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Noncommutative Uniform Algebras
Noncommutative Uniform Algebras Mati Abel and Krzysztof Jarosz Abstract. We show that a real Banach algebra A such that a2 = a 2 , for a A is a subalgebra of the algebra C (X) of continuous quaternion valued functions on a compact setk kX. ∈ H ° ° ° ° 1. Introduction A well known Hirschfeld-Zelazkoú Theorem [5](seealso[7]) states that for a complex Banach algebra A the condition (1.1) a2 = a 2 , for a A k k ∈ implies that (i) A is commutative, and further° ° implies that (ii) A must be of a very speciÞc form, namely it ° ° must be isometrically isomorphic with a uniformly closed subalgebra of CC (X). We denote here by CC (X)the complex Banach algebra of all continuous functions on a compact set X. The obvious question about the validity of the same conclusion for real Banach algebras is immediately dismissed with an obvious counterexample: the non commutative algebra H of quaternions. However it turns out that the second part (ii) above is essentially also true in the real case. We show that the condition (1.1) implies, for a real Banach algebra A,thatA is isometrically isomorphic with a subalgebra of CH (X) - the algebra of continuous quaternion valued functions on acompactsetX. That result is in fact a consequence of a theorem by Aupetit and Zemanek [1]. We will also present several related results and corollaries valid for real and complex Banach and topological algebras. To simplify the notation we assume that the algebras under consideration have units, however, like in the case of the Hirschfeld-Zelazkoú Theorem, the analogous results can be stated for none unital algebras as one can formally add aunittosuchanalgebra. -
Regularity Conditions for Banach Function Algebras
Regularity conditions for Banach function algebras Dr J. F. Feinstein University of Nottingham June 2009 1 1 Useful sources A very useful text for the material in this mini-course is the book Banach Algebras and Automatic Continuity by H. Garth Dales, London Mathematical Society Monographs, New Series, Volume 24, The Clarendon Press, Oxford, 2000. In particular, many of the examples and conditions discussed here may be found in Chapter 4 of that book. We shall refer to this book throughout as the book of Dales. Most of my e-prints are available from www.maths.nott.ac.uk/personal/jff/Papers Several of my research and teaching presentations are available from www.maths.nott.ac.uk/personal/jff/Beamer 2 2 Introduction to normed algebras and Banach algebras 2.1 Some problems to think about Those who have seen much of this introductory material before may wish to think about some of the following problems. We shall return to these problems at suitable points in this course. Problem 2.1.1 (Easy using standard theory!) It is standard that the set of all rational functions (quotients of polynomials) with complex coefficients is a field: this is a special case of the “field of fractions" of an integral domain. Question: Is there an algebra norm on this field (regarded as an algebra over C)? 3 Problem 2.1.2 (Very hard!) Does there exist a pair of sequences (λn), (an) of non-zero complex numbers such that (i) no two of the an are equal, P1 (ii) n=1 jλnj < 1, (iii) janj < 2 for all n 2 N, and yet, (iv) for all z 2 C, 1 X λn exp (anz) = 0? n=1 Gap to fill in 4 Problem 2.1.3 Denote by C[0; 1] the \trivial" uniform algebra of all continuous, complex-valued functions on [0; 1]. -
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. -
Class Notes, Functional Analysis 7212
Class notes, Functional Analysis 7212 Ovidiu Costin Contents 1 Banach Algebras 2 1.1 The exponential map.....................................5 1.2 The index group of B = C(X) ...............................6 1.2.1 p1(X) .........................................7 1.3 Multiplicative functionals..................................7 1.3.1 Multiplicative functionals on C(X) .........................8 1.4 Spectrum of an element relative to a Banach algebra.................. 10 1.5 Examples............................................ 19 1.5.1 Trigonometric polynomials............................. 19 1.6 The Shilov boundary theorem................................ 21 1.7 Further examples....................................... 21 1.7.1 The convolution algebra `1(Z) ........................... 21 1.7.2 The return of Real Analysis: the case of L¥ ................... 23 2 Bounded operators on Hilbert spaces 24 2.1 Adjoints............................................ 24 2.2 Example: a space of “diagonal” operators......................... 30 2.3 The shift operator on `2(Z) ................................. 32 2.3.1 Example: the shift operators on H = `2(N) ................... 38 3 W∗-algebras and measurable functional calculus 41 3.1 The strong and weak topologies of operators....................... 42 4 Spectral theorems 46 4.1 Integration of normal operators............................... 51 4.2 Spectral projections...................................... 51 5 Bounded and unbounded operators 54 5.1 Operations.......................................... -
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. -
1. Introduction
FACTORIZATION IN COMMUTATIVE BANACH ALGEBRAS H. G. DALES, J. F. FEINSTEIN, AND H. L. PHAM Abstract. Let A be a (non-unital) commutative Banach algebra. We consider when A has a variety of factorization properties: we list the (ob- vious) implications between these properties, and then consider whether any of these implications can be reversed in various classes of commu- tative Banach algebras. We summarize the known counter-examples to these possible reverse implications, and add further counter-examples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products. 1. Introduction Let A be a (non-unital) commutative Banach algebra. We wish to examine when A factors in a variety of senses. Our main results are counter-examples to a number of questions that have been raised. Indeed, we shall list seven such factorization properties, called (I)(VII), and note that each of these immediately implies the next one. We shall also, in x4, discuss two other `lo- cal' factorization properties, called (A) and (B) (where (A) ) (B)); these properties are relevant for Esterle's classication of commutative, radical Banach algebras that is given in [14]. We shall then discuss whether or not any of these implications can be reversed when we restrict attention to par- ticular classes of commutative Banach algebras. We shall show that several cannot be reversed, but we leave open other possible reverse implications. A summary in x6 describes our knowledge at the present time. We shall concentrate on two particular classes of commutative Banach algebras A: rst, on the case where A is semi-simple (so that A is a Banach function algebra), and in particular when A is a maximal ideal in a uniform algebra on a compact space, and, second, on the other extreme case where A 2010 Mathematics Subject Classication. -
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 . -
BANACH ALGEBRAS with UNIQUE UNIFORM NORM 1. Introduction A
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 2, February 1996 BANACH ALGEBRAS WITH UNIQUE UNIFORM NORM S. J. BHATT AND H. V. DEDANIA (Communicated by Palle E. T. Jorgensen) Abstract. Commutative semisimple Banach algebras that admit exactly one uniform norm (not necessarily complete) are investigated. This unique uniform norm property is completely characterized in terms of each of spectral radius, Silov boundary, set of uniqueness, semisimple norms; and its connection with recently investigated concepts like spectral extension property, multiplicative Hahn Banach extension property and permanent radius are revealed. Several classesofBanachalgebrashavingthispropertyaswellasthosenothaving this property are discussed. 1. Introduction A uniform norm on a Banach algebra (A, ) is a submultiplicative norm (not necessarily complete) satisfying the squarek·k property x2 = x 2 (x A). By |·| | | | | ∈ [1, Chapter 2], A is commutative and semisimple iff the spectral radius r (= rA) is a uniform norm iff A admits a uniform norm. A has unique uniform norm property ( UUNP)ifris the only uniform norm on A. This paper is concerned with Banach algebras with UUNP. Note that equivalent uniform norms are equal and any uniform norm r. A recent trend ([9], [10],|·|≤ [11], [14]) in Banach algebras is to investigate incomplete algebra norms; in the same spirit, it is interesting to characterize intrinsically UUNP and its influence on A. The spectral extension property (SEP) of [10] implies UUNP. In fact, it is claimed in [10] that A has SEP iff “every closed set of uniqueness contains the Silov boundary” (statement (2) of [10, Theorem 1]). However, there is a gap in the proof that this statement implies SEP (personal correspondence with Dr. -
Spectrum (Functional Analysis) - Wikipedia, the Free Encyclopedia
Spectrum (functional analysis) - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Spectrum_(functional_analysis) Spectrum (functional analysis) From Wikipedia, the free encyclopedia In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2, This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand 0 is in the spectrum because the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum. The notion of spectrum extends to densely-defined unbounded operators. In this case a complex number λ is said to be in the spectrum of such an operator T:D→X (where D is dense in X) if there is no bounded inverse (λI − T)−1:X→D.