The Jacobson Radical of Semicrossed Products of the Disk Algebra
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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. -
Midterm (Take Home) – Solution
Midterm (Take Home) – Solution M552 – Abstract Algebra 1. Let R be a local ring, i.e., a commutative ring with 1 with a unique maximal ideal, say I, and let M be a finitely generated R-modulo. (a) [10 points] If N is a submodule of M and M = N + (I · M), then M = N. [Hint: Last semester I proved Nakayma’s Lemma for ideals. The same proof works for [finitely generated] modules. [See Proposition 16.1 on pg. 751 of Dummit and Foote.] Use it here.] Proof. Since R is local, we have that its Jacobson radical is I. Now, since M is finitely generated, then so is M/N. [Generated by the classes of the generators of M.] So, M/N = (N +IM)/N = (IM)/N = I(M/N). [More formally, let α = m+N ∈ M/N, with m ∈ M. But, M = N + (I · M), and so there are n0 ∈ N, x0 ∈ I, and m0 ∈ M, such that m = n0 + x0m0. Hence, α = (n0 + x0m0) + N = x0m0 + N. Thus, α ∈ I(M/N), and hence M/N = I(M/N) [since the other inclusion is trivial].] By Nakayama’s Lemma, we have that M/N = 0, i.e., M = N. (b) [30 points] Suppose further that M is projective [still with the same hypothesis as above]. Prove that M is free. [Hints: Look at M/(I · M) to find your candidate for a basis. Use (a) to prove it generates M. Then let F be a free module with the rank you are guessing to be the rank of M and use (a) to show that the natural map φ : F → M is an isomorphism.] ∼ Proof. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
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 -
THE JACOBSON RADICAL for ANALYTIC CROSSED PRODUCTS Allan P
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DigitalCommons@University of Nebraska University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 2001 THE JACOBSON RADICAL FOR ANALYTIC CROSSED PRODUCTS Allan P. Donsig University of Nebraska-Lincoln, [email protected] Aristides Katavolos University of Athens, [email protected] Antonios Manoussos [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathfacpub Donsig, Allan P.; Katavolos, Aristides; and Manoussos, Antonios, "THE JACOBSON RADICAL FOR ANALYTIC CROSSED PRODUCTS" (2001). Faculty Publications, Department of Mathematics. 133. https://digitalcommons.unl.edu/mathfacpub/133 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. THE JACOBSON RADICAL FOR ANALYTIC CROSSED PRODUCTS ALLAN P. DONSIG, ARISTIDES KATAVOLOS, AND ANTONIOS MANOUSSOS Abstract. We characterise the Jacobson radical of an analytic crossed product C0(X) ×φ Z+, answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the d Jacobson radical of analytic crossed products C0(X) ×φ Z+. This consists of all elements whose ‘Fourier coefficients’ vanish on the recurrent points of the dynamical system (and the first one is zero). The multi-dimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom. There is a rich interplay between operator algebras and dynamical systems, going back to the founding work of Murray and von Neumann in the 1930’s. -
Directed Graphs of Commutative Rings
Rose-Hulman Undergraduate Mathematics Journal Volume 14 Issue 2 Article 11 Directed Graphs of Commutative Rings Seth Hausken University of St. Thomas, [email protected] Jared Skinner University of St. Thomas, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Hausken, Seth and Skinner, Jared (2013) "Directed Graphs of Commutative Rings," Rose-Hulman Undergraduate Mathematics Journal: Vol. 14 : Iss. 2 , Article 11. Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss2/11 Rose- Hulman Undergraduate Mathematics Journal Directed Graphs of Commutative Rings Seth Hausken a Jared Skinnerb Volume 14, no. 2, Fall 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a Email: [email protected] University of St. Thomas b http://www.rose-hulman.edu/mathjournal University of St. Thomas Rose-Hulman Undergraduate Mathematics Journal Volume 14, no. 2, Fall 2013 Directed Graphs of Commutative Rings Seth Hausken Jared Skinner Abstract. The directed graph of a commutative ring is a graph representation of its additive and multiplicative structure. Using the mapping (a; b) ! (a + b; a · b) one can create a directed graph for every finite, commutative ring. We examine the properties of directed graphs of commutative rings, with emphasis on the informa- tion the graph gives about the ring. Acknowledgements: We would like to acknowledge the University of St. Thomas for making this research possible and Dr. Michael Axtell for his guidance throughout. Page 168 RHIT Undergrad. Math. J., Vol. 14, no. 2 1 Introduction When studying group theory, especially early on in such an exploration, Cayley tables are used to give a simple visual representation to the structure of the group. -
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. -
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. -
Jacobson Radical and on a Condition for Commutativity of Rings
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 4 Ver. III (Jul - Aug. 2015), PP 65-69 www.iosrjournals.org Jacobson Radical and On A Condition for Commutativity of Rings 1 2 Dilruba Akter , Sotrajit kumar Saha 1(Mathematics, International University of Business Agriculture and Technology, Bangladesh) 2(Mathematics, Jahangirnagar University, Bangladesh) Abstract: Some rings have properties that differ radically from usual number theoretic problems. This fact forces to define what is called Radical of a ring. In Radical theory ideas of Homomorphism and the concept of Semi-simple ring is required where Zorn’s Lemma and also ideas of axiom of choice is very important. Jacobson radical of a ring R consists of those elements in R which annihilates all simple right R-module. Radical properties based on the notion of nilpotence do not seem to yield fruitful results for rings without chain condition. It was not until Perlis introduced the notion of quasi-regularity and Jacobson used it in 1945, that significant chainless results were obtained. Keywords: Commutativity, Ideal, Jacobson Radical, Simple ring, Quasi- regular. I. Introduction Firstly, we have described some relevant definitions and Jacobson Radical, Left and Right Jacobson Radical, impact of ideas of Right quasi-regularity from Jacobson Radical etc have been explained with careful attention. Again using the definitions of Right primitive or Left primitive ideals one can find the connection of Jacobson Radical with these concepts. One important property of Jacobson Radical is that any ring 푅 can be embedded in a ring 푆 with unity such that Jacobson Radical of both 푅 and 푆 are same.