On a Characterization of Maximal Ideals

1970 ◽  
Vol 13 (2) ◽  
pp. 219-220
Author(s):  
Jamil A. Siddiqi
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Analytic Functions ◽  
Entire Functions ◽  
Factorization Theorem ◽  
Codimension One ◽  
Complex Banach Algebra ◽  
Maximal Ideals ◽  
Invertible Elements

Let A be a commutative complex Banach algebra with identity e. Gleason [1] (cf. also Kahane and Żelazko [2]) has given the following characterization of maximal ideals in A.Theorem. A subspace X ⊂ A of codimension one is a maximal ideal in A if and only if it consists of non-invertible elements.The proofs given by Gleason and by Kahane and Żelazko are both based on the use of Hadamard's factorization theorem for entire functions. In this note we show that this can be avoided by using elementary properties of analytic functions.

Functionals on Real C(S)

1978 ◽  
Vol 30 (03) ◽  
pp. 490-498 ◽  
Author(s):  
Nicholas Farnum ◽  
Robert Whitley
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Codimension One ◽  
Maximal Ideals ◽  
Invertible Elements ◽  
Complex Valued

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property: (*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

2018 ◽  
Vol 17 (09) ◽  
pp. 1850169 ◽  
Author(s):  
Hossein Javanshiri ◽  
Mehdi Nemati
Keyword(s):  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Multiplier Algebra ◽  
Topological Center ◽  
Maximal Ideals ◽  
Open Questions ◽  
Arens Regularity ◽  
Amalgamated Duplication

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

scholarly journals Banach algebra elements whose spectra intersect R+

1974 ◽  
Vol 19 (1) ◽  
pp. 59-69 ◽  
Author(s):  
F. F. Bonsall ◽  
A. C. Thompson
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Real Numbers ◽  
Complex Banach Algebra ◽  
Image Position ◽  
Invertible Elements

Let A denote a complex Banach algebra with unit, Inv(A) the set of invertible elements of A, Sp(a) and r(a) the spectrum and spectral radius respectively of an element a of A. Let Γ denote the set of elements of A whose spectra contain non-negative real numbers, i.e.

scholarly journals Characterization of ideals of rhotrices over a ring and its applications

2021 ◽  
Vol 27 (1) ◽  
pp. 138-147
Author(s):  
Kailash M. Patil ◽  
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Higher Order ◽  
Unital Ring ◽  
Maximal Ideals ◽  
Unital Banach Algebra

We define higher order rhotrices over a commutative unital ring S and obtain a ring \mathcal{R}_n(S) of rhotrices of the order n \in \mathbb{N}. We characterize the ideals and maximal ideals of \mathcal{R}_n(S). As a particular case, we record ideals of rhotrix rings over integers and rhotrix algebras over complex plane \mathbb{C}. As an application, we characterize the maximal ideals of the commutative unital Banach algebra \mathcal{R}_n(\mathbb{C}).

Analysis on Sparse Parts in the Maximal Ideal Space of H∞

1992 ◽  
Vol 44 (4) ◽  
pp. 805-823
Author(s):  
Keiji izuchi
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Analytic Functions ◽  
Level Sets ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Bounded Analytic Functions ◽  
Douglas Algebras

AbstractAnalysis on sparse parts of the Banach algebra of bounded analytic functions is given. It is proved that Sarason's theorem for QC-level sets cannot be generalized to general Douglas algebras.

scholarly journals The Ultrametric Corona Problem and spherically complete fields

2010 ◽  
Vol 53 (2) ◽  
pp. 353-371 ◽  
Author(s):  
Alain Escassut
Keyword(s):  
Maximal Ideal ◽  
Analytic Functions ◽  
Unit Disc ◽  
Unique Continuation ◽  
Open Unit ◽  
Closed Field ◽  
Corona Problem ◽  
Open Unit Disc ◽  
Bounded Analytic Functions

AbstractLet K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the ‘open’ unit disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Multa(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Multa(A, ‖ · ‖) dense in Multm(A, ‖ · ‖)? Is it dense in Multm(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ‖ · ‖) in Multm(A, ‖ · ‖).

scholarly journals On approximation of homomorphisms of algebras of entire functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 158-162
Author(s):  
H.M. Pryimak
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Commutative Banach Algebra ◽  
Uniformly Continuous ◽  
Positive Results

It is known due to R. Aron, B. Cole and T. Gamelin that every complex homomorphism of the algebra of entire functions of bounded type on a Banach space $X$ can be approximated in some sense by a net of point valued homomorphism. In this paper we consider the question about a generalization of this result for the case of homomorphisms to any commutative Banach algebra $A.$ We obtained some positive results if $A$ is the algebra of uniformly continuous analytic functions on the unit ball of $X.$

scholarly journals Maximal ideal spaces of Banach algebras of derivable elements

1970 ◽  
Vol 11 (3) ◽  
pp. 310-312 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Present Note ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Supremum Norm ◽  
Simple Consequence ◽  
Maximal Ideals ◽  
Image Position

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

scholarly journals On the strong radical of certain Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 81-85 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Maximal Ideals

Let A be a complex Banach algebra. By an ideal in A we mean a two-sided idealunless otherwise specified. As in (7, p. 59) by the strong radical of A we mean theintersection of the modular maximal ideals of A (if there are no such ideals we set =A). Our aim is to discuss the nature of and the relation of to A for a specialclass of Banach algebras. Henceforth A will denote a semi-simple modular annihilatorBanach algebra (one for which the left (right) annihilator of each modular maximalright (left) ideal is not (0)). For the theory of such algebras see (2) and (9).

Two questions on domains in which locally principal ideals are invertible

2017 ◽  
Vol 16 (06) ◽  
pp. 1750112 ◽  
Author(s):  
Shiqi Xing ◽  
Fanggui Wang
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Negative Answer ◽  
Finitely Generated ◽  
Closed Set ◽  
Ideal Formula ◽  
Principal Ideals ◽  
Maximal Ideals

An LPI domain is a domain in which every locally principal ideal is invertible. In this paper, we give a negative answer to a question of Anderson–Zafrullah. We show that if [Formula: see text] is an LPI domain and [Formula: see text] is a multiplicatively closed set, then [Formula: see text] is not necessarily an LPI domain. Concerning a question of Bazzoni, we give a characterization of finite character for a finitely stable LPI-domain. We show that if [Formula: see text] is a finitely stable LPI-domain, then every nonzero nonunit element of [Formula: see text] is contained in only finitely many maximal ideals which are in [Formula: see text], and [Formula: see text] is of finite character if and only if each nonzero nonunit element is contained in only finitely many nonstable maximal ideals. A maximal ideal [Formula: see text] is in [Formula: see text] provided there is a finitely generated ideal [Formula: see text] such that [Formula: see text] is the only maximal ideal containing [Formula: see text].

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