 • Ordinary differential equations (Gelfand Levitan theory); • work on calculus of variations and soliton theory (Gelfand Dikii equations); • contributions to the philosophy of cusp forms; • Gelfand Fuks cohomology of foliations; • Gelfand Kirillov dimension; • integral geometry; • combinatorial definition of the Pontryagin class; • Coxeter functors; • generalised hypergeometric series; • Gelfand - Tsetlin patterns; • and many other results, particularly in the representation the- ory for the classical groups.’ When I was requested to write something about Gelfand, my first reaction was to say ‘how can I hope to do justice to his breadth and depth of mathematical contributions?’ and at once realised that no one can. -
MATHEMATICAL TRIPOS Part III PAPER 106 FUNCTIONAL ANALYSIS
MATHEMATICAL TRIPOS Part III Monday, 30 May, 2016 9:00 am to 12:00 pm PAPER 106 FUNCTIONAL ANALYSIS Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Coversheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Let A be a unital Banach algebra. Prove that the group G(A) of invertible elements of A is open and that the map x → x−1 : G(A) → G(A) is continuous. Let A be a Banach algebra and x ∈ A. Define the spectrum σA(x) of x in A both in the case that A is unital and in the case A is not unital. Let A be a Banach algebra and x ∈ A. Prove that σA(x) is a non-empty compact subset of C. [You may assume any version of the Hahn–Banach theorem without proof.] State and prove the Gelfand–Mazur theorem concerning complex unital normed division algebras. Let K be an arbitrary set. Let A be an algebra of complex-valued functions on K with pointwise operations and assume that · is a complete algebra norm on A. Prove carefully that supK|f| 6 f for all f ∈ A. 2 State and prove Mazur’s theorem on the weak-closure and norm-closure of a convex set in a Banach space. [Any version of the Hahn–Banach theorem can be assumed without proof provided it is clearly stated.] Let (xn) be a sequence in a Banach space X.