scholarly journals Estimations des solutions de l'équation de Bezout dans les algèbres de Beurling analytiques

2005 ◽  
Vol 96 (2) ◽  
pp. 307 ◽  
Author(s):  
O. El-Fallah ◽  
M. Zarrabi
Keyword(s):  
Banach Algebra ◽  
Existence Of Solutions ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Gelfand Transform ◽  
Maximal Ideals ◽  
Bezout Equation ◽  
Beurling Algebras

Let $A$ be a unitary commutative Banach algebra with unit $e$. For $f\in A$ we denote by $\hat f$ the Gelfand transform of $f$ defined on $\hat A$, the set of maximal ideals of $A$. Let $(f_1,\dots,f_n)\in A^n$ be such that $\sum_{i=1}^n\|f_i\|^2 \leq 1$. We study here the existence of solutions $(g_1,\dots,g_n)\in A^n$ to the Bezout equation $f_1g_1+\cdots+f_ng_n=e$, whose norm is controlled by a function of $n$ and $\delta=\inf_{\chi\in\hat A}(|\hat f_1(\chi)|^2+\cdots+|\hat f_n(\chi)|^2)^{1/2}$. We treat this problem for the analytic Beurling algebras and their quotient by closed ideals. The general Banach algebras with compact Gelfand transform are also considered.

Commutative Banach Algebras with Idempotent Maximal Ideals

1969 ◽  
Vol 9 (3-4) ◽  
pp. 275-286 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Complex Field ◽  
Linear Combinations ◽  
Maximal Ideals ◽  
Commutative Banach Algebras ◽  
Finite Linear

Letbe a commutative Banach algebra over the complex fieldC,Man ideal of. Denote byM2the set of all finite linear combinations of products of elements fromM.Mwill be termed idempotent ifM2=M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

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 Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles

1992 ◽  
Vol 44 (2) ◽  
pp. 342-356
Author(s):  
W. E. Pfaffenberger ◽  
J. Phillips
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Natural Classification ◽  
C Algebras ◽  
Gelfand Transform ◽  
Maximal Ideals ◽  
Almost Complex ◽  
Topological Obstruction ◽  
Algebra Homomorphisms

AbstractWe are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted ℳ, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

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 Tensor Products of Banach Algebras

1959 ◽  
Vol 11 ◽  
pp. 297-310 ◽  
Author(s):  
Bernard R. Gelbaum
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Integrable Functions ◽  
Group A

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.

ONE FUNCTIONAL OPERATOR INVERSION FORMULA

2001 ◽  
Vol 6 (1) ◽  
pp. 138-146 ◽  
Author(s):  
P. Plaschinsky
Keyword(s):  
Banach Algebra ◽  
Explicit Form ◽  
Maximal Ideal ◽  
Inversion Formula ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Functional Operator ◽  
Inversion Operator ◽  
Commutative Banach Algebras ◽  
Algebra Theory

Some results about inversion formula of functional operator with generalized dilation are given. By means of commutative Banach algebra theory the explicit form of inversion operator is expressed. Some commutative Banach algebras with countable generator systems are constructed, their maximal ideal spaces are investigated.

Amenability of Banach algebras

1989 ◽  
Vol 105 (2) ◽  
pp. 351-355 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Metric Space ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Commutative Banach Algebra ◽  
Cohomology Theory ◽  
Compact Metric Space ◽  
Amenable Banach Algebra ◽  
Second Dual

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

scholarly journals A Characterization of Bi-Jordan Homomorphisms on Banach Algebras

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Abbas Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms

For Banach algebras A and B, we show that if U=A×B is unital and commutative, each bi-Jordan homomorphism from U into a semisimple commutative Banach algebra D is a bihomomorphism.

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