Characterizing Derivations from the Disk Algebra to Its Dual
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Extreme and Exposed Representing Measures of the Disk Algebra
ANNALES POLONICI MATHEMATICI LXXIII.2 (2000) Extreme and exposed representing measures of the disk algebra by Alex Heinis (Leiden) and Jan Wiegerinck (Amsterdam) Abstract. We study the extreme and exposed points of the convex set consisting of representing measures of the disk algebra, supported in the closed unit disk. A boundary point of this set is shown to be extreme (and even exposed) if its support inside the open unit disk consists of two points that do not lie on the same radius of the disk. If its support inside the unit disk consists of 3 or more points, it is very seldom an extreme point. We also give a necessary condition for extreme points to be exposed and show that so-called BSZ-measures are never exposed. 1. Introduction and notational matters. In a previous paper [2] Brummelhuis and the second author studied representing measures for the disk algebra A = A(D)= {f ∈ C(D) | f is holomorphic on D}, where D is the unit disk in C. As usual A is endowed with the structure of a uniform algebra. Its spectrum equals D. That is, every homomorphism φ : A → C is of the form f 7→ f(x) for some x ∈ D. The Hahn–Banach theorem allows for a multitude of extensions of φ as a continuous linear functional on C(D). The dual space C(D)∗ can be identified with M, the space of complex Borel measures on D endowed with the weak∗ topology by the Riesz representation theorem. A representing measure µ for x ∈ D is a positive Borel measure on D such that \ f dµ = f(x) ∀f ∈ A. -
The Jacobson Radical of Semicrossed Products of the Disk Algebra
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2012 The aJ cobson radical of semicrossed products of the disk algebra Anchalee Khemphet Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Khemphet, Anchalee, "The aJ cobson radical of semicrossed products of the disk algebra" (2012). Graduate Theses and Dissertations. 12364. https://lib.dr.iastate.edu/etd/12364 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. The Jacobson radical of semicrossed products of the disk algebra by Anchalee Khemphet A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Justin Peters, Major Professor Scott Hansen Dan Nordman Paul Sacks Sung-Yell Song Iowa State University Ames, Iowa 2012 Copyright c Anchalee Khemphet, 2012. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my father Pleng and to my mother Supavita without whose support I would not have been able to complete this work. I would also like to thank my friends and family for their loving guidance and to my government for financial assistance during the writing of this work. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . v ABSTRACT . vi CHAPTER 1. -
Projective Coordinates and Compactification in Elliptic, Parabolic and Hyperbolic 2-D Geometry
PROJECTIVE COORDINATES AND COMPACTIFICATION IN ELLIPTIC, PARABOLIC AND HYPERBOLIC 2-D GEOMETRY Debapriya Biswas Department of Mathematics, Indian Institute of Technology-Kharagpur, Kharagpur-721302, India. e-mail: d [email protected] Abstract. A result that the upper half plane is not preserved in the hyperbolic case, has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case. Key words. EPH Cases, Projective Coordinates, Compactification, M¨obius transformation, Clifford algebra, Lie group. 2010(AMS) Mathematics Subject Classification: Primary 30G35, Secondary 22E46. 1. Introduction Geometry is an essential branch of Mathematics. It deals with the proper- ties of figures in a plane or in space [7]. Perhaps, the most influencial book of all time, is ‘Euclids Elements’, written in around 3000 BC [14]. Since Euclid, geometry usually meant the geometry of Euclidean space of two-dimensions (2-D, Plane geometry) and three-dimensions (3-D, Solid geometry). A close scrutiny of the basis for the traditional Euclidean geometry, had revealed the independence of the parallel axiom from the others and consequently, non- Euclidean geometry was born and in projective geometry, new “points” (i.e., points at infinity and points with complex number coordinates) were intro- duced [1, 3, 9, 26]. In the eighteenth century, under the influence of Steiner, Von Staudt, Chasles and others, projective geometry became one of the chief subjects of mathematical research [7]. -
Hyperbolic Geometry
Hyperbolic Geometry David Gu Yau Mathematics Science Center Tsinghua University Computer Science Department Stony Brook University [email protected] September 12, 2020 David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 1 / 65 Uniformization Figure: Closed surface uniformization. David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 2 / 65 Hyperbolic Structure Fundamental Group Suppose (S; g) is a closed high genus surface g > 1. The fundamental group is π1(S; q), represented as −1 −1 −1 −1 π1(S; q) = a1; b1; a2; b2; ; ag ; bg a1b1a b ag bg a b : h ··· j 1 1 ··· g g i Universal Covering Space universal covering space of S is S~, the projection map is p : S~ S.A ! deck transformation is an automorphism of S~, ' : S~ S~, p ' = '. All the deck transformations form the Deck transformation! group◦ DeckS~. ' Deck(S~), choose a pointq ~ S~, andγ ~ S~ connectsq ~ and '(~q). The 2 2 ⊂ projection γ = p(~γ) is a loop on S, then we obtain an isomorphism: Deck(S~) π1(S; q);' [γ] ! 7! David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 3 / 65 Hyperbolic Structure Uniformization The uniformization metric is ¯g = e2ug, such that the K¯ 1 everywhere. ≡ − 2 Then (S~; ¯g) can be isometrically embedded on the hyperbolic plane H . The On the hyperbolic plane, all the Deck transformations are isometric transformations, Deck(S~) becomes the so-called Fuchsian group, −1 −1 −1 −1 Fuchs(S) = α1; β1; α2; β2; ; αg ; βg α1β1α β αg βg α β : h ··· j 1 1 ··· g g i The Fuchsian group generators are global conformal invariants, and form the coordinates in Teichm¨ullerspace. -
Of Operator Algebras Vern I
proceedings of the american mathematical society Volume 92, Number 2, October 1984 COMPLETELY BOUNDED HOMOMORPHISMS OF OPERATOR ALGEBRAS VERN I. PAULSEN1 ABSTRACT. Let A be a unital operator algebra. We prove that if p is a completely bounded, unital homomorphism of A into the algebra of bounded operators on a Hubert space, then there exists a similarity S, with ||S-1|| • ||S|| = ||p||cb, such that S_1p(-)S is a completely contractive homomorphism. We also show how Rota's theorem on operators similar to contractions and the result of Sz.-Nagy and Foias on the similarity of p-dilations to contractions can be deduced from this result. 1. Introduction. In [6] we proved that a homomorphism p of an operator algebra is similar to a completely contractive homomorphism if and only if p is completely bounded. It was known that if S is such a similarity, then ||5|| • ||5_11| > ||/9||cb- However, at the time we were unable to determine if one could choose the similarity such that ||5|| • US'-1!! = ||p||cb- When the operator algebra is a C*- algebra then Haagerup had shown [3] that such a similarity could be chosen. The purpose of the present note is to prove that for a general operator algebra, there exists a similarity S such that ||5|| • ||5_1|| = ||p||cb- Completely contractive homomorphisms are central to the study of the repre- sentation theory of operator algebras, since they are precisely the homomorphisms that can be dilated to a ^representation on some larger Hilbert space of any C*- algebra which contains the operator algebra. -
Semicrossed Products of the Disk Algebra: Contractive Representations and Maximal Ideals
pacific journal of mathematics Vol. 185, No. 1, 1998 SEMICROSSED PRODUCTS OF THE DISK ALGEBRA: CONTRACTIVE REPRESENTATIONS AND MAXIMAL IDEALS Dale R. Buske and Justin R. Peters Given the disk algebra A(D) and an automorphism α, there + is associated a non-self-adjoint operator algebra Z ×α A(D) called the semicrossed product of A(D) with α. We consider those algebras where the automorphism arises via composi- tion with parabolic, hyperbolic, and elliptic conformal maps ϕ of D onto itself. To characterize the contractive representa- + tions of Z ×α A(D), a noncommutative dilation result is ob- tained. The result states that given a pair of contractions S, T on some Hilbert space H which satisfy TS = Sϕ(T), there exist unitaries U, V on some Hilbert space K⊃Hwhich dilate S and T respectively and satisfy VU =Uϕ(V). It is then shown that there is a one-to-one correspondence between the contractive + (and completely contractive) representations of Z ×α A(D) on a Hilbert space H and pairs of contractions S and T on H satis- fying TS = Sϕ(T). The characters, maximal ideals, and strong + radical of Z ×α A(D) are then computed. In the last section, we compare the strong radical to the Jacobson radical. I. Introduction. A semicrossed product of the disk algebra is an operator algebra associated to the pair (A(D),α), where A(D) is the disk algebra and α an automor- phism of A(D). Any such α has the form α(f)=f◦ϕ(f∈A(D)) for a linear fractional transformation ϕ. -
The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
Elliot Glazer Topological Manifolds
Manifolds and complex structure Elliot Glazer Topological Manifolds • An n-dimensional manifold M is a space that is locally Euclidean, i.e. near any point p of M, we can define points of M by n real coordinates • Curves are 1-dimensional manifolds (1-manifolds) • Surfaces are 2-manifolds • A chart (U, f) is an open subset U of M and a homeomorphism f from U to some open subset of R • An atlas is a set of charts that cover M Some important 2-manifolds A very important 2-manifold Smooth manifolds • We want manifolds to have more than just topological structure • There should be compatibility between intersecting charts • For intersecting charts (U, f), (V, g), we define a natural transition map from one set of coordinates to the other • A manifold is C^k if it has an atlas consisting of charts with C^k transition maps • Of particular importance are smooth maps • Whitney embedding theorem: smooth m-manifolds can be smoothly embedded into R^{2m} Complex manifolds • An n-dimensional complex manifold is a space such that, near any point p, each point can be defined by n complex coordinates, and the transition maps are holomorphic (complex structure) • A manifold with n complex dimensions is also a real manifold with 2n dimensions, but with a complex structure • Complex 1-manifolds are called Riemann surfaces (surfaces because they have 2 real dimensions) • Holoorphi futios are rigid sie they are defied y accumulating sets • Whitney embedding theorem does not extend to complex manifolds Classifying Riemann surfaces • 3 basic Riemann surfaces • The complex plane • The unit disk (distinct from plane by Liouville theorem) • Riemann sphere (aka the extended complex plane), uses two charts • Uniformization theorem: simply connected Riemann surfaces are conformally equivalent to one of these three . -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
Uniform Algebras As Banach Spaces
The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Uniform Algebras as Banach Spaces TW Gamelin SV Kislyakov Vienna Preprint ESI June Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat UNIFORM ALGEBRAS AS BANACH SPACES TW Gamelin SV Kislyakov June Abstract Any Banach space can b e realized as a direct summand of a uniform algebra and one do es not exp ect an arbitrary uniform algebra to have an abundance of prop erties not common to all Banach spaces One general result concerning arbitrary uniform algebras is that no prop er uniform algebra is linearly homeomorphic to a C K space Nevertheless many sp ecic uniform algebras arising in complex analysis share or are susp ected to share certain Banach space prop erties of C K We discuss the family of tight algebras which includes algebras of analytic functions on strictly pseudo con vex domains and algebras asso ciated with rational approximation theory in the plane Tight algebras are in some sense close to C K spaces and along with C K spaces they have the Pelczynski and the DunfordPettis prop erties We also fo cus on certain prop erties of C K spaces that are inherited by the disk algebra This includes a dis p cussion of interp olation b etween H spaces and Bourgains extension of Grothendiecks theorem to the disk algebra We conclude with a brief description of linear deformations of uniform algebras and a brief survey of the known classication results Supp orted in part by the Russian -
Semicrossed Products of the Disk Algebra
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 10, October 2012, Pages 3479–3484 S 0002-9939(2012)11348-6 Article electronically published on February 17, 2012 SEMICROSSED PRODUCTS OF THE DISK ALGEBRA KENNETH R. DAVIDSON AND ELIAS G. KATSOULIS (Communicated by Marius Junge) Abstract. If α is the endomorphism of the disk algebra, A(D), induced by composition with a finite Blaschke product b, then the semicrossed product + + A(D) ×α Z imbeds canonically, completely isometrically into C(T) ×α Z . Hence in the case of a non-constant Blaschke product b, the C*-envelope has the form C(Sb) ×s Z,where(Sb,s) is the solenoid system for (T,b). In the + case where b is a constant, the C*-envelope of A(D) ×α Z is strongly Morita equivalent to a crossed product of the form C0(Se) ×s Z,wheree: T × N −→ T × N is a suitable map and (Se,s) is the solenoid system for (T × N,e). 1. Introduction If A is a unital operator algebra and α is a completely contractive endomorphism, the semicrossed product is an operator algebra A×α Z+ which encodes the covari- ant representations of (A,α): namely completely contractive unital representations ρ : A→B(H) and contractions T satisfying ρ(a)T = Tρ(α(a)) for all a ∈A. Such algebras were defined by Peters [9] when A is a C*-algebra. One can readily extend Peter’s definition [9] of the semicrossed product of a C*-algebra by a ∗-endomorphism to unital operator algebras and unital completely contractive endomorphisms. -
Math 372: Fall 2015: Solutions to Homework
Math 372: Fall 2015: Solutions to Homework Steven Miller December 7, 2015 Abstract Below are detailed solutions to the homeworkproblemsfrom Math 372 Complex Analysis (Williams College, Fall 2015, Professor Steven J. Miller, [email protected]). The course homepage is http://www.williams.edu/Mathematics/sjmiller/public_html/372Fa15 and the textbook is Complex Analysis by Stein and Shakarchi (ISBN13: 978-0-691-11385-2). Note to students: it’s nice to include the statement of the problems, but I leave that up to you. I am only skimming the solutions. I will occasionally add some comments or mention alternate solutions. If you find an error in these notes, let me know for extra credit. Contents 1 Math 372: Homework #1: Yuzhong (Jeff) Meng and Liyang Zhang (2010) 3 1.1 Problems for HW#1: Due September 21, 2015 . ................ 3 1.2 SolutionsforHW#1: ............................... ........... 3 2 Math 372: Homework #2: Solutions by Nick Arnosti and Thomas Crawford (2010) 8 3 Math 372: Homework #3: Carlos Dominguez, Carson Eisenach, David Gold 12 4 Math 372: Homework #4: Due Friday, October 12, 2015: Pham, Jensen, Kolo˘glu 16 4.1 Chapter3,Exercise1 .............................. ............ 16 4.2 Chapter3,Exercise2 .............................. ............ 17 4.3 Chapter3,Exercise5 .............................. ............ 19 4.4 Chapter3Exercise15d ..... ..... ...... ..... ...... .. ............ 22 4.5 Chapter3Exercise17a ..... ..... ...... ..... ...... .. ............ 22 4.6 AdditionalProblem1 .............................. ............ 22 5 Math 372: Homework #5: Due Monday October 26: Pegado, Vu 24 6 Math 372: Homework #6: Kung, Lin, Waters 34 7 Math 372: Homework #7: Due Monday, November 9: Thompson, Schrock, Tosteson 42 7.1 Problems. ....................................... ......... 46 1 8 Math 372: Homework #8: Thompson, Schrock, Tosteson 47 8.1 Problems.