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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. -
Criss-Cross Type Algorithms for Computing the Real Pseudospectral Abscissa
CRISS-CROSS TYPE ALGORITHMS FOR COMPUTING THE REAL PSEUDOSPECTRAL ABSCISSA DING LU∗ AND BART VANDEREYCKEN∗ Abstract. The real "-pseudospectrum of a real matrix A consists of the eigenvalues of all real matrices that are "-close to A. The closeness is commonly measured in spectral or Frobenius norm. The real "-pseudospectral abscissa, which is the largest real part of these eigenvalues for a prescribed value ", measures the structured robust stability of A. In this paper, we introduce a criss-cross type algorithm to compute the real "-pseudospectral abscissa for the spectral norm. Our algorithm is based on a superset characterization of the real pseudospectrum where each criss and cross search involves solving linear eigenvalue problems and singular value optimization problems. The new algorithm is proved to be globally convergent, and observed to be locally linearly convergent. Moreover, we propose a subspace projection framework in which we combine the criss-cross algorithm with subspace projection techniques to solve large-scale problems. The subspace acceleration is proved to be locally superlinearly convergent. The robustness and efficiency of the proposed algorithms are demonstrated on numerical examples. Key words. eigenvalue problem, real pseudospectra, spectral abscissa, subspace methods, criss- cross methods, robust stability AMS subject classifications. 15A18, 93B35, 30E10, 65F15 1. Introduction. The real "-pseudospectrum of a matrix A 2 Rn×n is defined as R n×n (1) Λ" (A) = fλ 2 C : λ 2 Λ(A + E) with E 2 R ; kEk ≤ "g; where Λ(A) denotes the eigenvalues of a matrix A and k · k is the spectral norm. It is also known as the real unstructured spectral value set (see, e.g., [9, 13, 11]) and it can be regarded as a generalization of the more standard complex pseudospectrum (see, e.g., [23, 24]). -
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]. -
Reproducing Kernel Hilbert Space Compactification of Unitary Evolution
Reproducing kernel Hilbert space compactification of unitary evolution groups Suddhasattwa Dasa, Dimitrios Giannakisa,∗, Joanna Slawinskab aCourant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA bFinnish Center for Artificial Intelligence, Department of Computer Science, University of Helsinki, Helsinki, Finland Abstract A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). A key element of this approach is that Wτ is skew-adjoint (unlike regularization approaches based on the addition of diffusion), and thus can be characterized by a unique projection- valued measure, discrete by compactness, and an associated orthonormal basis of eigenfunctions. These eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, the regularized generator has a well-defined Borel functional calculus allowing tWτ the construction of a unitary evolution group fe gt2R on the RKHS, which approximates the unitary Koopman evolution group of the original system. We establish convergence results for the spectrum and Borel functional calculus of the regularized generator to those of the original system in the limit τ ! 0+. Convergence results are also established for a data-driven formulation, where these operators are approximated using finite-rank operators obtained from observed time series. An advantage of working in spaces of observables with an RKHS structure is that one can perform pointwise evaluation and interpolation through bounded linear operators, which is not possible in Lp spaces. -
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. -
The Notion of Observable and the Moment Problem for ∗-Algebras and Their GNS Representations
The notion of observable and the moment problem for ∗-algebras and their GNS representations Nicol`oDragoa, Valter Morettib Department of Mathematics, University of Trento, and INFN-TIFPA via Sommarive 14, I-38123 Povo (Trento), Italy. [email protected], [email protected] February, 17, 2020 Abstract We address some usually overlooked issues concerning the use of ∗-algebras in quantum theory and their physical interpretation. If A is a ∗-algebra describing a quantum system and ! : A ! C a state, we focus in particular on the interpretation of !(a) as expectation value for an algebraic observable a = a∗ 2 A, studying the problem of finding a probability n measure reproducing the moments f!(a )gn2N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π!(a) in the GNS representation of ! and thus it has important consequences for the interpretation of a as an observable. We n provide physical examples (also from QFT) where the moment problem for f!(a )gn2N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem n ∗ (a) for the sequences f!b(a )gn2N, being b 2 A and !b(·) := !(b · b). Letting µ!b be a n solution of the moment problem for the sequence f!b(a )gn2N, we introduce a consistency (a) relation on the family fµ!b gb2A. We prove a 1-1 correspondence between consistent families (a) fµ!b gb2A and positive operator-valued measures (POVM) associated with the symmetric (a) operator π!(a). In particular there exists a unique consistent family of fµ!b gb2A if and only if π!(a) is maximally symmetric. -
Asymptotic Spectral Measures: Between Quantum Theory and E
Asymptotic Spectral Measures: Between Quantum Theory and E-theory Jody Trout Department of Mathematics Dartmouth College Hanover, NH 03755 Email: [email protected] Abstract— We review the relationship between positive of classical observables to the C∗-algebra of quantum operator-valued measures (POVMs) in quantum measurement observables. See the papers [12]–[14] and theB books [15], C∗ E theory and asymptotic morphisms in the -algebra -theory of [16] for more on the connections between operator algebra Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout [1], is integrally related K-theory, E-theory, and quantization. to positive asymptotic morphisms on locally compact spaces In [1], Martinez and Trout showed that there is a fundamen- via an asymptotic Riesz Representation Theorem. Examples tal quantum-E-theory relationship by introducing the concept and applications to quantum physics, including quantum noise of an asymptotic spectral measure (ASM or asymptotic PVM) models, semiclassical limits, pure spin one-half systems and quantum information processing will also be discussed. A~ ~ :Σ ( ) { } ∈(0,1] →B H I. INTRODUCTION associated to a measurable space (X, Σ). (See Definition 4.1.) In the von Neumann Hilbert space model [2] of quantum Roughly, this is a continuous family of POV-measures which mechanics, quantum observables are modeled as self-adjoint are “asymptotically” projective (or quasiprojective) as ~ 0: → operators on the Hilbert space of states of the quantum system. ′ ′ A~(∆ ∆ ) A~(∆)A~(∆ ) 0 as ~ 0 The Spectral Theorem relates this theoretical view of a quan- ∩ − → → tum observable to the more operational one of a projection- for certain measurable sets ∆, ∆′ Σ. -
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. -
Banach J. Math. Anal. 6 (2012), No. 1, 45–60 LINEAR MAPS
Banach J. Math. Anal. 6 (2012), no. 1, 45–60 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/BJMA/ LINEAR MAPS PRESERVING PSEUDOSPECTRUM AND CONDITION SPECTRUM G. KRISHNA KUMAR1 AND S. H. KULKARNI2 Communicated by K. Jarosz Abstract. We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Sev- eral results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A, B be complex unital Banach algebras and > 0. Let Φ : A → B be an -pseudospectrum preserving linear onto map. Then Φ preserves spectrum. If A and B are uniform algebras, then, Φ is an isometric isomorphism. (2) Let A, B be uniform algebras and 0 < < 1. Let Φ : A → B be 0 an -condition spectrum preserving linear map. Then Φ is an -almost 0 multiplicative map, where , tend to zero simultaneously. 1. Introduction Let A be a complex Banach algebra with unit 1. We shall identify λ.1 with λ. We recall that the spectrum of an element a ∈ A is defined as −1 σ(a) = λ ∈ C : λ − a∈ / A , where A−1 is the set of all invertible elements of A. The spectral radius of an element a is defined as r(a) = sup{|λ| : λ ∈ σ(a)}. Date: Received: 20June 2011; Revised: 25 August 2011; Accepted: 8 September 2011. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 47B49; Secondary 46H05, 46J05, 47S48. Key words and phrases. Pseudospectrum, condition spectrum, almost multiplicative map, linear preserver, perturbation. 45 46 G. -
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 .