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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. -
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. -
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. -
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. -
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. -
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. -
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. -
Geometric Properties of the Disk Algebra
Geometric Properties of the Disk Algebra. Yun Sung Choi (Joint Work with D. Garcia, S.K. Kim and M. Maestre) POSTECH Worksop on Functional Analysis Valencia June 3-6, 2013 Worksop on Functional Analysis Valencia June 3-6, 2013 1 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Related to the numerical index, Daugavet property, Bishop-Phelps-Bollob´as property, etc. several geometrical properties of the space C(K)(resp.C(K, X )) on a Hausdor↵ compact set K (resp. with values in a Banach space X )have been obtained, Most of the results at one point or another use the classical Urysohn Lemma. Urysohn Lemma Theorem (Urysohn Lemma) AHausdor↵ topological space is normal if and only if given two disjoint closed subsets can be separated by a continuous function with values on [0, 1] and taking the value 1 on one closed set an 0 on the other. Worksop on Functional Analysis Valencia June 3-6, 2013 2 Yun Sung Choi (Joint Work with D. Garcia,Geometric S.K. Kim Properties and M. Maestre) of the Disk (POSTECH) Algebra. /20 Urysohn Lemma Theorem (Urysohn Lemma) AHausdor↵ topological space is normal if and only if given two disjoint closed subsets can be separated by a continuous function with values on [0, 1] and taking the value 1 on one closed set an 0 on the other. Related to the numerical index, Daugavet property, Bishop-Phelps-Bollob´as property, etc. several geometrical properties of the space C(K)(resp.C(K, X )) on a Hausdor↵ compact set K (resp. -
Pick Interpolation for a Uniform Algebra
FUNCTIONAL ANALYSIS AND OPERATOR THEORY BANACH CENTER PUBLICATIONS, VOLUME 30 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1994 PICK INTERPOLATION FOR A UNIFORM ALGEBRA J. WERMER Department of Mathematics, Brown University Providence, Rhode Island 02912-0001, U.S.A. I am reporting on joint work with Brian Cole and Keith Lewis. Let A be a uniform algebra on a compact Hausdorff space X, i.e. let A be an algebra of continuous complex-valued functions on X, closed under uniform convergence on X, separating points and containing the constants. Let M denote the maximal ideal space of A. Gelfand's theory gives that X may be embedded in M as a closed subset and each f in A has a natural extension to M as a continuous function. Set kfk = maxX jfj. We consider the following interpolation problem: choose n points M1;:::;Mn in M. Put I = fg 2 A j g(Mj) = 0; 1 ≤ j ≤ ng ; and form the quotient-algebra A=I. A=I is a commutative Banach algebra which is algebraically isomorphic to Cn under coordinatewise multiplication. For f 2 A, [f] denotes the coset of f in A=I and k[f]k denotes the quotient norm. We put, n for w = (w1; : : : ; wn) in C , n D = fw 2 C j 9 f 2 A such that f(Mj) = wj; 1 ≤ j ≤ n; and k[f]k ≤ 1g : Our problem is to describe D. It is easy to see that D is a closed subset of the closed unit polydisk ∆n in Cn and has non-void interior. -
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.