Lecture 3: Modules Over Operator Algebras David Blecher December
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Topological Divisors of Zero and Shilov Boundary
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 153 (2006) 1152–1163 www.elsevier.com/locate/topol Topological divisors of zero and Shilov boundary Alain Escassut Laboratoire de Mathématiques UMR 6620, Université Blaise Pascal, Clermont-Ferrand, Les Cézeaux, 63177 Aubiere cedex, France Received 17 February 2005; accepted 1 March 2005 Abstract Let L be a field complete for a non-trivial ultrametric absolute value and let (A, ·) be a com- mutative normed L-algebra with unity whose spectral semi-norm is ·si.LetMult(A, ·) be the set of continuous multiplicative semi-norms of A,letS be the Shilov boundary for (A, ·si) and let ψ ∈ Mult(A, ·si).Thenψ belongs to S if and only if for every neighborhood U of ψ in Mult(A, ·), there exists θ ∈ U and g ∈ A satisfying gsi = θ(g) and γ(g)<gsi ∀γ ∈ S \ U. Suppose A is uniform, let f ∈ A and let Z(f ) ={φ ∈ Mult(A, ·) | φ(f)= 0}.Thenf is a topo- logical divisor of zero if and only if there exists ψ ∈ S such that ψ(f) = 0. Suppose now A is complete. If f is not a divisor of zero, then it is a topological divisor of zero if and only if the ideal fAis not closed in A. Suppose A is ultrametric, complete and Noetherian. All topological divisors of zero are divisors of zero. This applies to affinoid algebras. Let A be a Krasner algebra H(D)without non-trivial idempotents: an element f ∈ H(D)is a topological divisor of zero if and only if fH(D) is not a closed ideal; moreover, H(D) is a principal ideal ring if and only if it has no topological divisors of zero but 0 (this new condition adds to the well-known set of equivalent conditions found in 1969). -
Weak and Norm Approximate Identities Are Different
Pacific Journal of Mathematics WEAK AND NORM APPROXIMATE IDENTITIES ARE DIFFERENT CHARLES ALLEN JONES AND CHARLES DWIGHT LAHR Vol. 72, No. 1 January 1977 PACIFIC JOURNAL OF MATHEMATICS Vol. 72, No. 1, 1977 WEAK AND NORM APPROXIMATE IDENTITIES ARE DIFFERENT CHARLES A. JONES AND CHARLES D. LAHR An example is given of a convolution measure algebra which has a bounded weak approximate identity, but no norm approximate identity. 1* Introduction* Let A be a commutative Banach algebra, Ar the dual space of A, and ΔA the maximal ideal space of A. A weak approximate identity for A is a net {e(X):\eΛ} in A such that χ(e(λ)α) > χ(α) for all αei, χ e A A. A norm approximate identity for A is a net {e(λ):λeΛ} in A such that ||β(λ)α-α|| >0 for all aeA. A net {e(X):\eΛ} in A is bounded and of norm M if there exists a positive number M such that ||e(λ)|| ^ M for all XeΛ. It is well known that if A has a bounded weak approximate identity for which /(e(λ)α) —>/(α) for all feA' and αeA, then A has a bounded norm approximate identity [1, Proposition 4, page 58]. However, the situation is different if weak convergence is with re- spect to ΔA and not A'. An example is given in § 2 of a Banach algebra A which has a weak approximate identity, but does not have a norm approximate identity. This algebra provides a coun- terexample to a theorem of J. -
Book of Abstracts for the 3Rd Conference of Settat on Operator Algebras and Applications, November 1–5, 2011
Book of Abstracts for the 3rd Conference of Settat on Operator Algebras and Applications, November 1{5, 2011 1. Speaker: Ahmed Al-Rawashdeh (United Arab Emirates University, Al Ain) Title: On Certain Projections of C*-Matrix Algebras ∗ Abstract: H. Dye defined the projections Pi;j(a) of a C -matrix algebra by ∗ −1 ∗ −1 Pi;j(a) = (1 + aa ) ⊗ Ei;i + (1 + aa ) a ⊗ Ei;j ∗ ∗ −1 ∗ ∗ −1 + a (1 + aa ) ⊗ Ej;i + a (1 + aa ) a ⊗ Ej;j; and he used it to show that in the case of factors not of type I2n, the unitary group determines the algebraic type of that factor. We study these projections and we show that in M2(C), the set of such projections includes all the projections. For infinite C∗-algebra A, having a system of matrix units, including the Cuntz algebra On, we have A ' Mn(A). M. Leen proved that in a simple, ∗ purely infinite C -algebra A, the ∗-symmetries generate U0(A). We revise and modify Leen's proof to show that part of such ∗-isometry factors are of the form 1−2Pi;j(!);! 2 U(A). In simple, unital ∗ purely infinite C -algebras having trivial K1-group, we prove that all Pi;j(!) have trivial K0-class. In particular, if u 2 U(On), then u can be factorized as a product of ∗-symmetries, where eight of them are of the form 1 − 2Pi;j(!). 2. Speaker: Pere Ara (Universitat Aut`onomade Barcelona, Spain) Title:(m; n)-dynamical systems and their C*-algebras Abstract: Given positive integers n and m, we consider dynamical systems in which n copies of a topological space is homeomorphic to m copies of that same space. -
Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc
Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc To cite this version: Jean-Louis Clerc. Geometry of the Shilov Boundary of a Bounded Symmetric Domain. journal of geometry and symmetry in physics, 2009, Vol. 13, pp. 25-74. hal-00381665 HAL Id: hal-00381665 https://hal.archives-ouvertes.fr/hal-00381665 Submitted on 6 May 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc today Abstract In the first part, the theory of bounded symmetric domains is pre- sented along two main approaches : as special cases of Riemannian symmetric spaces of the noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand. In the second part, an invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in C2 and also the triple Maslov index on the Lagrangian manifold. 1 Introduction The present paper is an outgrowth of the cycle of conferences delivred by the author at the Tenth International Conference on Geometry, Integrability and Quantization, held in Varna in June 2008. -
Extension of Characters and Generalized Shilov Boundaries
Revista de la 211 Union Matem~tica Argentina . Volumen 32, 1986. EXTENSION OF CHARACTERS AND GENERALIZED SHILOV BOUNDARIES Gustavo Corach and Alejandra M~estripieri ABSTRACT. We prove in a very simple way Shilov extension theorem (if A is a commutative Banach algebra, h a character in the Shilov boundary of A and B a commutative superalgebra of A then h admits an extension to B which can be chosen in the Shilov boundary of B) and use this proof t~ extend similar results to higher dimensional Shilov boundaries. 1. I NTRODUCT ION. Let A be a (complex commutative) Banach algebra with unity and X(A) the spectrum space of characters of A. The Shilov boundary ro(A) is a compact subset of X(A) which plays an important role in various problems, in particular in the search of existence of I-dimensional analytic structure on X(A). For higher dimensions ro(A) seems to be too small and Basener [2] and S;ibony [11} have defined a bigger subset of X(A), rn(A), which is an appropiate generalized Shilov boundary (see als0 [1,5,6,7,11] for more information on rnCA)). In this paper we are concerned w;ith the following property of r 0 (A), ca.lled Shilov extension theo'l'em: if A is a closed subalgebra of a Banach algebra B then every· chal'acter in r 0 (A) admits an extension to some character k of B. Moreover k can be chosen in ro(B), by a result of; Rickart [9,(3.3.25)]. -
Operator Algebras with Contractive Approximate Identities
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Functional Analysis 261 (2011) 188–217 www.elsevier.com/locate/jfa Operator algebras with contractive approximate identities David P. Blecher a,∗,1, Charles John Read b a Department of Mathematics, University of Houston, Houston, TX 77204-3008, United States b Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, United Kingdom Received 15 November 2010; accepted 24 February 2011 Available online 21 March 2011 Communicated by S. Vaes Abstract We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for ∗ C -algebras, and we give many other applications. © 2011 Elsevier Inc. All rights reserved. Keywords: Operator algebras; One-sided ideals; Hereditary subalgebra; Approximate identity; Peak set; Pseudo-invertible elements; Completely positive operator 1. Introduction An operator algebra is a closed subalgebra of B(H), for a Hilbert space H . We recall that by a theorem due to Ralf Meyer, every operator algebra A has a unique unitization A1 (see [30] or [10, Section 2.1]). Below 1 always refers to the identity of A1 if A has no identity. -
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.......................................... -
Noncommutative Functional Analysis for Undergraduates"
N∞M∞T Lecture Course Canisius College Noncommutative functional analysis David P. Blecher October 22, 2004 2 Chapter 1 Preliminaries: Matrices = operators 1.1 Introduction Functional analysis is one of the big fields in mathematics. It was developed throughout the 20th century, and has several major strands. Some of the biggest are: “Normed vector spaces” “Operator theory” “Operator algebras” We’ll talk about these in more detail later, but let me give a micro-summary. Normed (vector) spaces were developed most notably by the mathematician Banach, who not very subtly called them (B)-spaces. They form a very general framework and tools to attack a wide range of problems: in fact all a normed (vector) space is, is a vector space X on which is defined a measure of the ‘length’ of each ‘vector’ (element of X). They have a huge theory. Operator theory and operator algebras grew partly out of the beginnings of the subject of quantum mechanics. In operator theory, you prove important things about ‘linear functions’ (also known as operators) T : X → X, where X is a normed space (indeed usually a Hilbert space (defined below). Such operators can be thought of as matrices, as we will explain soon. Operator algebras are certain collections of operators, and they can loosely be thought of as ‘noncommutative number fields’. They fall beautifully within the trend in mathematics towards the ‘noncommutative’, linked to discovery in quantum physics that we live in a ‘noncommutative world’. You can study a lot of ‘noncommutative mathematics’ in terms of operator algebras. The three topics above are functional analysis. -
Banach Function Algebras with Dense Invertible Group
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 4, April 2008, Pages 1295–1304 S 0002-9939(07)09044-2 Article electronically published on December 21, 2007 BANACH FUNCTION ALGEBRAS WITH DENSE INVERTIBLE GROUP H. G. DALES AND J. F. FEINSTEIN (Communicated by N. Tomczak-Jaegermann) Abstract. In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov bound- ary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that tsr(A) ≥ tsr(C(ΦA)) whenever A is approximately regular. 1. Introduction Let A be a commutative, unital Banach algebra. The character space of A is denoted by ΦA,asin[8].Fora ∈ A, we denote the Gel’fand transform of a by a.WesaythatΦA contains analytic structure if there is a continuous injection τ from the open unit disk D to ΦA such that, for all a ∈ A, a ◦ τ is analytic on D. In [16], Stolzenberg gave a counter-example to the conjecture that, whenever a uniform algebra has proper Shilov boundary, its character space must contain analytic structure (see also [17, Theorem 29.19], [19] and [1]). Cole gave an even more extreme example in his thesis [3], where the Shilov boundary is proper and yet every Gleason part is trivial. It is elementary to show that the invertible group of A cannot be dense in A whenever ΦA contains analytic structure. -
A Covariance Function Approach to Prior Specification for Bayesian
Journal of Machine Learning Research 22 (2021) 1-57 Submitted 11/20; Revised 5/21; Published 6/21 The Ridgelet Prior: A Covariance Function Approach to Prior Specification for Bayesian Neural Networks Takuo Matsubara [email protected] School of Mathematics, Statistics & Physics Newcastle University Newcastle upon Tyne, NE1 7RU, United Kingdom Chris J. Oates [email protected] School of Mathematics, Statistics & Physics Newcastle University Newcastle upon Tyne, NE1 7RU, United Kingdom Fran¸cois-Xavier Briol [email protected] Department of Statistical Science University College London London, WC1E 6BT, United Kingdom Editor: Pierre Alquier Abstract Bayesian neural networks attempt to combine the strong predictive performance of neural networks with formal quantification of uncertainty associated with the predictive output in the Bayesian framework. However, it remains unclear how to endow the parameters of the network with a prior distribution that is meaningful when lifted into the output space of the network. A possible solution is proposed that enables the user to posit an appropriate Gaussian process covariance function for the task at hand. Our approach constructs a prior distribution for the parameters of the network, called a ridgelet prior, that approximates the posited Gaussian process in the output space of the network. In contrast to existing work on the connection between neural networks and Gaussian processes, our analysis is non-asymptotic, with finite sample-size error bounds provided. This establishes the universality property that a Bayesian neural network can approximate any Gaussian process whose covariance function is sufficiently regular. Our experimental assessment is limited to a proof-of-concept, where we demonstrate that the ridgelet prior can out-perform an unstructured prior on regression problems for which a suitable Gaussian process prior can be provided. -
Approximate Identities in Banach Function Algebras H. G. Dales
Approximate identities in Banach function algebras H. G. Dales, Lancaster National University of Ireland, Maynooth, 19 June 2013 In honour of the retirement of Anthony G. O'Farrell Work in progress with Ali Ulger,¨ Istanbul 1 Definitions Let K be a locally compact space. Then C0(K) is the space of all complex-valued, continuous functions on K that vanish at infinity. This is a commutative Banach algebra with respect to the uniform norm j · jK. A function algebra on K is a subalgebra of C0(K) such that, for x; y 2 K with x 6= y, there is f 2 A with f(x) 6= f(y), and, for each x 2 K, there is f 2 A with f(x) 6= 0. A Banach function algebra on K is a function algebra A that is a Banach algebra for a norm k · k, so that kfgk ≤ kfk kgk for f; g 2 A. Necessarily, kfk ≥ jfjK for f 2 A. The algebra A is a uniform algebra if it is closed in C0(K). 2 Natural Banach function algebras A Banach function algebra A on K is natural if every character on A has the form f 7! f(x) = "x(f) for some x 2 K. Equiv- alently, every maximal modular ideal has the form Mx = ff 2 A : f(x) = 0g for some x 2 K. Every commutative, semisimple Banach algebra is a Banach function algebra on its character space. 3 Approximate identities Let (A; k · k) be a natural Banach function algebra on K. -
Shilov Boundary for "Holomorphic Functions" on a Quantum Matrix Ball
Introduction Fock Representation Shilov Boundary Shilov boundary for "holomorphic functions" on a quantum matrix ball Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Department of Mathematics Chalmers University of Technology June 22, 2017 Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov. I It generalizes the maximus modulus priciple: I If f (z) 2 C(D) is an holomorphic function on the unit disc D; then max jf (z)j = max jf (z)j; z2D z2T where T is the unit circle. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov. I It generalizes the maximus modulus priciple: I If f (z) 2 C(D) is an holomorphic function on the unit disc D; then max jf (z)j = max jf (z)j; z2D z2T where T is the unit circle. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov.